A Communal Development of the Definitive Book on Statistical Causal Inference.
The purpose of this web site is to engage the analytic community in the collaborative development of a book, entitled Causal Inference via Causal Statistics: Causal Inference with Complete Understanding .
Interested parties can observe the evolution of the book on this web site. Then, if so motivated, you can participate in the book's development by questioning, critiquing, suggesting, and/or contributing via a discussion forum (e.g., Ning-Anyalytic Bridge Forum, Grad School Forum, etc.), or not. In either event, this should be a rewarding experience and an instructive experiment in communal participation.
Please note that this unprecedented development presentation is for experimental and educational purposes only. Be forewarned that the process may be AS UGLY AS MAKING SAUSAGE and I plan, for both pedagogical and efficiency reasons, to make no effort to tidy for your eyes.
The book is an outgrowth of my Ph.D. dissertation, entitled Foundations of Mathematical Epistemology: A Derivation of Causal Statistics, published in 1972. For a discussion of the background of and lead-up to the book, see www.causalstatistics.org
After completion of the dissertation and several years in academia, I went on to other things. Now, I am working toward retirement from financial pursuits, to reenter the non-pecuniary field of causal inference.
About 3 years ago I looked at the developments in the field of causal inference in the last 40 years, assuming that I would find great progress, but the situation was to the contrary. There is now a great deal more interest in causal inference than in the 1960s, but I don't see much real progress, with the notable exceptions of the work of a very few, like Judea Pearl and Donald B. Rubin.
Nevertheless, no one, other than myself, has taken the deductive approach, which I believe to be the optimal. My research is still the only effort to do what Euclid and Einstein did; i.e. go backwards, down to basics, start from definitions, axioms, primitives, etc. and derive a causal inquiring system. This approach gives a logical foundation to causal inference and allows a complete understanding of the inferential process, just as Euclid's and Einstein's deductive approaches to Geometry and Relativity, respectively, were the only routes to real understanding in their fields.
Their derivations effectively ended controversies concerning origins in their fields and their colleagues turned their thinking and research pursuits to other aspects of their disciplines. This is why I feel that the Causal Statistics path to causal inference, to be presented in the instant book, is the optimal approach and will eventually become the accepted paradigm.
The goal for my retirement is to make a sea change in the way non-experimental scientists conduct their research. Specifically, it is desired that social scientists, epidemiologists, and other non-experimental researchers--when appropriate--utilize Causal Statistics in the design, conduct, analysis, and reporting of their empirical research.
The book is a major step on the path to accomplishing this goal. It is intended that the book deliver a limited, yet understandable and definitive, presentation of Causal Statistics. These intentions will be accomplished by (1) drawing necessary extractions from the dissertation; (2) presenting background information, necessary for understanding; (3) exemplifying the application of Causal Statistics; (4) answering potential objections; (5) explaining the various components of Causal Statistics and how they relate to each other; (6) delineating how Causal Statistics fits into its philosophical, logical, and statistical environments; and (7) spelling out how Causal Statistics relates to the overall mosaic of human knowledge.
It is the intent that this book be the last word on statistical causal inquiring systems, like Einstein’s presentation of Relativity was pretty much the last word on the origin of the Lorentz Equations and the relationships among time, space, mass and velocity. In many ways, the planned book is more philosophical, conceptual, and explanatory than mathematical, although the mathematical formulation and use of Causal Statistics is presented, with examples.
The developing book will be presented, on this website, where you can watch the structure and content evolve over time. Presenting this development effort as it progresses is for experimental and educational purposes. Be forewarned that the process may be as ugly as making sausage and I plan, for both pedagogical and efficiency reasons, to make no effort to tidy or clean it for your eyes.
Super Rough Draft of
The Book
Causal Inference via Causal Statistics: Causal Inference with Complete Understanding
[with deductive certainty and no loose ends]
Preface
This book is intended for a broad range of readers, from causal inference specialists and research methodologists to the average undergraduate student with one course in statistics. In conformity with this minimal background requirement, the content will be presented and organized with the least knowledgeable reader in mind.
I will attempt to construct every explanation and argument from the ground up, with the exception of statistical presentations, which will build on top of the content of the typical college statistics course. More knowledgeable and experienced readers may wish to scan or skip some of the fundamental material which is redundant to their backgrounds.
The uses of this book are numerous and vary according to the backgrounds and interests of the different types of readers. Probably, the most extensive use of this book should be as a text in causal inference or Causal Statistics courses. The following is a list of categories of expected readers and some of the most important benefits they should draw from the book:
(1) Causal inference specialists can become familiar with the first and only causal inquiring system to give a complete and understandable foundation to non-experimental statistical causal inference.
(2) Non-experimental research methodologists can discover how to make causal inferences from non-experimental data with complete understanding (a) of all assumptions required, (b) of the meaning of and strictures on the resulting statistical causal connections, and (c) of ways in which the strictures can be relaxed.
(3) Statisticians can be introduced to the origin and nature of Causal Statistics and its relationship to the other two statistical paradigms, Classical and Bayesian Statistics.
(4) Non-experimental researchers can inform themselves about (a) an “algorithm” for making non-experimental causal inferences with complete understanding, (b) what assumptions are required, and (c) the techniques available to weaken these assumptions.
(5) Non-experimental research consumers can be introduced to theory construction, utilizing causal inferences, and the problems and limitations inherent in such an enterprise.
(6) Non-experimental science professors and practitioners can learn both about the power and the limitations of the theories that they are promulgating and applying, respectively.
(7) Philosophers interested in Epistemology should be informed about (a) the nature of mathematical and statistical knowledge, (b) the nature of statistical causal knowledge, (c) the nature of statistical causal theories, (d) the relationships and compatibilities among the three statistical paradigms, and (e) how causal inference and Causal Statistics fit into the mosaic of all human knowledge.
(8) Philosophers interested in causality will be introduced, through the book and references to my dissertation, (a) to the most complete and pragmatic formulation of causal philosophy and causal definitions (i.e., theoretical, operational, and statistical) in the 3000+ years of attempts on the concept and (b) to the operation of causality in our universe.
(9) Mathematicians, philosophers, and research methodologists interested in inductive and/or deductive logical systems and inference can learn about statistical causal inference, which is an unusual combination of both inductive and deductive reasoning, within the same logical/inferential line.
(10) College students, both undergraduate and graduate, from any field, can (a) learn about the aforementioned topics and/or (b) prepare for futures in or near any of the above specialties by reading the book or by taking a causal inference course in which this book is used as a text.
Concerning organization, it is my natural tendency to start at the beginning and proceed in chronological/logical order to the end/conclusion, Q.E.D. Although logical, this approach is pedagogically problematic. The problem is that, at best, the minimally qualified reader usually knows only generally where he/she is going, but not precisely. Hence, subtleties in the step-by-step presentation are not appreciated when they are read and, as a result, are not committed to memory and/or not placed in the proper juxtaposition to other elements, so the process flows in a well-reasoned, logically-tight continuum to the conclusion.
To overcome this common problem, I will deliver the punch line near the beginning. In Chapter I.1, I will give a brief summary of where we are going, i.e., Causal Statistics and its application in non-experimental, quasi-experimental, and in perfectly experimental (collectively referred to in this book as "non-experimental") research. Then, near the end, the punch line will be delivered again, in much greater detail and generality.
[There is another reason for the book to be organized in such a nonlinear manner. The arguments in favor of two given points may be recursive, for example, to understand the precise meaning and nature of causal inference and causal theories, one must have a clear statement and understanding of the definition of cause and to know what definition of cause is most appropriate and useful, one must understand causal inference and causal theory building. This problem of the need for recursive explanation requires a somewhat iterative or mutually asymptotic approach toward complete understanding.
Also, different levels of understanding of causal statistics are required to understand different arguments as one progresses through the text. Chapter 0 .1 gives only the briefest of descriptions of causal statistics. Chapter I.1 presents a two variable example of causal statistics; Chapter X presents a three variable example; and Chapter XI presents a 10 variable example.
Ultimately, the text deviates from linearity to aid in understanding.]
Over 40 years, I have received many questions and comments concerning my work in causal inference, some helpful, others motivated by misunderstanding and/or disbelief. It is intended that this book be the definitive word on non-experimental causal inference, that all outstanding issues be handled and handled with completeness and finality. Of course, this is “a consummation devoutly to be wished,” but unlikely to be actually realized for causal inference or probably for any discipline. Nevertheless, it is one of our objectives in this book and that explains why I will spend so much ink on the details of issues that are contextually related, but not central, to causal inference.
Additionally, as you read through the book, you will notice a number of asides, tangents, digressions, war stories, musings, thoughts, interpretations, etc. that are not generally a part of the logic of causal inference. As I write, I may gain or remember some insight or story I would like to share. The information may be valuable to you in some way in the non-causal-inference part of your life or it may not, but I promise not to charge you extra for it. Hey, I'm old, humor me. In any event, I will always try to label such digressions, so you can choose to read them or not.
[If this book is used for a textbook, instructors may desire that some sections be read out of order or may allow that others be skipped entirely. Obviously, if the instructor wishes to referrer to a standard section, it can be done by noting a standard section designation, e.g., III.C. 2. b. To referrer to nonstandard sections, I would suggest something like p. 25.2 - p.28 .1. This would designate page 25, paragraph 2 through page 28, paragraph 1.]
[Readers interested only in applying causal statistics algorithmically can focus on the chapters presenting example applications of causal statistics. Those interested in applying causal statistics with complete understanding should read all chapters.]
[I have put problems and questions at the end of each chapter. The simple problems are mainly for the purpose of putting your brain in gear to think about the issues brought up in the chapter. The harder questions and problems are for the purpose of challenging the student to think beyond the text and creatively apply the contents.]
As I began the book, I intended to make it gender-neutral. Yet, I found that keeping track of the he/she's, his/hers', etc. were quite distracting. Hence, I will not use the gender-neutral style, but please understand that this does not indicate a tolerance for gender chauvinism. After all, my mother was a mathematician.
The following are four alternative tables of contents. Table of Contents, Ver. 0 is the original table of contents which I developed for this book in 2008.
Then in September of 2009 I ran across a table of contents for a causal statistics textbook which I had sketched out in 1975, after developing and teaching a graduate course in causal statistics at the University of Hawaii; as far as I know the first causal statistics course ever taught. At that time, I wrote a first draft of the preface, table of contents, and part of the first chapter and had given that to a representative of Prentice-Hall to see if they would be interested in publishing the book.
When they got back to me they noted that there were no current courses in causal statistics and they were not interested in trying to build the market. They felt therefore that the proposed textbook didn't seem to them to be a financial winner. They mentioned that they did like my writing style and would like me to write a classical statistics textbook for them. I thanked them, but responded that I had no interest in such a project.
Anyway, when I rediscovered the 1975 textbook I realized that it was much more mathematical and application oriented and that the 2008 textbook was more philosophical, foundational, and derivational; focusing more on required assumptions and on the understanding of the nature of causality and causal inference.
Each approach has its advantages, so I combined the two tables of contents and produced Table of Contents, Ver. 1. Then I considered that the Ver. 1 book might be too long and that there might be different audiences for different portions of the book.
So I developed Table of Contents, Ver. 2 for a book entitled, Foundations and Understanding of Causal Statistics, and Table of Contents, Ver. 3 for a book entitled, Causal Statistics for Application. Foundations would have no prerequisites, but statistics through hypothesis testing or confidence intervals would be recommended. Causal Statistics for Application would require at least statistics through hypothesis testing or confidence intervals as a prerequisite.
All of these possible tables of contents are presented next in numerical order: Ver. 0, Ver. 1, Ver. 2, Ver. 3.
I. In a Nutshell, What is Causal Statistics and How Important Is It?
II. Origin, Background, Setting, and Context of Causal Statistics
III. The Needs for Non-Causal and Causal Inference
IV. Empirical Research, Associative and Causal (An Example)
A. Non-experimental Research, Associative and Causal (An Example)
B. Experimental Research, Associative and Causal (An Example)
V. Foundations of Causal Statistics (Philosophy)
A. Epistemology/Philosophy of Science
1. The Empiricists
2. The Rationalists
3. The Eclectics
4. Observability
5. Un-observability (Metaphysics)
B. Logic
1. Deductive
2. Inductive
3. Deductive/Inductive (Combination) Systems
C. Philosophy of Causality & Definition of “Cause”
D. Attempts at Causal Inference
1. God
2. Hobbs
3. Bacon
4. Mill
5. Conditionals, Counterfactuals, etc.
A. Nuclear Physics
B. Fundamental Particles
C. Quantum Mechanics
D. Statistical Mechanics
E. Etc.
F. Final Definitions of “Cause”
VII. Derivation of Causal Statistics
A. Deductive Logic
B. Research Methodology
C. The Derivation of Causal Statistics??
D. The Assumptions and the General Model of Causal Statistics
1. The First Assumption Set
2. The Second Assumption Set
3. The Third Assumption Set
E. Mathematical statistics (The General Form of Causal Statistics)
F. A Causal Calculus/Algebra
G. Inductive Logic in Causal Statistics
VIII. Causal Statistics: A Two Variable Example
Classical and Bayesian Statistical Analyses of the Example Data (The Mathematics and Interpretation of Classical and Bayesian Association)
History of Correlation
IX. Causal Statistics: a Three Variable Example
Classical and Bayesian Statistical Analyses of the Example Data (The Mathematics and Interpretation of Classical and Bayesian Association)
X. Causal Inference in Experimental Research
A. Non-Stochastic Causal Inference
1. Validity and Correctness
2. Causal Modeling and Theory Building
B. Statistical Causal Inference
1. Validity and Correctness
2. Causal Modeling and Theory Building
XI. Causal Inference in Non-experimental Research
A. The Relationship among the Three Statistical Paradigms
A. How does Causal Statistics fit into the Pantheon (Mosaic) of other Statistical Paradigms?
B. [A.] Understanding the Nature of Mathematical and scientific “Knowledge.”
?? B. Where Does the Dissertation Fit in?
A. [C.] The Logical Constructs within Statistics and their Relations to each other
XIV. The Design of Non-experimental Causal Studies and Causal Study Sequences
XV. History of Causal Philosophy and Causal Inference, as Seen from the High Ground
A. Conditionals, Counterfactuals, etc.
XVI. My Post Dissertation Attempts to do Causal Statistics and Obtain Funding
XVII. Why is it Taking so Long? And how much longer will it take?
XVIII. Where do we go from Here?
A. Get researchers using Causal Statistics
1. Develop a simple presentation of the Causal Statistics paradigm.
2. Answer all objections.
3. Publicize.
4. Free application consulting.
5. Book? With basics and examples
6. Funding: only after it’s no longer innovated and is acceptable. When it’s too late, there will then be adequate money.
Part I Introduction
I. In a Nutshell, What is Causal Statistics and How Important Is It?
II. Origin, Background, Setting, and Context of Causal Statistics
III. The Needs for Non-Causal and Causal Inference
IV. Empirical Research, Associative and Causal (An Example)
A. Non-experimental Research, Associative and Causal (An Example)
B. Experimental Research, Associative and Causal (An Example)
-- OR --
A. Experimental Research, Associative and Causal (An Example)
B. Non-experimental Research, Associative and Causal (An Example)
V. Example of Large Scale Causal Statistics Study
A. Applicability of Causal Statistics
B. Application of Causal Statistics
C. Assumptions Required, 3 Assumption Sets
D. Further Complications
1. Non Linearity
2. Time Lags ( e.g., psychological problems in kids occur later or was it an operationalization error or measurement error or a study design error that we didn't see psychological problems earlier)
3. Show various techniques for increasing identification ( e.g., results of previous studies, assumptions, etc.) and the effects of ~ 0 causal parameters.
E. Reporting
F. Follow on Studies
Part II Foundations
VI. Foundations of Causal Statistics (Philosophy)
A. Epistemology/Philosophy of Science
1. The Empiricists
2. The Rationalists
3. The Eclectics
4. Observability
5. Un-observability (Metaphysics)
B. Logic
1. Deductive
2. Inductive
3. Deductive/Inductive (Combination) Systems
C. Philosophy of Causality & Definition of “Cause”
Hume
Mill
Definitions of "Cause"
D. Attempts at Causal Inference
1. God
2. Hobbs
3. Bacon
4. Mill
5. Conditionals, Counterfactuals, etc.
A. Nuclear Physics
B. Fundamental Particles
C. Quantum Mechanics
D. Statistical Mechanics
E. Etc.
F. Final Definitions of “Cause”
VIII. Causal Inference in Experimental Research
A. Non-Stochastic Causal Inference
1. Validity and Correctness
2. Causal Modeling and Theory Building
B. Statistical Causal Inference
1. Validity and Correctness
2. Causal Modeling and Theory Building
IX. Causal Inference in Non-experimental Research
X. Derivation of Causal Statistics
A. Deductive Logic
B. Research Methodology
C. The Derivation of Causal Statistics??
D. The Assumptions and the General Model of Causal Statistics
1. The First Assumption Set
2. The Second Assumption Set
3. The Third Assumption Set
E. Mathematical statistics (The General Form of Causal Statistics)
F. A Causal Calculus/Algebra
G. Inductive Logic in Causal Statistics
XI. History of Causal Philosophy and Causal Inference, as Seen from the High Ground
A. Conditionals, Counterfactuals, etc.
Part III Simple (or Linear) Single Equation Models
XII. Causal Statistics: A Two Variable Example
Classical and Bayesian Statistical Analyses of the Example Data (The Mathematics and Interpretation of Classical and Bayesian Association)
A. Curve Fitting
1. Scatter Diagrams
2. Possible Criteria for Curve Fitting
3. The Least Squares Criteria
B. Statistical Inference for a Fixed Independent Variable
1. The Fundamental Assumptions of Regression Analysis
2. The Fundamental Assumptions of Causal Statistics
3. Causal Parameter Estimation
4. The Means, Variance, and Distributions of Causal Parameter Estimators
5. Confidence Intervals for Causal Parameters
6. Hypothesis Tests for Causal Parameters
7. Confidence Intervals for Causal Predictions
C. Statistical Inference for Random Independent Variables
D. Application to a Bivariate Normal Population
E. Interpretation of Results
XIII. Correlation
A. Preliminary Remarks
B. Simple Correlation
1. The Population Correlation Coefficient
2. The Sample Correlation Coefficient
3. Interpretation of the Correlation Coefficient
C. Relationship between the Correlation Coefficient and the Causal Parameter
D. Partial Correlation
E. Relationship between the Partial Correlation Coefficient and the Partial Causal Parameter
F. Multiple Correlation
G. Canonical Correlation
H. History of Correlation
XIV. Causal Statistics: a Three Variable Example
Classical and Bayesian Statistical Analyses of the Example Data (The Mathematics and Interpretation of Classical and Bayesian Association)
XV. Multivariable Models
A. Preliminary Remarks
B. Graphical Representation of the Three Dimensional Case
C. Assumptions
D. Causal Parameter Estimation
E. Confidence Intervals and Hypothesis Testing
F. Multicollinearity
G. Stepwise Parameter Estimation
H. Computer Analysis
I. Interpretation of Results
PART IV Complexities
XVI. Nonlinear Models
A. Preliminary Remarks
B. Nonlinearity in the Variables
C. Nonlinearity in the Causal Parameters
D. Differential Models
E. Intractable Nonlinear Models
XVII. Relaxation of Various Assumptions
A. Preliminary Remarks
B. Heteroschedasticity
C. Specification Error
D. Data Snooping
E. Omitted Relevant Variables
F. Included Irrelevant Variables
G. Superfluous Variables
H. Serial correlation in the Error Term
I. Lagged Variables
J. Correlation between the Error Term and an Independent Variable
K. Measurement Error
L. Transcription Error
M. Causal Relationships between Independent Variables
N. Reciprocal Causation
XVIII. Special Topics and Techniques in Causal Inference
A. Preliminary Remarks
B. Operational Variables
C. Proxy Variables
D. Ordinal Variables
E. Dichotomous Variables
F. Qualitative Variables
G. Moderator Variables
H. Scanning
I. Dummy Variables Used as a Jackknife
J. Time as an Independent Variable
K. Inclusion of Associative Variable
L. Dummy Variables for Seasonal Variations
M. Clustered Residuals
XIX. Performing a Causal Analysis Study
A. Preliminary Remarks
B. State the Problems
C. Construct a Hypothesized Model
D. State the Assumption Set
E. Identify the Equations
F. Operationalize Variables
G. Collect Data
H. Estimate Causal Parameters
I. Interpret Results
J. Write-up Study
K. Criticize and Redo Study
Part V Simultaneous Equation Models
XX. Limited Information Estimation Techniques
A. Preliminary Remarks
B. Ordinary Least Squares
C. Indirect Least Squares
D. Instrumental Variables
E. Two Stage Least Squares
F. Least-Variance Ratio
XXI. Identification
A. Preliminary Remarks
B. Under-identification
C. Identification by Addition of Exogenous Variables to the Model
D. Identification Using A Priori Information
E. Necessary Conditions for Identification
F. Necessary and Sufficient Conditions for Identification
G. Over-identification
XXII. Full Information Estimation Techniques
A. Preliminary Remarks
B. Three Stage Least Squares
C. Iterated Three Stage Least Squares
D. Full Information Maximum Likelihood
XXIII. Comparison of Estimation Techniques
A. Preliminary Remarks
B. Small Sample Distributions
C. Summary of Overall Comparisons
XXIV. The Design of Non-experimental Causal Studies and Causal Study Sequences
Part VI Conclusions
A. The Relationship among the Three Statistical Paradigms
A. How does Causal Statistics fit into the Pantheon (Mosaic) of other Statistical Paradigms?
B. [A.] Understanding the Nature of Mathematical and scientific “Knowledge.”
?? B. Where Does the Dissertation Fit in?
A. [C.] The Logical Constructs within Statistics and their Relations to each other
XXVI. My Post Dissertation Attempts to do Causal Statistics and Obtain Funding
XXVII. Why is it Taking so Long? And how much longer will it take?
XXVIII. Where do we go from Here?
A. Get researchers using Causal Statistics
1. Develop a simple presentation of the Causal Statistics paradigm.
2. Answer all objections.
3. Publicize.
4. Free application consulting.
5. Book? With basics and examples
6. Funding: only after it’s no longer innovated and is acceptable. When it’s too late, there will then be adequate money.
Foundations and Understanding of Causal Statistics
Part I Introduction
I. In a Nutshell, What is Causal Statistics and How Important Is It?
II. Origin, Background, Setting, and Context of Causal Statistics
III. The Needs for Non-Causal and Causal Inference
IV. Empirical Research, Associative and Causal (An Example)
A. Non-experimental Research, Associative and Causal (An Example)
B. Experimental Research, Associative and Causal (An Example)
-- OR --
A. Experimental Research, Associative and Causal (An Example)
B. Non-experimental Research, Associative and Causal (An Example)
V. Example of Large Scale Causal Statistics Study
A. Applicability of Causal Statistics
B. Application of Causal Statistics
C. Assumptions Required, 3 Assumption Sets
D. Further Complications
1. Non Linearity
2. Time Lags ( e.g., psychological problems in kids occur later or was it an operationalization error or measurement error or a study design error that we didn't see psychological problems earlier)
3. Show various techniques for increasing identification ( e.g., results of previous studies, assumptions, etc.) and the effects of ~ 0 causal parameters.
E. Reporting
F. Follow on Studies
Part II Foundations
VI. Foundations of Causal Statistics (Philosophy)
A. Epistemology/Philosophy of Science
1. The Empiricists
2. The Rationalists
3. The Eclectics
4. Observability
5. Un-observability (Metaphysics)
B. Logic
1. Deductive
2. Inductive
3. Deductive/Inductive (Combination) Systems
C. Philosophy of Causality & Definition of “Cause”
Hume
Mill
Definitions of "Cause"
D. Attempts at Causal Inference
1. God
2. Hobbs
3. Bacon
4. Mill
5. Conditionals, Counterfactuals, etc.
A. Nuclear Physics
B. Fundamental Particles
C. Quantum Mechanics
D. Statistical Mechanics
E. Etc.
F. Final Definitions of “Cause”
VIII. Causal Inference in Experimental Research
A. Non-Stochastic Causal Inference
1. Validity and Correctness
2. Causal Modeling and Theory Building
B. Statistical Causal Inference
1. Validity and Correctness
2. Causal Modeling and Theory Building
IX. Causal Inference in Non-experimental Research
X. Derivation of Causal Statistics
A. Deductive Logic
B. Research Methodology
C. The Derivation of Causal Statistics??
D. The Assumptions and the General Model of Causal Statistics
1. The First Assumption Set
2. The Second Assumption Set
3. The Third Assumption Set
E. Mathematical statistics (The General Form of Causal Statistics)
F. A Causal Calculus/Algebra
G. Inductive Logic in Causal Statistics
XI. History of Causal Philosophy and Causal Inference, as Seen from the High Ground
A. Conditionals, Counterfactuals, etc.
Part III(VI) Conclusions
A. The Relationship among the Three Statistical Paradigms
A. How does Causal Statistics fit into the Pantheon (Mosaic) of other Statistical Paradigms?
B. [A.] Understanding the Nature of Mathematical and scientific “Knowledge.”
?? B. Where Does the Dissertation Fit in?
A. [C.] The Logical Constructs within Statistics and their Relations to each other
XIII(XXVI). My Post Dissertation Attempts to do Causal Statistics and Obtain Funding
XIV(XXVII). Why is it Taking so Long? And how much longer will it take?
XV(XXVIII). Where do we go from Here?
A. Get researchers using Causal Statistics
1. Develop a simple presentation of the Causal Statistics paradigm.
2. Answer all objections.
3. Publicize.
4. Free application consulting.
5. Book? With basics and examples
6. Funding: only after it’s no longer innovated and is acceptable. When it’s too late, there will then be adequate money.
Causal Statistics for Applications
Part I(III) Simple (or Linear) Single Equation Models
I(XII). Causal Statistics: A Two Variable Example
Classical and Bayesian Statistical Analyses of the Example Data (The Mathematics and Interpretation of Classical and Bayesian Association)
A. Curve Fitting
1. Scatter Diagrams
2. Possible Criteria for Curve Fitting
3. The Least Squares Criteria
B. Statistical Inference for a Fixed Independent Variable
1. The Fundamental Assumptions of Regression Analysis
2. The Fundamental Assumptions of Causal Statistics
3. Causal Parameter Estimation
4. The Means, Variance, and Distributions of Causal Parameter Estimators
5. Confidence Intervals for Causal Parameters
6. Hypothesis Tests for Causal Parameters
7. Confidence Intervals for Causal Predictions
C. Statistical Inference for Random Independent Variables
D. Application to a Bivariate Normal Population
E. Interpretation of Results
II(XIII). Correlation
A. Preliminary Remarks
B. Simple Correlation
1. The Population Correlation Coefficient
2. The Sample Correlation Coefficient
3. Interpretation of the Correlation Coefficient
C. Relationship between the Correlation Coefficient and the Causal Parameter
D. Partial Correlation
E. Relationship between the Partial Correlation Coefficient and the Partial Causal Parameter
F. Multiple Correlation
G. Canonical Correlation
H. History of Correlation
III(XIV). Causal Statistics: a Three Variable Example
Classical and Bayesian Statistical Analyses of the Example Data (The Mathematics and Interpretation of Classical and Bayesian Association)
IV(XV). Multivariable Models
A. Preliminary Remarks
B. Graphical Representation of the Three Dimensional Case
C. Assumptions
D. Causal Parameter Estimation
E. Confidence Intervals and Hypothesis Testing
F. Multicollinearity
G. Stepwise Parameter Estimation
H. Computer Analysis
I. Interpretation of Results
PART II(IV) Complexities
V(XVI). Nonlinear Models
A. Preliminary Remarks
B. Nonlinearity in the Variables
C. Nonlinearity in the Causal Parameters
D. Differential Models
E. Intractable Nonlinear Models
VI(XVII). Relaxation of Various Assumptions
A. Preliminary Remarks
B. Heteroschedasticity
C. Specification Error
D. Data Snooping
E. Omitted Relevant Variables
F. Included Irrelevant Variables
G. Superfluous Variables
H. Serial correlation in the Error Term
I. Lagged Variables
J. Correlation between the Error Term and an Independent Variable
K. Measurement Error
L. Transcription Error
M. Causal Relationships between Independent Variables
N. Reciprocal Causation
VII(XVIII). Special Topics and Techniques in Causal Inference
A. Preliminary Remarks
B. Operational Variables
C. Proxy Variables
D. Ordinal Variables
E. Dichotomous Variables
F. Qualitative Variables
G. Moderator Variables
H. Scanning
I. Dummy Variables Used as a Jackknife
J. Time as an Independent Variable
K. Inclusion of Associative Variable
L. Dummy Variables for Seasonal Variations
M. Clustered Residuals
VIII(XIX). Performing a Causal Analysis Study
A. Preliminary Remarks
B. State the Problems
C. Construct a Hypothesized Model
D. State the Assumption Set
E. Identify the Equations
F. Operationalize Variables
G. Collect Data
H. Estimate Causal Parameters
I. Interpret Results
J. Write-up Study
K. Criticize and Redo Study
Part III(V) Simultaneous Equation Models
IX(XX). Limited Information Estimation Techniques
A. Preliminary Remarks
B. Ordinary Least Squares
C. Indirect Least Squares
D. Instrumental Variables
E. Two Stage Least Squares
F. Least-Variance Ratio
X(XXI). Identification
A. Preliminary Remarks
B. Under-identification
C. Identification by Addition of Exogenous Variables to the Model
D. Identification Using A Priori Information
E. Necessary Conditions for Identification
F. Necessary and Sufficient Conditions for Identification
G. Over-identification
X1(XXII). Full Information Estimation Techniques
A. Preliminary Remarks
B. Three Stage Least Squares
C. Iterated Three Stage Least Squares
D. Full Information Maximum Likelihood
XII(XXIII). Comparison of Estimation Techniques
A. Preliminary Remarks
B. Small Sample Distributions
C. Summary of Overall Comparisons
XIII(XXIV). The Design of Non-experimental Causal Studies and Causal Study Sequences
Rough Draft
A critical introduction to the methods used to collect data in social science: surveys, archival research, experiments, and participant observation. Evaluates "facts and findings" by understanding the strengths and weaknesses of the methods that produce them. Case based.
Make statistical def of cause
Simi exp’al, quasi exp’al, etc
Who should be able to understand this book
CAUSAL INFERENCE VIA CAUSAL STATISTICS
The remainder of this web page is dedicated to accomplishing the goal of making a sea change in the way non-experimental scientists conduct their research. Specifically, the goal is that social scientists, epidemiologists, and other non-experimental researchers--when appropriate--utilize Causal Statistics in the design, conduct, analysis, and reporting of their empirical research. The more precise purpose of this Exposé is to tackle Objective 4, above, i.e., to present Causal Statistics in an easily accessible (intellectually) way, even multiple ways.
This article draws extractions from the dissertation and presents a minimal, yet sufficient, formulation of Causal Statistics, along with examples of its application in non-experimental research. Additionally, the presentation is interspersed with background information (necessary for understanding), answers to potential objections, explanations of how the various components being presented relate to each other, and delineations of how these components fit into their philosophical, logical, and statistical environments and into the overall mosaic of human knowledge.
This presentation will proceed via the following sections:
{Click to go back to: Table of Contents, I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII, XIII, XIV, XV, XVI, XVII, XVIII}
I. In a Nutshell, What Is Causal Statistics and How Important Is It?
“Causal Statistics is a mathematical inquiring system which enables empirical researchers to draw causal inferences from non-experimental data, based upon the minimum required assumptions, explicitly stated.” 1968
"Causal Statistics is the only causal inquiring system which is a deductive mathematical construct, in the sense that Euclidian geometry is an axiomatic, deductive, logical construct. It's derivation is founded in causal philosophy, physics, epistemology, symbolic logic, statistics, and non-experimental research methodology." 1976
“The development and utilization of Causal Statistics will eventually be as important to the non-experimental sciences as the codification and utilization of the scientific method was to the physical (i.e., experimental) sciences." 1969
“100 years from now, research results and theories in the non-experimental sciences will consist mostly of large arrays of variables, connected by multi-equation causal models, inferred from a single large or a compounded succession of smaller applications of Causal Statistics in empirical research studies.” 2007
{Click to go back to: Table of Contents, I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII, XIII, XIV, XV, XVI, XVII, XVIII}
II. Origin, Background, Setting, and Context of Causal Statistics
In 1967, I was at the University of California (Berkeley), finishing an M.S. in Nuclear Engineering and beginning a Ph.D. in Business Administration. The dissertation research plan was to carry out a non-experimental management study, in an attempt to determine what managerial behaviors caused increased productivity of scientists and engineers engaged in physical sciences research and development.
. [causal inferences were difficult to impossible to draw from non-experimental studies.] [ I had expected a completely understandable, logically tight causal inquiring system to be one of the ]
As work on the dissertation progressed, I was amazed to discover that there was no well-defined technique which social scientists used to make causal inferences. My sojourn in the physical sciences led me to think that such an important and needed component of research methodology would be a standard arrow in the quiver of all social science researchers and, if it wasn't, the whole field of social science would be moving heaven and earth to discover and/or develop such an arrow. [As a recent convert from the physical sciences,] [as a recent refugee from the physical sciences,] [What seemed even more amazing to me was that no one seemed to be working on the development of an understandable, logically tight causal inquiring system.]
Well, there was no such standard arrow. As I would discover later, there were ongoing efforts on the parts of some nonexperimental research methodologists to develop causal inquiring systems, but the frenzied, almost frantic, intensity which I expected was not observable. Most researchers and research consumers seemed to accept the limitations on drawing causal inferences as a given, as part of the nature of the field. Many even felt that any discussion of causality was somehow inappropriate and even unscientific. Even the social scientists and others who wished to have a complete, understandable, and well-founded causal inquiring system generally felt that such a thing was impossible. [There any significant pressure or effort to develop one. So I set about finding out what work had been done in causal inference. ]
After some time I found that the available causal inquiring systems broke down into three, more or less distinct, mathematical forms, all forms of classical statistics. All three were attempts to stretch classical statistics in a way that causal inferences could be produced. One of the problems with these attempts was that they were basically intuitive and not based on logical derivation or deduction, resulting in an understandable, logically tight causal inquiring system.
"Cause" is not a term defined in classical statistics; therefore, causal inferences cannot be established from classical statistics. Each of the three forms of causal inquiring system utilized a slightly different mathematical form from classical statistics. Each then gave an intuitive, handwaving argument to arrive at its causal inquiring system, utilizing the initial mathematical form. No wonder the systems, assumptions, definitions, etc. were not completely understandable; they were never actually stated nor was any continuous logical argument to derived them from classical statistics ever made.
So what you had was the domain of classical statistics and then three, somewhat overlapping causal inquiring systems some distance away, in the logical space, from classical statistics and no real, logical, deductive connection between classical statistics and the causal inquiring systems.
This is not to impugn the work done by these authors. Even to see the problem put them far ahead of 90% of the other professionals in the field. Then, to develop a causal inquiring system, no matter how intuitive, was godlike compared to everyone else. The point that I am making is only that these causal inquiring systems were not derived analytically, but intuitively, and therefore could not be applied with complete understanding or confidence.
What I did was (1) in a sense, to fill that opening (Actually, I started from scratch and fill the space from zero to the general form of causal statistics with definitions assumptions the deck of logic etc.) with deductive logic; (2) derive a general, rather than specific, form for causal inquiry; (3) generate all the required definitions; (4) produce all the required assumptions; and (5) explain all of the above in a manner hopefully understandable to a careful reader.
Now, in this book, I am engaged in simplifying and extracting all of the above from the dissertation; adding some explanations, global and epistemological information; and presenting a complete, understandable, and well-founded algorithm (which I call causal statistics) for making causal inferences from nonexperimental studies.
{2,5 Extant Causal Inquiring Systems
2.5.1 Summary
At present, there are three, more-or-less distinct,
causal inquiring systems. They are path analysis,
econometrics, and the Simon-Blalock approach. In actual
fact they are virtually identical to one another.
2.5.2 Path Analysis
Path analysis was introduced in a phenomenally
innovative paper by Sewall Wright* in 1921. Since that
*Wrlght, Sewall: "Correlation and Causation,"
J. of Agricultural Research, Vol. 20, 1921, pp.
557-85.
time additional innovations and, also, acceptance have
been amazingly slow. Path analysis considered only oneway
causation until 1954 when John Tukey** introduced
**Tnkey, John Wilder: "Causation, Regression, and
Path Analysis," in Oscar Keinpthorne, T. A.
Bancroft, J. W. Gowen, and J. L. Lush, eds.,
Statistics and Mathematics in Biology, Ames: Iowa
State College Press, 1954, pp. 35-66
two-way path analysis. This is the only innovation in
path analysis of major importance since 1921.
Basically, path analysis is a linear regression or
simultaneous linear regression technique in which the
coefficients are causal, assuming that the basic
assumptions of the model are valid. These coefficients
cj’s are called path regression coefficients. See
WrIght* 1960 for a summary of path analysis.
*Wright, Sewall: "Path Coefficients and Path
Regressions: Alternative or Complementary
Concepts?" Biometrics, Vol. 16, 1960, pp. 189-202.
2.5.3 Econometrics
Econometrics employs regression and simultaneous
equation models. It is far more advanced mathematically
than path analysis, but there are comparatively few
papers in the field which consider the causal implica
tions of these mathematical techniques.
Econometricians try to avoid the word "cause"
because of’ their misinterpretation of Humian philosophy
on the subject. Due to their avoidance of this word,
econometricians have failed to consider sufficiently a
many of the causal implications and proertIes of
econometrics and b many of the problems and benefits
connected with causal prediction.
Two good econometric references are Johnston** and
Goldberger***.
**Johnston, J.: Econometric Methods. New York,
McGraw-Hill, 1963.
***Goldberger, Arthur S. z Econometric Theory. New
York, John Wiley & Sons, 1964.
2.5.4 The Simon-.Bialock Approach
The Simon-Blalock approach began with a 1954 paper
by Herbert Simon*. This paper served as the foundation
*Simon, Herbert A.: "Spurious Correlation: A
Causal Interpretation," J. of the American Statis
tical Association, Vol. 9, 19514, pp. 467-47.
for a great deal of later work by Hubert Blalock.
Basically, this approach gives causal Interpreta
tion to some of the more elementary f’ormalizations of
econometrics. An exception is Blalock** 1969 in which
**Blalock, Hubert N., Jr.: Theory Construction:
From Verbal to Mathematical Formulati, Englewood
Cliffs, Prentice-Hall, 1969.
he gives preliminary consideration to some simple
differential equation models.}
mathematical treatise on the subject was published by Sewall Wright, with his innovative, 1921 article, “Correlation and Causation”, in J. of Agricultural. Research, Vol. 20, 1921, pp. 557-85. Dr. Wright presented a regression analysis approach to causal inference. That should have gotten the ball rolling, but amazingly it didn't.
33 years later, John Tukey published “Causation, Regression, and Path Analysis,” in Oscar Kempthorne, T. A. Bancroft, J. W. Gowen, and J.L. Lush, eds., Statistics and Mathematics in Biology Ames: Iowa State College Press, 1954, pp. 35-66 and Herbert Simon published “Spurious Correlation: A Causal Interpretation,” in J. of the American Statistical Association, Vol. 49, 1954, pp. 467-479.
These and other causal inquiring systems were incomplete and their foundations in philosophy and axiomatic logic not established. Nevertheless, these insightful initial efforts should have triggered a tidal wave of research into causal inquiring systems and their foundations.
Yet, there was no more than a diminishing ripple on the ponds of non-experimental research, statistics, and research methodology.
Econometrics, with its multi-equation and sometimes recursive forms, was mathematically superior to, i.e., more general than, the other forms of the day for making causal inferences. Econometrics employs regression and simultaneous equation models. It is more advanced mathematically than path analysis, but the number of papers in that field which consider the causal implications of these mathematical techniques is small.
Econometricians try to avoid, the word “cause” because of their misguided belief that Hume put a stake in the heart of causality. Due to their avoidance of this word, econometricians have failed to consider sufficiently (a) many of the causal implications and properties of econometrics and (b) many of the benefits that could be gleaned from facing cause inference from econometric analysis honestly and straight forwardly.
In 1968, I attempted to utilize the aforementioned systems to draw causal inferences in my nonexperimental management study, but I found that I could not apply these systems for causal inference with complete understanding, insight, or confidence. For example, many of the assumptions implicit in the various systems were unknown, the nature of statistical causal connections was not understood, etc.
13 years after Tukey’s and Simon’s first articles, I stumbled into the field of causal inquiring systems because of a desire to do good empirical research, rather then me-too research, with inferior and inappropriate statistical tools.
Recognizing that causal results were, by far, the most desired and that most research in the social sciences, epidemiology, management, etc. was non-experimental; I realized that the development of a non-experimental causal inquiring system, which could be applied with complete understanding and confidence, was of transcendent importance. Hence, I discontinued the original empirical study and turned my efforts to the development of a definitive causal inquiring system.
This statement rolls off the tongue very easily, but what kind of ego maniac would so cavalierly set about solving one of the most important problems in philosophy and THE most important and difficult problem in the social sciences, epidemiology, and statistics: first the problem of causality--a concept challenged with only a modicum of success by philosophers for over 3000 years--and second, the problem of non-experimental causal inference, a task never adequately handled by social scientists, epidemiologists, nor statisticians? Answer: a young graduate student who didn’t know better than to believe that he could ultimately solve any problem that had a rational solution.
{ In 1967-8 I suspected that the problem of causal inference must be pretty difficult, in that it was the most important problem in the field and it hadn't been solved. Even so I didn't realize how extremely hard it was; I spent at least two man years posing questions to myself, thinking, doing library research, and writing down the answers.
In another sense, this was an easy project for me because of the sheer joy of attacking such difficult and such interesting unsolved problems and, over time, seeing them yield to the relentless assault of pure reason. . I had no idea how much I could learn out of my own head by just thinking.
The concept of a theoretical dissertation was appealing to me from the first time I heard that such a thing was possible, sitting on the football/softball intramural field at Berkeley and talking to a Ph.D. candidate in nuclear engineering who was doing a theoretical dissertation. It turned out to be all I imagined and more.
I attacked the problem with all the confidence of one who didn't know better.}
Aside: I would note that any researcher, who doesn't believe he can solve his research problem, is probably right. He probably couldn't solve it. Einstein, Enrico Fermi, and R. A. Fisher certainly attacked their respective, important problems with the belief that they could solve them.
{I studied causal philosophy from Plato to Hume.}
After trying many approaches to the development of a complete causal inquiring system, I eventually did as Euclid did for Geometry and Einstein did for Relativity and no one had or has done for causality. I went back to basics and--beginning with axioms, definitions, primitives, etc., about the nature of the universe and the nature of empirical research--performed a logical derivation of the general formulation of a causal inquiring system, I called Causal Statistics. This research was reported in my Ph.D. dissertation, entitled Foundation of Mathematical Epistemology: A Derivation of Causal Statistics, published in 1972.
During the time I was working on the dissertation and after its completion -- when I was looking for funding to carry the research to the next stage--I was amazed at the negative reactions of many ostensibly intelligent people (professors, funders, etc.) to my research. Over 40 years, I have heard it all. It couldn't be done; it needn't be done; Hume has already considered it and proved that it couldn't and needn't; it should be done, but someone else should fund its further development (e.g., it's too interdisciplinary, too different, too eclectic, too innovative, or conflicting with accepted paradigms to be funded here or evidently anywhere); it's too risky (i.e., it might fail) for results oriented departments and especially for the untenured; etc.; etc.; etc.
There were about as many different, negative criticisms as there are people reacting and, in the final analysis, almost none of the negative opinions held any water. If there had been some convergence of opinion about what was wrong with my research, the critiques would have been more believable.
Nevertheless, I analyzed every criticism and modified my results for those very few with merit. Even so, such corrections almost never satisfied the doubter whose critique was accepted and corrected. These critics would then come up with some new, off-the-wall criticism. I called these critiques the “yes-buters.” "Yes, but what about this new criticism?"
Over the years, I've had many different thoughts about why people with the right educational background, have produced so many negative and incorrect reactions. I have come to several different conclusions: (1) Many just can't think out of the box, even when led by the hand. (2) Many fear, or just naturally resist, the new and different, i.e., anything more than 1 millimeter deviant from what they were taught at their professor’s knee. (3) Most don't have the breadth of mind to comprehend the whole of a large and complicated project (what I call a mega project) all at one time. They can only see distinct pieces. (4) A surprisingly large number just don’t have the mental where-with-all to even make the aforementioned three errors, e.g., "We like the way we've been doing statistics previously."
{In all fairness, might some of the blame for the retarded acceptance of causal statistics properly be placed at my door?
Could I have explain things better? With regard to the dissertation itself, I think things were explained quite well. But concerning the relationship of causal statistics to other statistical paradigms, my early explanations were lack.
Could I have published in journals to get the word out there? Maybe, but there were no journals in any way related to or interested in causal inference. Getting something in a journal which the publisher and referees feel are off-topic, is difficult in the best of circumstances. Doing it with a paper on causal inference, a topic of anathema to most academics, was multipally harder, apropos my experience with funding applications.
Would it have been better to apply causal statistics to nonexperimental research data and use the resulting report as an example of the benefits of causal statistics? I did exactly that with a 10 variable analysis of DDT, DDE, hypertension, pesticides, etc. on the data of a nationwide EPA study. They had collected data for about 10 years and, in all that time, had been incapable of sorting out the causal connections. I did exactly that in about two months and wrote a research report. The local scientists were happy, but those in Washington couldn't be less interested. Again, I was amazed. Is all the world mad or just most of it?}
As a consequence, we are 40 years down the road and Causal Statistics is still a Ph. D. dissertation and generally unavailable to non-experimental researchers: 40 years of lost time, billions of wasted dollars in the non-experimental sciences, untold waste in competent human resources--i.e., social science and epidemiological researchers getting 10% of what they could get from their research efforts--and many lives lost (e.g., from lung and other environmental cancers) and destroyed (e.g., due to the dearth of good causal research and theories to correct criminal, social, and health problems).
If I had been able to spend a substantial portion of the last 35 years carrying out the beyond-dissertation steps in the Causal Statistics project, I believe that today Causal Statistics would be in broad usage. But, for the lack of less funding than that required for one government secretary, surprisingly little progress toward codifying an understandable, complete, and algorithmic arrow for causal inference has been made in the last 35 years. Such collective stupidity on the part of funders and funding agencies is beyond belief.
At least one other explanation is possible, maybe I'm insane. The mental case is always the last to know. Well, then my dissertation chairman, Professor C. West Churchman and the dissertation committee must also have been insane. Further, if I was insane then, I still am. The research makes just as much sense to me now as it did in 1972.
As I said before, the dissertation derived the general form of Causal Statistics. Originally, I had planned to get research funding to extract an application oriented formulation of Causal Statistics from the dissertation. When the funding didn't materialize, I went on to other things, with the belief that some extremely analytic and dedicated researcher would either do the extraction or develop his/her own application oriented formulation of Causal Statistics.
About two years ago, as I started moving toward retirement, I looked at the developments in the field over the past 35 years and found that a great deal had been written about causal inference and causal theory construction. Virtually all attempts were laudable and most were correct, as far as they went. These writers were obviously people who were (1) sharp enough to see the need and (2) insightful enough to say something meaningful and correct about the subject. Nevertheless, they were about where I was in 1968, a year after I began researching the subject and before realizing the need to return to fundamentals and take the deductive approach.
A few methodologists--like Pearl, Rubin, and others--have gone further, but no one has even come close to the total package, i.e., an understandable, complete, algorithmic causal inquiring system for nonexperimental research. Neither has anyone taken the deductive approach, which I believe to be the optimal. My research is still the only effort to do what Euclid and Einstein did; i.e., go backwards, down to basics, start from definitions, axioms, primitives, etc. and derive the whole field. This approach gives a logical foundation to causal inference and allows a complete understanding of the inferential process, just as Euclid's and Einstein's deductive approaches to Geometry and Relativity, respectively, were the only routes to real understanding in their fields. Their derivations effectively ended controversies concerning origins in their fields and their colleagues move on to research in other aspects of their fields.
Not all deductive constructs are as earth shattering as Euclid's and Einstein's. Alfred North Whitehead and Bertrand Russell used symbolic logic to derive arithmetic and algebra in their book, Principia Mathematica. Few people who use arithmetic or algebra refer to or even know about their work.
In one sense, the derivation of causal statistics is of greater utility to its field than the work of Whitehead and Russell was to their field. Euclid's deductive construct was earth shattering in terms of its importance to deductive and symbolic logic, but less so to the field of plane geometry. Interestingly, Euclid's derivation of plane geometry was more important for its contribution to the development of non-Euclidian geometries then for Euclidian geometry.
As everyone knows, Einstein's derivation of relativity was and is unequaled in its effect on the field of physics and in popular culture (the public consciousness). At the turn of the 20th century, physics was where nonexperimental causal inference was in 1968. Most of the mathematical formulations have been developed, but they could not be applied with confidence or understanding because no one understood why the formulas were as they were. Einstein's logical construct, relativity, derived the Lorentz equations and specified the assumptions and concepts on which they were based. After Einstein published his work on relativity, physicists progressively moved toward the accept that's of relativity and its assumptions as the intellectual foundations for the Lorentz equations and moved on to other problems in physics.
In the above sense, the derivation of causal statistics is not comparable in importance to the derivations of Euclid and Einstein. Yet, in the effect on human lives in the long run, the derivation of causal statistics will arguably be more important than any of the others. If causal statistics were used in all appropriate nonexperimental research, the increase in medical, social, etc. knowledge would be so great and the application of this knowledge so earth shattering in its effects that, in that sense, the derivation of causal statistics would surpass the others.
Now you should have a ballpark understanding of the content of the 1972 dissertation and where the field of causal inquiring systems is today. About two years ago I started a website called causalstatistics.org. The initial purpose of the web site was to make my dissertation easily accessible. A downloadable, selectable, and searchable copy of the dissertation is presented on that website.
The dissertation presents Causal Statistics at a level that extremely analytic and dedicated researchers could apply the paradigm in non-experimental research and obtain valid causal inferences. Nevertheless, greater simplification is necessary for the vast majority of social science researchers to utilize Causal Statistics with complete understanding and confidence.
{ The remainder of this web page is dedicated to accomplishing the goal of making a sea change in the way non-experimental scientists conduct their research. Specifically, the goal is that social scientists, epidemiologists, and other non-experimental researchers--when appropriate--utilize Causal Statistics in the design, conduct, analysis, and reporting of their empirical research. The more precise purpose of this Exposé is to tackle Objective 4, above, i.e., to present Causal Statistics in an easily accessible (intellectually) way, even multiple ways.
This article draws extractions from the dissertation and presents a minimal, yet sufficient, formulation of Causal Statistics, along with examples of its application in non-experimental research. Additionally, the presentation is interspersed with background information (necessary for understanding), answers to potential objections, explanations of how the various components being presented relate to each other, and delineations of how these components fit into their philosophical, logical, and statistical environments and into the overall mosaic of human knowledge.}
But, as long as I'm at it, I might as well take the opportunity to handle all other relevant issues that I can think of. Hence, the presentation will be interspersed with background information (necessary for understanding), answers to potential objections, explanations of how the various components being presented relate to each other, and delineations of how these components fit into their philosophical, logical, and statistical environments and into the overall mosaic of human knowledge. (Moved this paragraph to preface?)
[the purpose of this book is to design an algorithm for properly applying causal statistics in non-experimental research so that, after reading this book, any non-experimental researcher will be able to apply causal statistics to their work correctly and with understanding. In furtherance of that purpose, the book should answer any questions and handle any problems that the researcher is
Hence, as my work has progressed, my ultimate goal has become more far-reaching. The goal has evolved toward making a sea change in the way non-experimental scientists conduct their research. Specifically, it is desired that social scientists, epidemiologists, and other non-experimental researchers, when appropriate, utilize Causal Statistics in the design, conduct, analysis, and reporting of their empirical research.
In an effort to accomplish this goal, I have established five objectives (“Objectives” are steps on the path toward accomplishing the overall goal.):
1. To make the dissertation readily available to all (accomplished via the presentation of the dissertation at causalstatistics.org),
2. To extract from the dissertation portions that are, in sum, necessary and sufficient for formulating a physics/logically/epistemology/ statistically/research methodology/philosophically based causal inquiring system,
3. To utilize these extractions to formulate Causal Statistics in a complete, coherent, and interrelated (i.e. with consideration of how Causal Statistics related to its epistemological environment) form,
4. To present this formulation of Causal Statistics in an easily accessible (intellectually) way (even multiple ways) to present and future research methodologists, to the researchers themselves, and to research consumers. (The initial presentation will be accomplished through the development of this book),
5. To challenge non-experimental scientists and research methodologists to do the hard work to study, understand, analyze, critique, extend, and apply Causal Statistics
[GOAL:] The initial impetus for this book is to extract from the dissertation the elements necessary for a minimal, yet sufficient and usable, formulation of Causal Statistics and present it herein, along with examples of its application in non-experimental research.
{But, as long as I'm at it, I might as well take the opportunity to handle all other relevant issues that I can think of. Hence, the presentation will be interspersed with background information (necessary for understanding), answers to potential objections, explanations of how the various components being presented relate to each other, and delineations of how these components fit into their philosophical, logical, and statistical environments and into the overall mosaic of human knowledge.
[the purpose of this book is to design an algorithm for properly applying causal statistics in non-experimental research so that, after reading this book, any non-experimental researcher will be able to apply causal statistics to their work correctly and with understanding. In furtherance of that purpose, the book should answer any questions and handle any problems that the researcher is likely to encounter in either application or understanding.
In a]}
Take your recent example. A researcher found a correlation between lack of sleep and poor grades, on the part of children. The newscaster said, “You should make sure your children get lots of sleep so they will get better grades."
Is it possible that parents who don't care enough to make sure that their children get sufficient sleep also don't care to push or to help them make better grades? Or could it be a spurious correlation due to a genetic factor?
What about correlations between book ownership and reading scores of children? Or between....
What about the causes and the effects of mental illness? What about the environmental causes cancer? What about the effects, of lax law enforcement in low income areas and on young people, on the future criminal behavior of these young people. Etc., etc., etc.
I have virtually never seen a non-experimental study reported in the media correctly. The original research report almost never uses the word cause, but the media people almost invariably give an inappropriate causal interpretation, even if the word cause is not used by them either.
[ IV. A Brief History of Non-Experimental Causal Inference
WP5
Dissertation, Part I
WP7 ]
[V. The Need for a New Causal Inference Tool
Give Researchers algorithm for applying Causal Statistics
WP6 Paragraph 1-4
WP7 ]
[I.1 Philosophical and Logical Foundations of Causal Statistics: A Two Variable Example
Identification pb
Recursive equations
Endogenous
Exogenous, etc Dissertation, Chapter 13
Estimations
Parameter interpretations
Error handling
Structure and structural Change
Start with general form of Causal Statistics
Recommend econometric sources and note their avoidance of “cause”
Simultaneous equation parameters partial out the considered correlation
Give researchers algorithm for applying Causal Statistics
Consider an empirical study in which variables B (i.e., number of book owned by the family) and R (i.e., reading capability of the child) are found to be correlated. If we assume (1) that no outside variables causally affect both B and R in such a way as to change their correlation, (2) that R does not cause B, and (3) that causal relationship are linear; one can validly conclude, subject to the usual statistical error, that B causes R with standardized strength equal to the correlation coefficient.
A “valid” causal inference is a causal inference which is a logically necessary result of the definitions stated, the assumptions made, and the data collected. In other words, a “valid” causal connection results from a causal inference arrived at by the proper application of Causal Statistics.
Note that a valid causal inference does not necessarily result in a correct causal connection. If one or more of the assumptions is incorrect, the causal connection will be incorrect. The greater the degree to which the assumptions are in error, generally, the greater the error in the causal connection drawn. This second assumption set can be very restrictive and questionable, yet, that is not the fault of Causal Statistics. It is, if you will, the fault of logic and the universe we exist in. Gravity can be inconvenient, if you want to fly, but the field of physics is not at fault. In fact physics can assist in overcoming the problem; same with Causal Statistics.
Questions which I had about causal inquiring systems that existed in 1968.
When assumptions were these systems based on?
To the systems you'll causal parameters? Only causal parameters? Partially causal parameters? What determines these things?
What is the definition of cause?
Why are causal inferences drawn so effortlessly from experimental studies? and seem almost impossible to draw from nonexperimental studies?
Many people assert that Hume sounded the death knell for causality and proved that it was a word and concept which should not be used. Hume showed that causal connections cannot be proved.
Our causal connections immutable?
How can you have causal connections in the social sciences which are separated in time and space?
What is statistical causality? Sometimes it causes and sometimes it doesn't?
Are all causes scientific laws built into this fabric of the universe?
Our mathematical connections causal laws? Like, people who are taller in inches are also taller in feet.
Our definitional connections causal laws?
How about variables or objects, they are almost always composed of smaller variables or objects? Are the universes causal laws working between macro variables? What if the structure of the macro variables are different from one situation to the next, like thrown rocks or depression?
Many people argued that causality was incompatible with theory building in the social sciences. Are they correct or not?
In causal inference, you are not discovering fundamental laws of the universe, but regularities observable in the macro world, flowing from the basic micro causal laws of the universe. These macro causal inferences are like the lumpy nature of objects in the universe, like asteroids, planets, stars, galaxies, galaxy clusters, etc.
The concept of regularity or lumpiness is not limited to causal inference. Consider a correlation. The Association discovered between two variables is not likely a connection built into the fundamental micro laws of the universe. It is simply a regularity, observable in the macro universe.
What's the relationship between macro causal laws and statistical causal laws?
Another reason I couldn't use the existing techniques for drawing causal inferences for my dissertation, was that I didn't know how to determine if regression parameters were causal or not. You get a regression parameter of 3. Is it causal or associative? Or is it somewhat causal and the rest associative, and if so, how much?]
{Click to go back to: Table of Contents, I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII, XIII, XIV, XV, XVI, XVII, XVIII}
III. The Needs for Non-Causal and Causal Inference
Dissertation in Library of Congress and University of California Berkeley Graduate Library
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A government planner, as a consumer of social science research, might want to predict recidivism rates as a function of the hours per week an inmate spends in self-improvement and educational programs, in order to plan for future prison space requirements. For a non-experimental study to produce the information that the government planner needs, the researcher need only obtain the association (i.e., correlation) between prison program participation and recidivism (a dichotomous variable) or the amount of time prior to return (a continuous variable). In either event, the data analysis would require only a straight forward application of Associative Statistics, a subset of Classical Statistics, and no information about causation is needed.
From the correlation or, more desirably, the regression results, government planners could predict recidivism rates and, from that and other information, prison space requirements.
But more importantly, government planners, counselors, administrators, etc. would like to know how to intervene to reduce the recidivism rate. Such manipulation would require knowledge of the causes, both positive and negative, of recidivism rate for one or more of the variables which government can control.
The demand by research consumers for prediction, without intervention, is miniscule compared to the demand (generally, unsatisfied in the Social Sciences) for research results enabling prediction, with intervention or manipulation, i.e., causal results. Accurate causal theories enable us to control the future rather than just forecast it.
Unfortunately, no complete or totally understandable research tool is available or drawing valid causal inferences from non-experimental data. The dominant, almost exclusive, inquiring tool is Classical Statistics, which yields correlations and/or regression coefficients. And, as basic statistics texts will tell you, if they deal with the subject at all, correlation does not imply causation. “Cause” is not even a term in the vocabulary of Classical Statistics; therefore, Classical Statistics alone could not draw valid causal inferences. Yet, the need for causal results and theories is so great that many researchers and research consumers have incorrectly purported to do just that, utilize Classical Statistics to draw valid causal inferences from non-experimental data.
Now, stop a moment to let this lunacy sink in. Causal inferences are the most desired and needed conclusions in the non-experimental sciences. But, there is no complete or logically consistent inquiring tool available to draw the needed valid causal inferences from non-experimental data. [Yet, little funding or effort is being expended to remedy this gross and debilitating short coming. Is such mass stupidity possible? Regretfully Dorothy, it is.]
Presented with these facts, any intelligent person would imagine that a tremendous amount of research money and effort would have been devoted to the development of a complete, understandable, and valid causal inquiring system. But this intelligent person would be, not only wrong, but grossly wrong. No significant money and only slightly more effort has been expended on the methodological problem of causal inference.
Aside: One might argue to the contrary, in that a significant portion of my dissertation was supported by the National Aeronautics and Space Administration through the Space Sciences Laboratory at the University of California (Berkeley). While true, and even though I had worked at the Manned Spacecraft Center (now the Johnson Space Center) for NASA in the late 60's, in the Theoretical Physics Branch analyzing the Bremsstrahlung and other radiation hazards to Apollo astronauts, it was not NASA's intent to fund me to develop Causal Statistics. The original funding was for an empirical study to determine the causes (both positive and negative) of productivity in scientific research and development.
Since the study was non-experimental, causation was difficult to establish. I attempted to use the causal inquiring systems available at that time, but could not apply them with understand, insight, or confidence--e.g., the assumptions implicit in the various systems were unknown, there were no proofs of the appropriateness of such systems, and it was unclear how to disentangle causal components from associative components in coefficients (e.g., regression coefficients) inferred.
I then turned my efforts to the development of a causal inquiring system which could overcome the aforementioned problems. As this work proceeded, I saw a far-reaching importance of this line of statistical research. Its significance dwarfed that of the original R & D study. For this reason the R & D study was discontinued, with the causal statistics project taking its place.
At this point, NASA would likely have cut my funding. But Professor C. West Churchman--the head of Berkeley's Space Sciences Laboratory, my dissertation chairman, and all around intellect and great human being, renewed my funding, probably without saying a word to NASA.
Before completing the dissertation I left Berkeley to teach for a year at the University of Hawaii. In the next year, I completed the dissertation funded by unemployment and food stamps.
As a further aside, I would have gladly continued my research at that time had any government or private organization been willing to fund me at the level of my unemployment and food stamps. I searched extensively, but no one was willing: 40 years of lost time, billions of wasted dollars in the non-experimental sciences, and many lives lost (e.g., from lung and other cancers) and destroyed (e.g., due to the dearth of good causal research and theories to correct criminal and social problems). All for the lack of less funding than that required for one welfare client. Such collective stupidity is beyond belief... [unless one understands that the intelligence of an organization is generally inversely proportional to the number of people in it.]
To me, this state of affairs almost defies explanation, but not quite. The reason for “not quite,” is that I have endured 40 years of ostensibly intelligent people telling me it couldn't be done; it needn’t be done; Hume has already considered it and proved that it couldn’t and needn’t; it should be done but someone else should fund it (i.e., it’s too interdisciplinary, eclectic, different, innovative, or conflicting with accepted paradigms to be funded here or evidently anywhere); it’s too risky (i.e., it might fail) for results oriented departments, especially for the untenured; etc.; etc.; etc. The need for a usable, complete, understandable, and valid causal inquiring system is beyond question to clear thinkers, those one in every million or so. “I think, therefore I see the need.” This is the need which Causal Statistics is designed to fill.
It is my belief that, 100 years from now, research results and theories in the non-experimental sciences will consist of large arrays of variables, connected by multi-equation causal models, inferred from a single large or a succession of smaller, Causal Statistics based, empirical research study(s) This, of course, assumes that I am able to complete my work on the objectives (stated above) of this web site.
Should I die before completing this work and be unable to fund the effort in my will, I estimate it would be in the range of 300 years before this type of advanced, non-experimental research and theory construction will be common place. This conclusion is drawn inductively from my observations of the glacial progress in the field of causal inference in the last 100 years.
A 200 year delay sounds pretty extreme, especially considering that, in recent years, a great deal has been written about causal inference and causal theory construction, as one can see by Googling “causal inference.” Virtually all attempts are laudable and most are correct, as far as they go.
These writers are people who are (1) sharp enough to see the need and (2) insightful enough to say something meaningful and correct about the subject. This is where I was in 1967, before realizing the need to return to fundamentals.
My research in causal inference methodology is the only effort to do what Euclid and Einstein did, i.e., go backwards, down to basics, and start from definitions, assumptions, etc. and derive the whole field; giving a logical foundation and allowing a complete understanding of causal inference. Further, I see not even a tendency, on the part of any writer in the field, to so much as glanced in the direction of the deductive approach to causal inference, which is the only real route to a complete and understandable causal inquiring system for the non-experimental sciences, just as Euclid’s and Einstein’s deductive approaches to Geometry and Relativity, respectively, were the only routes to real understanding in their fields.
Whenever this advanced research and theory building paradigm does enter common research usage, that research methodology and its foundation, i.e., Causal Statistics, will be generally considered to be the most important developments ever, for progress in the non-experimental sciences.
[III. Causality and the Empirical Research Environment ]
Prior to a detailed presentation of Causal Statistics, it is necessary to understand the empirical research environment which gives rise to the need for Causal Statistics and to see how Causal Statistics fits into it.
The empirical research environment includes not only the nature and types of empirical research, but also the methodological and analytical tools available, the types of findings desired, and the uses to which research findings are put.
Aside: It is my tendency to present a subject by starting at the beginning and proceeding in chronological/logical order to the end/conclusion, Q.E.D. Although logical, this approach is pedagogically problematic. The problem is that, at best, the reader usually knows only generally where he is going, but not precisely. Hence, subtleties in the step-by-step presentation are not appreciated when they are read, and, as a result, not committed to memory and/or not place in the proper juxtaposition to other elements for reaching a well-reasoned, logically tight conclusion. Hence, you might consider reading the concluding chapter and then return to chapter II with a comprehension of exactly where you are going.
[III. A. Prediction, without Intervention]
As indicated by the government planner example at the beginning of Section I.B., non-experimental research, to enable research consumers to make forecasters, without manipulation of any of the variables, requires no knowledge of causal connections between the variables and is comparatively simple. All the scientist needs to do is use Classical Statistics to obtain correlations or regression coefficients, pretty much a straight forward calculation within the domain of Classical Statistics.
A research consumer might desire a confident interval or a hypothesis test on the results, which is only a little more complicated calculation with a Classical Statistics computer algorithm.
For prediction, without intervention causal information is not needed and, therefore, neither is Causal Statistics.
[Prediction, without intervention is in small demand. Prediction, without intervention is not so simple in a game against a thinking opponent or if the structure has changed.]
[ III. B. Prediction, with Intervention]
As invented in Section I.B., the demand by research consumers and theoreticians for results to support prediction, without intervention, is miniscule compared to the demand for support of prediction, with intervention. To support prediction, with intervention, these results must be causal connectors and presently these are no available tool for drawing valid causal connections from non- experimental date.
It is beyond incredulity that, in the 150 year history of the modern social sciences, so little money, thinking, and work have been devoted to the development of methodological tools and techniques for making valid causal inferences. This is an indictment of funders, methodologists, and methodology developers.
In the absence of such tools and techniques, Social Science researchers have been confronted with a classic dilemma. Associative results lead to unsatisfied and unhappy theoreticians and practitioners and invalid causal inferences lead to failure in the long run. Researchers who take the latter path, i.e., draw invalid causal inferences, are usually either unscrupulous or naive in the ways of causal inference. The ultimate result of such invalid causal conclusions is incorrect theories and failed interventions and programs.
As a result of this unhappy, no-win dilemma, social scientists have largely concerned themselves with definitions, concepts, associative theories, analytically developed theories or models (like in Economics), controversial experiments, case studies, and questionable causal studies and theories (usually based on improperly drawn and highly suspect causal inferences). All of these, except the invalid causal inferences, are valuable. But, if the social scientists were able to supplement their formulations with validly drawn causal connections and large multi-variable causal theories, the overall power and value of the Social Sciences could more then doubled.
This is the contribution which Causal Statistics is poised to make.
[?The dominant mode of theory building in the physical and biological sciences is based on “knowledge” of the cause and effect relationships among variables. The social and other non-experimental sciences would like to do the same, but generally can’t, because of the difficulty, in obtaining valid causal inferences from non- experimental data. Hence, social scientists concern themselves with definitions, concepts, associative theories, analytically developed theories or models (like in Economics) contrived experiments, case studies, and questionable causal theories (usually based on improperly drawn and highly suspect causal inferences).
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IV. Empirical Research, Associative and Causal (An Example)
A. Non-experimental Research, Associative and Causal (An Example)
Consider a typical non-experimental study. The original research, carried out in the 1960s, found a correlation between children’s reading performance (R) and the number of books he/she owned (B).
Drawing:B and R are correlated
In the research results, it was correctly reported that kids who owned books were better readers. Technically, this is and associative statement, indicating only that R and B are correlated.
In fact, this associative statement communicates something much different to the typical human mind, namely that B causes R. Over the eons, evolution has selected in favor of brains which draw causal inferences, even if such causal inferences are not always valid. (This is consistent with Hume's statement)
Drawing: B causes R and are correlated
This associative statement certainly communicated a causal conclusion to a certain group of people, who then proceeded to develop a program and establish an organization called Reading is Fundamental (RIF) which has been in existence for 41 years and is “the nations largest children’s literacy organization.”
RIF was formed to increase book ownership so Johnny would read better, based on the invalid inference, not explicitly stated, that B causes R.
40 years, 300 million books distributed, and millions of people duped by just one of thousands and probably tens of thousands of inappropriate causal inferences.
How did the founder of RIF know that correlation between B and R wasn’t due to the fact that R caused B? Looking a little further a field, how did they know that socioeconomic level of the child (S) didn’t cause both B and R, leading to a spurious (i.e., non-causal) correlation between B and R?
Drawing: B and R and SES, with all causal connections and the correlation
Both of these additional causal theories would lead to an observed Association between B and R. In fact, if the causal connection work chosen properly, each of the aforementioned three causal theories could produce the exact correlation between B and R, measured in the 1960s study. Further, it may be that any two or even all three of these causal connections do exist and they take on causal connection magnitudes which mutually generate the observed correlation between B and R.
Going even further, it may be that there are additional unconsidered variables, like SES, say X., which causes both B and R, leading to the exact or some portion of the observed correlation between B and R.
Drawing: B and R and SES and X., with all causal connections and the correlation
Each of the above for causal theories and an infinite number of combinations on them could lead to the observed correlation between B and R. This is referred to as an infinite number of observationally equivalent explanations or theories. Observationally equivalent outcomes are observables, like an association or correlation, which could result from various, unobservable underlying mechanisms.
As you can see, associations and correlations are comparatively simple, pretty much a result of observation only. Causation and causal inferences are horses of a different feather. Hey, I never promised you a Rose Garden or smelless stable for your feathered horses.
How can we ever sort out all of these unobservable, underlying causal mechanisms to determine the correct one? It's very difficult, but not impossible, at least theoretically.
In general terms, one must enter enough information into the system and thereby limit to one the number of possible causal theories which could give rise to the observed correlation. Such information could be assumptions or results from prior emperical studies.
The kinds of information put into the system might be, “assume there is no variable in, x, except for SES, outside the system of study, which causes both B and R.” or “previous research has determined that a one unit increase in R causes a three unit increase in B.”
The input of such information will limit the number of causal theories which could produce the observed correlation. When the data and input information reduce to one the number of possible causal theories capable of producing the observed correlation, one can then infer, based on the data and input information, that, for example, B causes R.
The reader might be and probably should be disconcerted by the state of affairs, recognizing that the assumptions necessary to make a causal inference from nonexperimental data are likely to be extensive, restrictive, and highly suspect.
All I can say about this is, "Guilty as charged." I'm sorry, but that is the nature of the universe we live in. I only promised to do the best that could be done, not to do better than is logically possible.
Causal inferences are very powerful pieces of information, giving those who apply such knowledge far greater power (reference the power of experimental/physical science knowledge versus nonexperimental/social science knowledge) than those with only associative knowledge. In nonexperimental research, such great output power generally only comes at great input costs.
Even with causal statistics, nonexperimental valid causal inference is still generally very difficult and, without causal statistics, valid causal inference is next to impossible.
After all, how could philosophers spend over 3000 years and make little progress and how could social science methodologists and researchers, epidemiology methodologists and researchers, etc. spend well over 100 years wrestling with the problem of nonexperimental causal inference and not solve it? Because it is highly eclectic, plus extremely difficult.
IV. B. Experimental Research, Associative and Causal (An Example)
The 1965 RIF study was nonexperimental, but such research could be carried out as an experimental study. In a perfectly designed experimental study, a random sample of students would be drawn from the population. By “population,” I do not mean the population of the United States or the world. I mean the group of all children to which the study is intended to infer, i.e., the group to which you would like the findings of the study to apply.
The random sample of students would then be randomly assigned to two groups, one the treatment group and the other of the control group. Books would be given to the treatment group and no books would be given to the control group. Otherwise, both groups would be treated exactly the same.
After some time, the researchers will return and measure the reading performances in both groups. If the average reading performance of the treatment group was determined to be significantly greater than the average reading performance of the control group, the researchers would undoubtedly conclude that receiving and/or owning books causes improved reading performance.
Significance: By “significantly greater,” I mean statistically significantly greater. In classical statistics the researchers might determine that they difference between the means of the two sample groups is statistically significant at the 0.5 level. This would mean that.... In Bayesian Statistics,....
Note that here the researchers have no difficulty in drawing a causal inference and extending it to the population. This is the huge advantage of experimental research over nonexperimental research.
Causal Inference: Note for future reference that the act of drawing the causal inference concerning the sample groups does not flow from classical or Bayesian statistics, but the act of extending the causal conclusion to the population is accomplished by applying either classical or Bayesian statistics. The act of drawing the causal conclusion for the sample must come from something other than classical or Bayesian statistics because neither of those ever uses or defines the concept of cause. This matter will be discussed in detail later.
In this experiment the researchers chose only to do a post test or post measurement and no pretest or pre-measurement. The researchers could have chosen to....
{ In the first experiment, above, the research was perfectly controlled. The only two relevant variables that changed were force and acceleration. In non-experimental research….}
{ A management researcher, unsatisfied with the problems inherit in non-experimental studies, could avail himself of the option to perform an experimental study, likely in a lab and using students. Such a study would be experimental and allow for valid causal inferences, but the artificial, over simplified environment would greatly limit the generalizability(sp?) of any causal findings. This is often the dilemma in the non-experimental sciences, (1) to experiment in an artificial and limited environment and obtain causal inferences of questionable generalizability or (2) to carry out a non-experimental study in a context rich environment and relinquish the likelihood of reaching valid causal conclusions.
Enter Causal Statistics. The proper use of Causal Statistics can distinctly diminish these problems for non-experimental researchers. I developed Causal Statistics 40 years ago, spend 10 years, when I wasn't teaching, trying to push the idea and to get funding, without notable success, and eventually went on to other things.}
{Solving the problem of nonexperimental causal inference would have been much easier if 1) I had begun with a far simpler problem of experimental causal inference (on the other hand, at that point, it was not at all clear that there was any problem with experimental causal inference. But I should have tried to understand causal inference in a simple situation before moving to the extreme complexity and difficulties inherent in nonexperimental causal inference.) and 2) I had realized at the outset that, in order to obtain a complete understanding of the process of causal inference, I needed to return to fundamentals, go back to the beginning, and arrive causal statistics as a logical/deductive system.
Of course, this brings me back to my undergraduate days in chemical engineering. There was a homily been making the rounds among the chemical engineering students (circulating through the halls of the chemical engineering department) that the solution to a complicated issue was not the problem. Once the problem had been structured properly, the solution was obvious or at least mechanistic.
This homily also applies to the causal statistics problem. If I had known points 1) and 2) above at the outset the solution may not have been easy, but it would've been a whole lot easier and I would have spent far less time wandering around in the desert.}
{Click to go back to: Table of Contents, I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII, XIII, XIV, XV, XVI, XVII, XVIII}
V. Foundations of Causal Statistics (Philosophy) xx
At its core, causal statistics is the product of the alchemy (admixture) of a large number of apparently disparate fields
[VI. Advancing by Returning to the Beginning
The dissertation. Started the process.
Here we shall extract and surpass.
Induction vs. deduction.
[ III. Foundations of Causal Statistics]
At its core, Causal Statistics is a philosophically, logically, and statistically based causal inquiring system. Knowledge, techniques, and original developments from these three disparate fields are blended to form the foundation upon which Causal Statistics is constructed.
Causal Statistics is an axiomatic/deductive construct, in the way that Euclidian Geometry is an axiomatic/deductive construct. Like Geometry, Causal Statistics begins with definitions and basic assumptions. From these “first principles” and through the application of logic, the construct of Causal Statistics was derived deductively in my 1972 dissertation, entitled Foundations of Mathematical Epistemology: a Derivation of Causal Statistics, presented above.
But why would research methodologists, empirical researchers, theoreticians, and research and theory consumers want to have Causal Statistics derived? especially since the derivation resulted in mathematical forms very similar to those of regression analysis and econometrics.
Similarly, one might ask why is it desirable for Euclid to derive Geometry (Certainly the theorems came before the axioms and derivations.), for Einstein to derive Relativity, for Whitehead and Russel to derive Arithmetic and Algebra in Principia Mathematica, and others to derive Classical and Bayesian Statistics?
Consider Einstein’s contribution with his derivation of Relativity. Much of the mathematical formulations of Special Relativity, the Lorentz Equations, were already known from experimentation. But nobody understood why these equations were the way they were; i.e., why, for fast moving objects, their length shortened; their mass increased; and time slowed down. So the equations couldn’t be applied with understanding. Einstein, rather judiciously, formulated a set of assumptions (e.g., the measured velocity of light is constant, no matter what the velocity of the observer) about the nature of the universe and derived the Lorentz Equations and then went beyond the empirically known, to derive E=mc².
Causal inference was in a somewhat analogical situation, in that those few people who dared consider causation thought they knew some of the mathematical formulations, but they couldn’t be sure and couldn’t apply these formulations with understanding. My derivation of Causal Statistics, in combination with my present task, i.e. the extraction from the dissertation of a generally usable formulation of a complete causal inquiring system, should enable researchers to make causal inferences with understanding and confidence. The derivation within the dissertation is logically sufficient for this, but the extraction of the essence of Causal Statistics from the dissertation is far too complicated for most non-experimental research methodologists and researchers. So, I will do the extraction herein and present Causal Statistics with an intellectually accessible structure, which can be readily applied.
The application of this Causal Statistics paradigm to non-experimental data will enable researchers to draw valid causal inferences with complete understanding of the basic definitions and assumptions utilized, the use of prior information, and the proper interpretation of the resulting causal coefficients.
[IV. Foundation, Codification, and Exemplification of Causal Statistic]
At its core, Causal Statistics is a philosophically, logically, and statistically based causal inquiring system. Knowledge, techniques, and original insights in these three disparate fields are blended to form a foundation from which Causal Statistics can be derived.
My dissertation formally derived the mathematical structure of Causal Statistics from a particular, useful definition of cause and a set of basic assumptions about the nature of the universe, e.g., the basic laws upon which the universe operates are micro causal laws acting between adjacent fundamental particles.
It is important to note that the structure of Causal Statistics is not dependent on the exact assumption set used in the dissertation. I could have assumed that the most fundamental operation of the universe was (1) quantum mechanical in nature, with the micro causal laws being stochastic in nature or (2) based on micro causal interactions between string theory “particles.” Such varied micro causal assumptions would lead deductively, to the same mathematical macro structure for Causal Statistics, because random fluctuations at the micro level would average to virtually zero at the macro level, in accordance with the Law of Large Numbers.
That is a mathematical/statistical way of looking at the jump from the micro world to the macro world. Another approach to the same conclusion would be: if the three different theories about the fundamental operation of the universe (exact, Quantum Mechanical, and String Theory micro causes) lead to different macro mathematical formulations of Causal Statistics, they would also lead to differences in the operation of the macro universe. Therefore, by studying the macro universe, one could infer back to which of the three theories of the fundamental operation of the universe was correct.
But we can’t do that, so the three theories must be observationally equivalent at the macro level and the macro mathematical structure of Causal Statistics would also be the same for all three. Hence, the derivation of Causal Statistics is quite robust, with regard to differences in the fundamental micro assumptions about the nature of the universe.]
A. Epistemology/Philosophy of Science
1. The Empiricists
2. The Rationalists
3. The Eclectics
4. Observability
5. Un-observability (Metaphysics)
The first part of Chapter 7 constructs a natural
istic metaphysics of’ the universe. This basic causal
117
theory of the operation of the universe is then in
vestigated for practicality. In the latter portion
of the chapter, axioms formalizing this metaphysics
are forwarded, and from these axioms continuous causal
micromatheniatics is derived.
The author realizes that this is not the only
metaphysics consistent with observed phenomena. Others
are possible. For example, a Supreme Being might
control i.e., dictate all behavior within the universe.
These alternative metaphysical theories will not be
presented within this work or compared with our natural
istic metaphysics because such considerations would
be only remotely related to our major objective, that
of deriving causal statistics.
The metaphysics present here was chosen because
it seems to be the simplest, most pragmatic, and most
applicable. It explains completely the apparent causal
nature of the universe.
7,1 PresentatIon of S ome Bai c Definitions
7,1,1 Characteristics and Variables
The universe, as we perceive It, is simply a
conglomeration of characteristics or lIties. As
[XII. B. [A.] Understanding the Nature of Mathematical and scientific “Knowledge.”
?? B. Where Does the Dissertation Fit in?
In science and math, the big, encompassing theories and mostly logical, deduct paradigms; like Causal Statistics in my dissertation, like Euclid derived Geometry, like Einstein derived the Theory of Relativity, like Whitehead and Russel derived Arithmetic and Algebra in Principia Mathematica, and others to derive Classical and Bayesian Statistics.
The interested reader might ask, “So what?” and the uninterested reader has probably already moved on to taking out the trash or to a porn site. Anyway, for you that me left, back to, “So what?”
Consider Einstein’s contribution with his derivation of Relativity. Much of the mathematical formulation of Special Relativity, the Lorentz Equations, were already known from experimentation. But nobody understood why these equations were the way they were; i.e., why, for fast moving objects, their length shortened; their mass increased; and time slowed down. So the equations couldn’t be applied with understanding. Einstein, rather judiciously, formulated a set of assumptions (e.g., the measured velocity of light is constant, no matter what the velocity of the observer) about the nature of the universe and derived the Lorentz Equations and then went beyond the empirically known, to derive E=mc².
I did a similar thing in deriving a causal inquiring paradigm, which I call Causal Statistics. The application of the Causal Statistics paradigm to non-experimental data can now enable researcher to draw valid causal inferences with complete understanding of the basic definitions utilized, the assumptions made, the use of prior information, and the proper interpretation of causal coefficients. See Working Papers #___ for a simple, but complete, application oriented formulation of Causal Statistics.]
{[VIII. Extractions and Additions\
Give Researchers algorithm for applying Causal Statistics
VIII. A. Philosophical and Epistemological Foundations (Background)
[IV. A. The Philosophy and Nature of Causality
IV. A. 1 Background]
Causality is a philosophical subject of long standing, with definitive of cause, typologies of causation, methods for establishing cause, etc. all over the ballpark.
2300 years ago, Aristotle posited four types of causes. His typology was largely based on the Greek language usage of the word and has little relevance to the conception of causality useful for developing scientific theories.
VIII. A. 1. [ IV. A. 1. a.] The Empiricists
Extreme Empiricism is a theory of knowledge which holds that all knowledge comes from experience, i.e., though sense impressions of the natural world.
In the 17the Century, empiricism was centered in Britain, with devotes like Lock, Hume, and Berkeley. Lock recorded the first explicit formulation of the doctrine of empiricism. Lock felt that at birth the mind was a clean slate (tabula rasa) upon which experience writes. Such beliefs would deny innate ideas, like the concept of causality. So, if the concept of causality is not innate, it must somehow spring from he natural world, through the senses, and into the mind as a fully formed concept, since, in the extreme empiricist's view, rational thinking or analysis can add nothing to an idea or concept that was not already there in the sense impressions.
VIII. A. 2. [IV. A. 1. b.] The Rationalists
In its purest (i.e., most extreme) form, rationalism looks to reason alone as the source of all knowledge. For rationalist philosophers, Euclidian Geometry was the paragon of their discipline, beginning with definitions and axioms and using deductive reasoning to derive Geometry.
In this process, they saw no use of information obtained through the senses from the physical universe. Hence, rationalism was their science a method, based on pure reason. Causality is not observable.
Extreme rationalist philosophers believed (1) the concept of causality to be a prior knowable, without reference to experience and (2) that knowledge of specific cause and effect relationships was determined a prior]??
Experiment: push ball 1 into ball 2 once. Assume nothing caused both balls to move. It could be that a que from below, pushing thru the table. Therefore, we cant deduce causal connection between ball 1 and 2.
VIII. A. 3. [IV. A. 1. c.] The Eclectics
[IV. B.???? Certainty of any Causal Connection and the Concept of Causality]
Hume presented two different arguments, both or which yield the same conclusion, namely that it is impossible to be certain that any two objects are related causally. This conclusion is considered by an amazingly large number of scholars to wield a death blow to the concept of causality.
Even though correct, Hume's conclusion is not necessarily a fatal blow to the usefulness of the concept. What percentage of the "knowledge" employed daily is know with certainty and/or exactitude? Epistemology recognizes relatively few items of certain knowledge and, in fact, there may be none. Can you be sure that the sun will rise tomorrow?
If Bayesian Statistics were employed to infer from a sample correlation to its population correlation, the hypothesis that the population correlation is zero, may be rejected at the .05 level. Have we established, with certainty, that the population correlation is not zero. No! But, based on the assumption of no measurement error and other assumptions, we can conclude, with a confidence of 95% that the population correlation is greater than zero. This is not certainty, but useful information.
Hume believed that a billiard ball could be perceived through the senses, without further mental or logical manipulation. If a rolling billiard ball hits another and the second ball moves off toward a pocket, the observer will likely attribute cause and effort to the event. As before, the balls are perceived through the senses, but the attribution of cause and effect is a result of metal processing and not simple sensation or observation. Hume would say that causality cannot be observed and is, therefore, a mental, i.e., rational, construct or concept. That being said, the inference of a specific cause and effect relationship is the result of the alchemy of (1) the rational concept of causality (and its definition); (2) various assumptions about the nature of our universe, about the relationships among variables (both considered and unconsidered), and about the research design (e.g., no measurement error); (3) the observable co variations among the study variables (i.e., data) (empiricism).
[ 4. Causal Usefulness]
Hume denied our ability to obtain certain knowledge of a causal connection. Here, I would agree with Hume and go much further to assert that we cannot obtain certain knowledge of almost any thing, including associations. Associations possess statistical error, measurement uncertainties, sampling error, etc.
But, from a pragmatic point of view, Hume admits the usefulness and universality of the concept of causality:
"...it may still, perhaps, be rash to conclude positively that the subject, therefore, pass all human comprehension....It is certain that the most ignorant and stupid peasants--nay, infants; nay, even brave beasts--improve by experience, and learn the qualities of natural objects, by observing the effects which result from them. When a child has felt the sensation of pain from touching a flame of a candle, he will be careful not to put his hand near any candle; but will expect a similar effect from a cause which is similar in its sensible qualities and appearance.”*
------------------------------------------------------------------------------------------------------------*Hume, David: ENQUIRIES, Second Edition, Oxford at the Clarendon Press, MDCCCCII, p. 38-39. ------------------------------------------------------------------------------------------------------------
This is the point at which many scholars misinterpret Hume. They see his conclusion that there can be no certainty of causal connections, but do not comprehend the distinction he draws between certainty and usefulness.
In statistical terms, we cannot prove causal connections, but we can be 99+% confident of the usefulness of a causal influence, based on an appropriate set of assumption.
Hume explains the seeming conflict between the philosophic and the pragmatic points of view, asserting that, based upon the experience of a constant conjunction between flame and heat, "the mind is carried by custom to expect heat"** from a flame. ------------------------------------------------------------------------------------------------------------**Ibid., p.46 ------------------------------------------------------------------------------------------------------------ "All inferences from experience, therefore, are effect of custom, not of reasoning."*** ------------------------------------------------------------------------------------------------------------***I bid., p.43
------------------------------------------------------------------------------------------------------------
Hume asserts that the effect of custom upon the mind, in overcoming reason is "an operation of the soul."**** ------------------------------------------------------------------------------------------------------------****Ibid., p.46 ------------------------------------------------------------------------------------------------------------
With today’s understands, we would simply state that human and even animal minds evolved to infer causal connections, even though uncertain, because such inferences were useful for their survival.
[5. Observability
6. Observationally Equivalent Concepts/Theories]
Causality is one concept to explain the observed activity in the universe. An alternative concept is God. One could assert that God controls everything and there is no cause and effect relationship operating between the 2 billiard balls. God only makes it look that way.
These are two observationally equivalent theories of the operation of the universe. Either concept could be correct and the observed universe would look the same.
This puts the lie to the empiricists’ assertion that causality and causal connections are observable. Obviously, causality is a mental construct, generated to explain (account for) the observed behavior of objects within the natural world.
Hence, the empiricists and rationalists were both partly correct and partly incorrect. The concept of causality is a non-observable, rational construct, but the concept was created to explain observed activity in the universe. Therefore, the concept of causality ultimately arises from both rational and empirical inputs.
As with the concept of causality, specific causal connections cannot be directly observed either, but nor can valid specific causal correlations be inferred without empirical input.
Sample correlations are empirical results. One could correctly argue that the calculations of correlation coefficient are a mental process. But associates can be perceived without the calculation of correlation coefficients. On the other hand, rationalists might claim that remembering multiple observations and putting the multiple observations together to perceive an association are mental acts. I would agree with this, but a causal inference is a different level of mental processing. Association or correlation requires memory and calculation. Causal inferences require these plus an appropriate definition of cause plus assumptions about the causal nature of the universe plus, in non-experimental research, assumptions about unconsidered variables and the relationship of all variables to each other.
What the construct of Causal Statistics does is to routinize (or codify or make into an algorithm) the handling of data (mostly correlations), the definition of cause, the assumptions, and the calculations required to arrive at valid causal inferences.
One could say that valid causal inferences could be drawn without Causal Statistics, simply using Classical Statistics to calculate correlations and then by doing all of the other above activities correctly without reference to the construct of Causal Statistics. This is true and that is what researchers have been trying to do, using Classical Statistics in the absence of the knowing that Causal Statistics exists, without much success. First, in the best cases, they don’t know all of the inputs required. More likely, they aren’t even aware that these additional inputs are required.
Attempting non-experimental causal inferences without Causal Statistics is like not using algebra to determine four unknowns, given sufficient relationships among the variables. But if algebra exists, no one would attempt the task without using algebra. Some with Causal Statistics.
Other of observationally equivalent theories are possible. Consider the possibility that there is no causality and no God and that the behavior of the universe is simply a result of extremely highly improbable random occurrence.
One such random occurrence might be that a billiard ball roles up and stops next to (i.e., touching) a second billiard ball and, on a random fluke, the second billiard ball immediately begins to rolled away toward a pocket. No causality, no God; just random occurrence. Note again that causal connections cannot be observed, only inferred.
Bayesian Statistics would consider it highly improbable that the operation of our universe could be based on the occurrence of so many highly improbable fluke random events, like the probability of getting heads one Trillion Trillion… times in a row. Highly unlikely, but the possibility cannot be rules out with certainty.
VIII. A. 4. [7.] Definition of Cause
VIII. B. [B.] Logic
V.B. Logic
1. Deductive
2. Inductive
3. Deductive/Inductive (Combination) Systems
Secondly, Causal Statistics is a logical construct, in the sense that Euclidean Geometry is a logical construct. It is founded on appropriately chosen definitions and assumptions and its final formulation is derived through deductive logic.
The necessary assumptions can be grouped into three sets. The three assumptions sets could be broadly, but somewhat imperfectly (i.e., over simplified), labeled as follows: fundamental assumptions, isolating assumptions, and statistical assumptions.
VIII. B. 1. The First Assumptions Set
The first and most fundamental assumption set postulates that we live in a causal universe, where cause and effect governs the behavior of variables, at least at the macro level. These assumptions are presented in chapters 7, of the dissertation, below.
VIII. B. 2. Second Assumptions Set
The second assumption set is for the purpose of isolating and/or for limiting the interactive freedom of variables considered in a non-experimental study. These assumptions must be judiciously chosen in ways that will allow valid causal inferences.
VIII. B. 3. The Third Assumption Set
The third assumption set deals with the typical concerns of any statistical study: measurement error, sampling bias, etc. With regard to this set, there is no real difference between Causal Statistics and Classical Statistics. ]}
B. Philosophy of Causality xx
V.C. Philosophy of Causality & Definition of “Cause”
Causality is a philosophical subject of long standing, with definitions of cause, typologies of causation, methods for establishing causes, etc. all over the ballpark.
2300 years ago, Aristotle posited four types of causes. His typology was largely based on the Greek language usage of the word and has little relevance to the conception of causality useful for developing scientific theories.
In the intervening millennia, many other philosophers followed without
appreciable convergence of opinion. As an indication
of the fundamental nature of the controversy, note that
seldom have philosophers agreed even on the definition
of the word "cause"; that is, if they attempt to define
it at all. Given this disagreement over the concept
itself, it is not surprising that conclusions about
the concept (or, actually, concepts) of causality are
many and varied.
Of all these philosophers, probably the most insightful and influential was the British empiricist David Hume (1711-1776). His most influential conclusion concerning causality was that it is impossible to be certain that any two objects are related causally. You arrived at this conclusion twice through two separate deductive arguments.
Argument (5-1):
Premise (1,1): All knowledge of causal(H) relationships
"is not, in any instance, attained by reasonings
a priori, but arises entirely from experience, when
we find that any particular objects are constantly
conjoined with each other"* In other words, causal
*Hume, David: ENQUIRIES Concerning the Human Under
standing, Second Edition, Oxford at the Clarendon
Press, MDCCCCII, p. 27.
conclusions are arrived et by way of an inductive
argument, based on experience.
Premise (1,2)(Note: The 1 refers to the argument
number and the 2 refers to the premise number.):
In no case is the conclusion of an inductive ar
gument (in this case, an induction from experience)
certain.
Conclusion (1,1): Therefore, it is impossible to
the certain that any two objects are related causally(H). Or, as Hume puts the conclusion, "In vain do you pretend
to have learned the nature of bodies from your past
experience."*
*Ibid., p. 38
What is Hume's definition of cause?
What do you think about Hume's definition of cause and why?
Write down your definition of cause.
How does your definition of cause compare to the commonly understood definition of cause?
Is your definition of cause suitable for use in the sciences?
Hume's second argument is based on his belief that natural laws may and do change.
Argument (5-2):
Premise (2,1)1 (Same as Premise (1,1)) All knowledge
of causal(H) relationships "is not, in any instance,
attained by reasonings a priori, but arises entirely
from experience, when we find that any particular
objects are constantly conjoined with each other."
In other words, causal conclusions are arrived
at by way of an inductive argument, based on
experience.
Premise (2,2): ".,.all inferences from experience
suppose, as their foundation, that the future will
resemble the past...""
**Ibid.„ 37,
Premise (2,3): It is not certain "that the future
will resemble the past."*** (Hume argues that
***Ibid., p. 38.
natural laws may and do change.)
Conclusion (2,1): Therefore--even if we could
determine that two objects were related causally(H)
in the past--it would be impossible to be certain
that these two objects would continue to be related
causally(H) in the present or future.
Do you buy these two arguments? Are they valid?
What does "valid" mean?
How damning are Hume's conclusions to the concept of causality? How detrimental are these conclusions to the drawing causal inferences in emperical research? And the utilization of the concept of causality in theory building?
Xx B. 1. Recasting Hume's Arguments in Current Day Statistical Terms
Arguments (5-1) and (5-3) could be presented in a
form, different from the previous form, in order to bring
out the statistical implications of the arguments.
Arguments (5-1) and (5-3), rewritten for this purpose,
follow: Experience of two variables gives us only sample
information (data) about the world, because the population
of all data on these two variables contains all
possible past, present, and future experience of them.
When we infer--based upon sample data (our experience)--
that two variables are associated, our conclusion is
subject to sampling error. Therefore, we cannot be
certain that an association between two variables really
exists for the population as a whole. Causal(H or P)
relationships are a subset of all associational relationships,*
Therefore, we cannot be certain that two
*Simon Herbert: op.cit., p. 230
variables are causally(H or P) related because of the
possibility of sampling error.
The implication for statistical analysis exhibited
in Arguments (5-2) and (5- )6) are equally clear. We
cannot be certain that the laws of the universe will not
change in the future. Thus, inferences based on past
statistical data may lead to invalid conclusions about
the present and future.
B. 2. Logical Implications of Hume's C onclusions xx
Now, we answer the questions posed at the end of
Section 5.2. How damning are Hume's conclusions to the
concept of causality(H or.P) and how detrimental are
they to the pragmatic utilization of tho oonoopt? These
questions seek to determine the logical impact of Hume's
conclusions as opposed to the actual impact. The actual
impact is the result of overreaction to and misunderstanding
of Hume's position, on the part of many scholars.
Hump believes his conclusions to be absolutely
damning to any certainty or proof of causal(H) connections,
which is precisely what his conclusions state. In this
belief, Hume is speaking as a philosopher who wants
"to learn the foundations of this inference." From
*Hume: op.cit., p. 38.
this point of view, I would agree with him and extend
his belief to the common usage of the term and, therefore,
to cause(P).
80
81
But, from a pragmatic point of view, Hume admits
the usefulness of the concept of causality.
"...it may still, perhaps, be rash to conclude
positively that the subject, therefore, pass all
human comprehension....It is certain that the most
ignorant and stupid peasants--nay, infants; nay,
even brate beasts--improve by experience, and learn
the qualities of natural objects, by observing the
effects which result from them. When a child has
felt the sensation of pain from touching a flame
of a candle, he will be careful not to put his
hand near any candle; but will expect a similar
effect from a cause which is similar in its sensible
qualities and appearance."*
*Ibid., p. 38-39,
This is the point at which many scholars misinterpret
Hume. They see his conclusions that there can be no
certainty of causal connections, but do not comprehend
the distinction he draws between certainty and usefulness.
In statistical terms, we cannot prove causal connection,
but we can be 99+% confident of the usefulness of the
concept (see Section 4.6). And, as George Stigler**
**Stigler, George J.: The Theory of Priced 3rd ed.,
New York, Macmillan, 1966, p.6.
would say, if no other theory with greater confidence
(predictive ability) is available, then this theory is,
at present, the most useful; so use it.
Hume explains the seeming conflict between the
philosophic and the pragmatic points of view asserting
that, based upon the experience of a constant conjunction
between flame and heat, "the mind is carried by custom
82
to expect heat" from a flame. "Al]. inferences from
*Hume: op.cit., p.46
experience, therefore, are effect of custom, not of
reasoning."** Hume asserts that the effect of custom
**Ibid., p.43
upon the mind, in overcoming reason is "an operation
of the soul."***
***Ibid. p.46
This resolution of the apparent conflict approaches,
if not duplicates, Kant's explanation of the belief in
causal relationships. Kant states that the belief that
two variables are related causally is a priori synthetic
knowledge.**** The similarity seems even more reasonable
****Kant, Immanuel: Critique of Pure Reason,
Translated by Norman Kemp Smith, N.Y., St. Martin's Press, 1965, pp. 44-45,
when we see that Hume describes the mental effects of
"custom" as "a principle of human nature." *****
*****Hume: op.cit., P. 43
Not only are causal(P) "laws" uncertain, but
associations (i.e., correlations) and specific causal(P)
events are also uncertain. The reformulation of Arguments
(5-1) and (5-3) in Section 5.3 states specifically that
8 3
we cannot be certain of observed association. The
association could be due to sampling error. In fact
both of Hume's arguments apply to associations.
Given that Hume's arguments can conclude that
associations are uncertain, even stronger arguments
than those of Hume could be formulated to reject the
certainty of causality. This is due to the fact that
even if we were certain of an association between two
variables, we could not be certain that a causal(P)
connection existed between them. But this potential
strengthening of the argument against certainty of the
causal(P) concept, does not harm the usefulness of
the concept.
Neither can we be certain that a specific event
causes(P) another specific event, even if we can experiment.
Say we manipulate X and observe a change in
Y. The change in Y may have been a "random" fluctuation
in Y or a spurious variable could be causing(C) both
our manipulation of X and the change in Y.
B.3. [5.5] Summary xx
In summary, there are two errors to which all causal (H
or P) inferences arc subject.
First, the data upon which the inference Is based
may not be a representative sample of the population
due to sampling error. In other words a sample correlation
may occur due to sampling error even though there
is no population correlation.
Second, even if a relationship between two objects
or variables existed in the past, the natural laws of
the universe may .change, resulting in no or a different
relationship between the variables in the present and/
or future.
For these reasons it is im -possible to be certain
that any two variables are related causally(H or P)
or even correlated. But from the pragmatic point of
view, we (and Hume) accept the usefulness, indeed the
necessity, of the concept of causality.
C. The Definition of Cause xx
Dissertation, Chapter 4
Give theoretical definition of cause; then, micro definition of cause, with one fundamental particle and only one initial state. Then present the concept and definition of macro causes.
The concept of causality has been analyzed in the
writings of almost every important philosopher without
appreciable convergence of opinion. As an indication
of the fundamental nature of the controversy, note that
seldom have philosophers agreed even on the definition
of the word "cause"; that is, if they attempt to define
it at all. Given this disagreement over the concept
itself, it is not surprising that conclusions about
the concept (or, actually, concepts) of causality are
many and varied.
Therefore, an
analysis of the causal writings of these two great
men should be quite helpful in our attempt to understand
the conceptions and misconceptions about causality.
After hundreds of years of discussion, analysis,
and use--"cause" remains a word without a definition.
At least, there is no appreciable agreement among
philosophers and scientists upon any particular
definition.
Consider the following, statement made by Julian
Simon in a 1970 article,
...no perfect or near-perfect definition of
'cause and effect relationships' has yet been
created.*
*Simon, Julian L.: "The Concept of Causality
in Economics", Kyklos, No.2, 1970, pp. 232-233.
Simon goes on to say that this is not surprising
due to the complex and abstract nature of the relationship.
Bunge, in discussing the meaning--i.e.,
definition--of "cause", calls it the "thorniest of
words".**
**Bunge, Mario: Causality, Harvard University
Press, Cambridge, Mass., 1959, p.31.
In this chapter we present and analyze the
definitions of cause forwarded by Galileo, Hume,
and Mill. Their definitions are operational rather
than theoretical (i.e. ontological) and encompass
all combinations of necessary and/or sufficient
conditions. Each of these definitions is shown to
be inconsistant with the common usage of the term
"cause", We then forward a theoretical definition
of cause (called cause(P)) which is consistent with
the common usage of the term.
4.2 Galileo's Definition of Cause
In 1632 Galileo described the causal relationship
as "a firm and constant connection".*** This
***Galileo (1632): Dialogo sopra i due massimi
sistemi del mondo, giornata 4a, in Opere, Vol.
7, p. 471. Quoted from Bunge, op. cit., p.4.
is equivalent to saying that an event, A, is the
cause of an event, B, if and only if A is a necessary
and sufficient condition for B. Presumably this
necessary and sufficient relationship must hold for
any and all conditions, i.e., irrespective of the
states of other events. We will denote this definition
of cause by "cause(G)".
Note that cause(G) is very restrictive and does
not conform to the common usage of the term, "cause".
Not that there is anything necessarily holy about
common usages in general, but in this case the common
usage of the word--denoted by "cause(C)"--is the
most useful and applicable meaning of the term.
No explicit definition of cause(C) is given because
it is the goal of this chapter to find or formulate
a definition which conveys the common usage meaning
of cause.
It is not necessary that cause(G) and cause(C)
should be identical in every respect because cause(G)
is an operational definition and cause(C) could be
considered to have a theoretical definition. (This
is analogous to the relationship between I.Q. and
intelligence.) But a good operational definition
will conform to (i.e., be consistent with) the
theoritical definition, In other words, if and
46
only if A causes(C) B, then A should cause(G) B.
In this case the face validity of cause(G) would
be perfect.
Cause(G) is inconsistent with cause(C) in many
respects, but we will only consider two of the
shortcomings here. Other sources of inconsistency,
which apply to cause(G) and cause(C), will be
discussed later during the analysis of Hume's and
Mill's definitions. The two shortcomings considered
here are not discussed later because they are not
valid criticisms of either Hume's or Mill's definitions.
Plus rest of chap 4.
Aside:
{A measure of intelligence is one's ability to adapt to new and different, novel situations}
{A genius is a person who aims at a target that no one else can see and hits it}
{True genius is not complicated, blazing intelligence. It is simply crystal clear thinking along with creativity, the ability to visualize a megaproject all at one time (a large memory cache), and the willingness to truly consider ideas which deviate from the norm, accepted paradigms, and the contemporary dogma.}
Sections B and C should be collapsed into a new section B.
C. Definition and Philosophy of Cause xx
Readers Note:
It may seem unusual for a book on causal statistics to devote a significant amount of print to the history and development of causal philosophy and definitions of cause. "Why not just give us the bottom line on each issue and then move on to drawing causal inferences?"
There are several answers to this question. One, in most cases it is difficult to have a deep understanding of an issue without observing some of the missteps and also some of the proper decisions in the development of the solution. Two, it is not possible to present in this book every situation which you might encounter in your dealings with causality, but hopefully I can teach you how to think about these issues, so you can solve your own novel problems when they arise.
Pedagogical question for the reader:
Before beginning this section, write down your definition of cause, without referring to a dictionary or the remainder of this section.
Start with Aristotle, etc. [Aristotelianism. any of the four things necessary for the movement or the coming into being of a thing, namely a material (material cause), something to act upon it (efficient cause), a form taken by the movement or development (formal cause), and a goal or purpose (final cause).
Cause(C): a person or thing that acts, happens, or exists in such a way that some specific thing happens as a result; the producer of an effect: You have been the cause of much anxiety. What was the cause of the accident?
From ol dictionary.]
Then maybe Galileo, Hume, Mill, someone with clear necessary and/or sufficient definition, etc. Discuss common definition.
Discuss necessary and sufficient conditions.
Present definitions of cause posited in dissertation. In 1972 I thought that it was only a matter of time until the underlying causal, deterministic nature of quantum mechanics was discovered. In the succeeding 35 years, no such nature has been discovered and there has been a quite credible proof (actually originating in 1967) that such hidden variables cannot exist.
Present and analyze Pearl's definition.
Discuss theoretical and operational definitions here or in the next section on definition of cause
V.D. Attempts at Causal Inference
1. God
2. Hobbs
3. Bacon
4. Mill
5. Conditionals, Counterfactuals, etc.
{Click to go back to: Table of Contents, I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII, XIII, XIV, XV, XVI, XVII, XVIII}
VI. Foundations of Causal Statistics (Physics, Metaphysics, and New Definitions of “Cause”, for Application in Research and Science)
Causality is generally thought, by those who believe in it at all, to be built into the fabric of the universe. By this I mean that the operation of the universe is either totally or partly based on natural causal laws.
Does this mean that a causal connection between book ownership and reading capability, in either direction, or a causal connection from in-prison-program-participation to recidivism rates are causal laws of the universe? No. These are not laws of the universe, but macro causal connections which are aggregates of a huge number of micro-causal connections and chains.
Macro causal connections are more like regularities in the universe rather than laws, like macro variables are regularities or lumpiness in the universe. These macro causal connections are not universal or immutable. In one cultural setting or for a given population, in-prison-program-participation may decrease recidivism rates, but in another cultural setting or for another population, say hardened terrorists, in-prison-program-participation may have no effect.
So, where are and what are the causal laws of the universe? Recently I have given this matter considerable thought and have come to the decision that there are two possible types of causal laws built into the universe.
{ Thus, the current logic of correspondence principle between classical and quantum mechanics is that all objects obey laws of quantum mechanics, and classical mechanics is just a quantum mechanics of large systems (or a statistical quantum mechanics of a large collection of particles). Laws of classical mechanics thus follow from laws of quantum mechanics at the limit of large systems or large quantum numbers. Wiki}
This odd conclusion is necessitated by Einstein's nemesis, quantum mechanics. Experiments have shown that at small scales (at the level of molecules and below) the universe behaves stochastically, in accordance with specific probability distributions.
Physicists have argued for almost 100 years whether the universe is really random at these small scales or whether there are underlying deterministic mechanisms that we can't see which are determining the apparently random behaviors, outcomes. This is like flipping a coin....
In the Dissertation I postulated that the universe is composed of only one type of fundamental particle, always in the same initial condition, and infinitesimal in size. The mechanism by which this arrangement could lead to the stochastic character of quantum mechanics would be convoluted and contrived, inconsistent with Occam's razor and the general desire for elegant simplicity in physics.
If the metaphysics were changed to postulate 1. fundamental particles are a finite size. 2. There are various types of fundamental particles or there is one type, but initial conditions of this one type of fundamental particle are not always the same and therefore effect the way the causal impulse is communicated through it making an identical cause lead to different outcomes depending on the type or initial state of the intermediary fundamental particle.
This would mean that similar initial causes could lead to different outcomes because the intervening fundamental particles through which the causal impulse is transmitted might differ. The longer this chain, the more relevant the law of large numbers, and the more stable, consistent, and predictable the outcome. This is simply and elegantly consistent with the observed randomness of quantum mechanics.
Even so, after an exhaustive and continuing search, no hint of these hidden, underlying, deterministic mechanisms has been found. Hence, it seems very possible that the universe is truly stochastic at small scales. If that is the case, what happens to the concept of causality?
One way to handle this concern is by postulating that the causal laws of the universe are stochastic in accordance with given probability distributions. For example, consider a proton speeding through a particle accelerator in striking a target....
Note that he first postulated metaphysics leads to quantum mechanics and the second postulated metaphysics is quantum mechanics, so both must deal with the question, if quantum mechanics is stochastic, how do we get to a macro causal world?
This is really two questions. First, how does the real world get from micro randomness to macro determinism? And second, how do we do that with our derivation of causal statistics?
The answer is really the same for both, the law of large numbers. Events in the macro world are aggregates, usually huge aggregations, of micro-events. As the number of stochastic micro-events composing a macro event increases and standard deviation of the average outcome decreases and, at usual macro levels, is effectively zero.
Aside: probability of the glass jumping up off the table is due to apparently random fluctuation of particles, but not to quantum mechanics. What highly improbable randomness at the macro level would be due to quantum randomness?
Hints, I am postulating that the universe has micro-causal laws at small scale, but macro causal connections are not laws of the universe, just regularities or lumpiness for a given population.
Assumption 5-1: The universe is composed of one or more types of fundamental particles, each of which can take on two or more states.
VI. A. Nuclear Physics
VI. B. Fundamental Particles
VI. C. Quantum Mechanics
Does a single electron, going through two grates of a diffraction grating, break into quanta or as the breakup continuous?
If the universe is stochastic, causal " laws" still operate between fundamental particles and their stochastic causal mechanisms determine the apparently random behavior of larger (composite) particles, e.g., electrons or protons.
VI. D. Statistical Mechanics
VI. E. Etc.
VI. F. Final Definition of “Cause”
Distinguish the effects of quantum mechanics from those of statistical mechanics.
New Definitions of "Cause," For Application in Research and Science xx
This book is about causal statistics. The purpose of causal statistics is to draw causal inferences so that theoreticians can develop valid causal theories, allowing research consumers to utilize these causal inferences and causal theories. Hence, any definition of cause developed here should be oriented toward usefulness in research and science.
In the dissertation and in a previous chapter, we assumed that (1) our universe is made up of fundamental objects or particles of zero size and (2) the operation of the universe is ultimately based on causal interactions between adjacent fundamental particles. It was further assume for simplicity that there was only one type of fundamental particle and that the initial condition of each fundamental particle was always in the same state.
Based on the presentation in the foregoing chapter, it would seem appropriate to make some modifications to these assumptions.
Expand micro-assumptions to assume two or more types of fundamental particles and or two or more initial states.
Define quantum cause: causation between particles in the quantum size range, i.e., causation with a quantum probability distribution superimposed upon its effect.. Generally, quantum probabilities are only significant for particles smaller than a large molecule. Quantum effects are inversely proportional to sizes of particles.
No need to change the original definition of cause, because the original definition says nothing about completely deterministic causes or probabilistic causes. The original definition does not rule out a probability distribution in the effect.
{Click to go back to: Table of Contents, I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII, XIII, XIV, XV, XVI, XVII, XVIII}
VII. Derivation of Causal Statistics
A. Deductive Logic
Consider rules of deductive inference from kids geometry book.
The beginning: definitions, postulates, axioms, etc.
Analogous to Euclid, Einstein, Whitehead and Russell, Probability Theory, etc.
[I began, as Euclid began with Geometry, by using a Meta language to state primitives, definitions, axioms, etc. From there, the whole of Causal Statistics was derived using both verbal and symbolic logic. Such an approach, allows complete understanding of the paradigm to anyone who is willing to put in the effort to follow the steps of its development, i.e., the derivation of Causal Statistics.
For a less rigorous, but adequate understanding of Causal Statistics -- i.e., an understanding adequate for a knowledgeable application of Causal Statistics to non-experimental research and utilization of the conclusions -- note the following.]
VII. B. Research Methodology
VII. C. The Derivation of Causal Statistics??
Micro Cause, micro variable, discrete causes (quantum mechanics), macro variable, macro causal chain, error term, general form of causal statistics
VII. D. The Assumptions and the General Model of Causal Statistics XX
1. The First Assumption Set
2. The Second Assumption Set
3. The Third Assumption Set
Additionally, Causal Statistics is based on three sets of assumptions. The first and most fundamental set postulates that we live in a causal universe, at least at the macro level, where cause and effect governs the behavior of all variables.
The second assumption set is for the purpose of isolating the system of variables considered by the specific research study. As an example of some of the possible assumptions in Set #2, note the assumptions discussed above, in conjunction with the RIF program. These assumptions were necessary to make the causal inference valid, although in no way do these assumptions make the causal conclusion correct, if one or more of these assumptions are incorrect. We can be confident that the causal conclusion is correct, only if the assumptions are correct.
The second assumption set need not totally rule out all influences from outside variables on variables inside the system of study. These assumptions need only prohibit outside influences that would replace just certain types of influences. To understand which set two assumptions are required and which are not, one must understand the components of association.
The association between two variables can have four primary sources; association due to (1) causal connection (e.g., B causes R and/or R causes B), (2) spurious correlation (e.g., S causes both R and B or S causes X and B and X causes R), (3) definitional overlay of internal variables (e.g., book ownership and school book ownership), and (4) statistical error. An observed association will often be a combination of all of these sources.
Association due to causal connection, source (1), is what researchers are looking for. Spurious correlation, source (2), is what researchers want to avoid and is what assumption set two needs to rule out. To be a little more precise, some spurious correlations are acceptable and others are not. It is not acceptable for any external variable(s) to cause a spurious correlation within an internal variable pair, if the variable pair, is the subject of a causal hypothesis test in the study. Such a spurious correlation must be eliminated by assumptions or by bringing external variables into the study. External sources of spurious correlation within any non-hypothesis variable pair are not a problem to the study and need not be eliminated.
Definitional overlap, source (3), is simply an error in research design and should be avoided. Statistical error, source (4), is inevitable in statistical studies and should be handled with the usual techniques of Classical Statistics. After all that’s what Classical Statistics is for and not for causal inference.
If the set two assumptions are properly designed and all else is handled correctly, the causal inferences drawn should be “valid.” But if these assumptions are not satisfied (i.e., false), the causal conclusion may be incorrect or at least the accuracy of the causal inferences will be diminished. The extent of diminution is usually related to the degree of error in the isolating assumptions.
The first two assumption sets are required only by Causal Statistics and not by Classical Statistics, but there is a third set required in both paradigms. This third set assumes no measurement error, no sampling bias, etc. Since the third assumption set impacts both Associative and Causal Statistics equally and our purpose here is to highlight the differences, we will simply consider the third set to be satisfied and not deal with these assumptions further.
In Classical Statistics, with the third set of assumptions satisfied, a positive result supports a non-causal hypothesis, e.g., an association, a mean value, etc., about the population, subject only to the error inherent in random sampling (i.e., statistical error). The probability of statistical error can be calculated and specified precisely. Such associative results are very close to simply reporting on what was observed, i.e., only slightly more inferential than reporting on what was observed.
In Causal Statistics, one or more causal hypothesizes are proposed; a study is designed; data are collected; the causal hypothesizes are either confirmed or not confirmed; subject (1) to all three assumption sets (although we have, for purposes of this discussion, stipulated that the third assumption set is satisfied.), (2) to a given definition of “cause,” (3) to calculated probabilities of statistical error, and (4) to the requirement that all assumptions be explicitly stated.
Causal connections are not directly observable and Causal Statistics requires more input to obtain its conclusions (i.e., the causal inferences). Associations are directly observable and Associative Statistics requires no additional inputs, except assumption set number three, (which was already considered satisfied) to obtain its conclusions, i.e., inferences to population associations. The additional [burden inherent] inputs required by Causal Statistics are a useful definition of “cause” and two additional sets of assumptions.
Many people will initially be uncomfortable with Causal Statistics’ (1) additional input requirements and the greater domain for error introduced by these added inputs, i.e., the definition of “cause” and the extra two assumption sets. But, people should not be discouraged by this, because in all optimized systems (like the various statistical paradigms), no additional benefits are free. You have to give up something to get something.
In the case of Causal Statistics verses Associative Statistics, when we accept a larger domain for error, we get causal inferences, generally, a much more useful (powerful) result for research consumers. Looked at from an information theory point of view, a causal inference contains much more information than an associative inference and, in optimized systems, greater information output always requires greater information input and greater inputs almost always increase the chances for error.
Way too often, researches use Associative Statistics and claim to make causal inferences. Since Associative Statistics doesn’t dictate the heavy inputs required by Causal Statistics, these researchers erroneously believe that they have drawn causal inferences without the additional input burden and without the added error risk. Would, that [It would be wonderful if] causal inference in non-experimental research were so simple. Unfortunately, valid causal inference is orders-of-magnitude more difficult and complicated than this. Nevertheless, researchers around the world deceive themselves and their consumers, on a daily basis, with such invalid causal inferences. The world wide waste of research money, human resources, and lost research benefits is truly beyond comprehension.
[Examples]
If Causal Statistics were used, the research would be significantly more difficult, but the resulting causal inferences would be valid. A well designed Causal Statistics study would probably consider more variables in the system, because the researchers would want to make assumption set number two as weak as possible, and therefore would obtain more robust, more powerful causal inferences. Follow-on studies would typically build on the first study and consider even more variables to further weaken the required set number two assumptions.
Such follow-on studies should be able to incorporate data and use the findings from the earlier study to further strengthen the findings from the second study. Bayesian statistics likely will be useful for this, but I will have to spend some quality time thinking about this issue and, presently, I do not have it.
Such applications of Causal Statistics would yield valid causal inferences. Validity does not guarantee the correctness of causal conclusions from Causal Statistics, just as validity of associative inferences from the application of Classical Statistics does not guarantee the correctness of associative inferences about the population association.
I’m sorry to have to inform you of this, but we live in a universe where guaranteed correctness, i.e., perfect knowledge or 100% certainty, is virtually, if not actually unattainable. For example, Newton’s Laws of Motion and Gravitation [?], once thought to be perfect, were shown by Einstein to be imperfect and ultimately incorrect. They were supplanted by Einstein’s Theory of Relativity which was thought to be perfect. But more recently it has been shown that Einstein’s Theory breaks down at the micro level and in black holes, again retreating from perfection.
Like everything else in this universe, Causal Statistics does not generate certain or perfect knowledge, but, if applied correctly, it does yield Preado Optimal causal inferences, i.e., causal conclusions that are the best and most reliable outputs attainable, given the inputs. If more input information (i.e., data, assumptions, etc.) is injected into the study, better and more reliable outputs (i.e., causal conclusions) are attainable.
So, for those of us in this universe -- in contrast to those people in a parallel universe with a different system of logic -- Causal Statistics is a Preado Optimal system for drawing causal inferences from non-experimental data. Its causal conclusions give the greatest probability of successful application by research consumers and social science practitioners; far greater than the probability of success with invalid causal conclusions obtained from the inappropriate application of Classical Statistics. [Smoking causes cancer and welfare examples]
Image the total benefits to society, if all non-experimental researchers used Causal Statistics when appropriate and these results were applied by research consumers…, “a consummation devoutly to be wished.” It's
RIF exemplifies two errors in causal inference that are common in non-experimental research. The first error is that causal inferences are often drawn, both explicitly and implicitly, by scientists who almost as often deny, cover up, and/or are unaware that, in their results, causal connections between variables are either stated or implied.
The RIF error was accomplished by (1) not specifically stating that B causes R, but saying that kids who own books are better readers, misleading the majority of human minds to jump to their own invalid causal inferences, and (2) then proceeding to develop a program and establish an organization, as if it had been proved that B causes R.
Other researchers hide the fact that they are making causal inferences (typically in an invalid manor) by not using the word “cause,” but by using synonyms like “yields,” “results in,” “produces,” “brings forth,” “brings out,” “creates,” “effectuates,” “elicits,” “is due to,” “generates,” “induces,” “leads to,” “makes,” and more.
The second error made, by those who claim (either explicitly or implicitly and either knowingly or unknowingly) to have discovered a causal connection, is that the assumptions, upon which the causal conclusions are based, are virtually never stated or even known. In the RIF example, they could validly conclude that B causes R if they were willing to state the assumptions (1) that R did not cause B and (2) that no other variables caused both R and B. Under these assumptions, a researcher could make the valid causal inference that B causes P, but few would be deceived into accepting the conclusion because most people would be unwilling to accept one or both of the required assumptions and, of course, if even one of the assumptions is invalid, then the conclusions are invalid.
Consider an empirical study in which variables B (i.e., number of book owned by the family) and R (i.e., reading capability of the child) are found to be correlated. If we assume (1) that no outside variables causally affect both B and R in such a way as to change their correlation, (2) that R does not cause B, and (3) that causal relationship are linear; one can validly conclude, subject to the usual statistical error, that B causes R with standardized strength equal to the correlation coefficient.
A “valid” causal inference is a causal inference which is a logically necessary result of the definitions stated, the assumptions made, and the data collected. In other words, a “valid” causal connection results from a causal inference arrived at by the proper application of Causal Statistics.
Note that a valid causal inference does not necessarily result in a correct causal connection. If one or more of the assumptions is incorrect, the causal connection will be incorrect. The greater the degree to which the assumptions are in error, generally, the greater the error in the causal connection drawn. This second assumption set can be very restrictive and questionable, yet, that is not the fault of Causal Statistics. It is, if you will, the fault of logic and the universe we exist in. Gravity can be inconvenient, if you want to fly, but the field of physics is not at fault. In fact physics can assist in overcoming the problem; same with Causal Statistics.
[VII. Attempting to Move Beyond the Beginning]`
Dissertation in Library of Congress and University of California Berkeley Graduate Library
Best year of my life. I learned so much out of my own head, I could almost become a rationalist.
WP6
During the 10 years I was working on and with Causal Statistics, beginning 40 years ago, I expected, with increasing disillusionment, that funding agencies, like the NSF, would shower me with research money and that non-experimental research methodologists, researchers, and research consumers would immediately grasp the importance and benefits of the subject and extract from the dissertation an application oriented formulation of Causal Statistics because of the great need for and the exceptional importance of this inquiring system. I expected that they would analyze all the philosophical details, expand and further develop the formulation of Causal Statistics, and apply it in almost all non-experimental research studies.
Boy! Was I mistaken. Most scientists showed little or no interest or understanding. The most common reactions were avoidance or ridiculous, naive criticisms. Research funders were polite, but felt that I should try some one else, anyone else, but them. Causal Statistics was much too eclectic for them.
I began wondering of they were all insane. Oh, that didn’t sound to good. Insane people usually think everyone else is crazy. Maybe it was me.
After ten years, I give up and went on to other things. Now, I’m back. When I looked at what’s been done in the intervening years, I was surprised. There is much more talk now, but not that much improvement in insight or over all progress.
As I read over the dissertation, I was motivated anew at the need for and importance of Causal Statistics. Some times you read something you wrote 40 years ago and it seems hopelessly off the beam and naive. There was a little of that, but, overall, I was struck by the insights and clarity with which the whole thing was pulled off and the feeling that the axiomatic/deductive approach to the development and overall understanding of Causal Statistics was the best one and that this approach is far superior to anything in the literature.
Maybe I was crazy then, but if so, I still am. I believe 100 years from now, assuming I don’t die before I can get the word out, that Causal Statistics will be considered the most important contributor in the development of the non-experimental sciences.
I’m back, but back with some differences. I now don’t see the reticent scientists and funders as insane. I now capabilities, their understanding of the needs, their ability to grasp such a large and complicated system as Causal Statistics, and their ability to rap their minds around such an eclectic, revolutionary, all-encompassing approach to the subject.
(Should have written papers, published empirical c. s. res., etc. That didn’t help S. Wright and others.)]
E. Mathematical statistics (The General Form of Causal Statistics)
F. A Causal Calculus/Algebra
G. Inductive Logic in Causal Statistics
{Click to go back to: Table of Contents, I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII, XIII, XIV, XV, XVI, XVII, XVIII}
VIII. Causal Statistics: A Two Variable Example XX
{V. Classical and Bayesian Statistical Analyses of the Example Data (The Mathematics and Interpretation of Classical and Bayesian Association) xx
History of Correlation
At the outset of this section, let me be clear that neither classical nor Bayesian statistics can produce valid causal inferences. "Cause" is not defined in either paradigm, like red is not defined in Euclidian geometry.
In this section, I will present the mathematics of associative statistics. The mathematics of causal statistics is not significantly different; it's the assumptions, the vocabulary [definitions, defined terms (like the concept of line or point), concepts and definitions not present in classical or Bayesian statistics], the input information, the understandings, the interpretations, and the results that are different.
Present a lot of mathematics of Association in general, e.g., nonlinear association, nonparametric Association, etc.
[Explain analysis of variance and analysis of covariance]
Along with the mathematics of associative statistics, I will present all of the assumptions associated with these mathematical/statistical forms, i.e., assumption set 3.
This is a chapter which presents the mathematics and interpretations of classical and Bayesian associative statistics. In the beginning of my research, I asked myself many times, "how do I know if a regression parameter or a correlation is causal or associative?"
Eventually, I came to half of the answer in a flash. I wasn't burning my brain to try to answer the question; I simply stumbled on to the question of fresh while dealing with another issue. At that moment half of the answer came to me in a stroke of intuition, but I had to spend some time thinking about it to be confident that my intuition was correct.
The first half of the answer is that a calculated regression parameter is nothing more than a representation of an observed Association. Causal connections cannot be observed; only associations can be observed. Therefore, the initial calculation of a correlation or regression parameter is an indication of observed linear association and indicates nothing about causation.
I arrived at the second half of the answer by a couple of flashes of intuition; no brain burning from just a brain that over time has acquired all the needed information and automatically brings information together to arrive at the correct intuition. It's like people who write music or lyrics or poetry sometimes say. It's like they are not creating the output at all; it's more like being channeled through them and they are writing it down. Sometimes they can't even write it as fast as is trying to slow out of their brain.
Your Anyway, the second half of the answer is that additional information; in the forms of assumptions, prior research results, etc.; must be added to the analytical system to decrease the number of possible causal mechanisms which could generate all or a portion of the observed correlation. When enough information has been added to reduce the possible causal mechanisms to one, the researcher can then infer that causal mechanism is responsible for some or all of the observed correlation, based on the input information and the observed data.
At that point, the correlation or regression parameter becomes at least partially causal. Any portion of the regression parameter which remains associative, might be attributable to some outside variable, x, which causes a portion of the correlation to be spurious.
[When it did come to me it came right out of the blue.]
[It's like opening the top of my head and throwing information in and then letting my brain grind around on it on its own time. Sometime; one day, two days, a few days; later the answer simply pops out.]}
Think back to the two-variable example of Chapter IV. It is observed that book ownership (B) and reading performance (R) are correlated. If our study
Figure: B and R, with correlation
is limited to these two variables, there are only two causal connections that could have produced the observed correlation, B causes R and R causes B. Neither of these causal connections, as indeed no causal connection, is
Figure: B and R are correlated and cause each other
directly observable. They can only be inferred from a combination of 1) postulates, 2) assumptions, 3) findings from prior empirical research studies, if available (e.g., the findings from a previous causal study that a one unit increase in reading score causes a 0.5 unit increase in the number of books own), 4) prior theories, 5) time precedence, and 6) data and/or statistics based on the data (like the correlation between B and R).
Initially, let's deal with the case in which R does not cause B, but it is possible
Figure: B and R are correlated and B causes R
for B to cause R. Say that the correlation between B and R is 0.5. Setting up a regression equation with R as the dependent variable, we obtain
R = a + bB + e,
where a and b are regression constants derived from sample data and e is the regression error term.
As shown before
b = 0.5 (12/2) = 3.
This means that, in this study, a correlation coefficient of 0.5 is equivalent to a regression parameter of 3. Substituting 3 for b in the above regression equation, tells us that, for a person who's book ownership is one greater than the book ownership of the second person, the first persons expected reading score will be three points higher than the expected reading score of the second person.
Note that this is a statistical prediction, which by its very nature contains the possibility, indeed the likelihood, of some error. In fact, for the two specific people, i.e., person 1 and person 2, the predicted scores are 80 and 83, respectively. But because of the random error term, the actual scores may be say 81 and 76 respectively.
But as more students are selected from the population, the average of the deviations from the predicted values should get smaller and smaller as the number of students sample gets larger.
Further note, I went to considerable effort to state the meaning of the regression equation in a non-causal manner. Even so, inadequately prepped, most people would still jump to a causal interpretation of the statement.
At this point, I have only applied regression analysis to the observed data. I have made no input of assumptions or knowledge which would enable me to go beyond Association to causation.
If I wish to move toward causal inference, I must input assumption sets 1, 2, and 3 into the analysis. As before, assumption set 1 is.... Assumption set 3 is....
Note here, that if the researcher found the correlation between R and B to be 0.5, we would have an interesting situation. It would turn out that the causal regression parameter of 3, found in an earlier causal research study, would lead to this exact .05 correlation. What would that mean? It would mean that there would be no correlation left over to result from B causing R, therefore we could conclude that, under the proper assumptions, B does not cause R.
{ Conclusions
Causal inference from experiments, even highly stochastic experiments, are much more difficult and much more complicated. These difficulties have lead to philosophical, definitional, and validity concerns that have permeated work with and the understanding of causality in non-experimental research.
The essence of this difference in causal inference between experimental and non-experimental studies, is (1) the difference in assumption (3), above, and (2) the usually diminished ability of the researcher to control (e.g., hold constant) other variables.
In experimentation, assumption (3) simply asserts that nothing causes both the experimenter to manipulate the independent variable (force) and the change in the dependant variable (acceleration), resulting in a non-causal correlation between the two variables; an eminently reasonable and acceptable assumption.
But, in non-experimentation, assumption (3) becomes, “assume that, during the study, no other variable or variables caused both independent and dependent variables, resulting in a spurious (i.e., non-causal) correlation between them;” a vastly more suspect assumption.
In the first experiment, above, the research was perfectly controlled. The only two relevant variables that changed were force and acceleration. In non-experimental research….
In fact, causal inferences are generally so easy to draw from experiments that, most of the time, no or only minor thought is required and, in stochastic experimental, only rudimentary statistical analysis is required. In such cases, Classical Statistics has generally been use to assist in making these causal inferences from experimental data, but Classical Statistics alone would not be sufficient for drawings valid causal inferences from even experimental data because “causality” is not defined in or considered by Classical Statistics.
Therefore, something additional is necessary. It turns out that that that extra something is human’s natural, innate intuition for making causal inferences. It is so natural for us that we don’t even realize that we are making a leap of intuition to causality, something, as Hume said, that we cannot sense directly, but can only infer.
But, such intuition cannot be a logical basis for valid causal inference. Hence, as was shown earlier, valid causal inferences from experimental data must additionally be based on a generally acceptable set of assumptions.
Causal statistics, unlike any other causal inquiring system, is a deductive system. See preface.
[II.D. Non-Experimentation]
In non-experimental research, everything changes. Causal inference is extremely difficult, leading to philosophical, definitional, and validity concerns that permeate work with and the understanding of causality in non-experimental research.
In the Social Sciences and in Epidemiology, experimentation is seldom feasible. In these and other non-experimental sciences, the situation with regarding to drawing causal inferences is vastly different. It is much more difficult and much more complicated. These difficulties lead to philosophical, definitional, and validity concerns that permeate work with and the understanding of causality in non-experimental research.
The essence of this difference in causal inference between experimental and non-experimental data is (1) the difference in assumption (3), above, and (2) the usually diminished ability of the researcher to control (e.g., hold constant) other variables.
In experimentation, assumption (3) simply asserts that nothing causes both the experimenter to manipulate the independent variable (force) and the change in the dependant variable (acceleration), resulting in a non-causal correlation between the two variables; and eminently reasonable and acceptable assumptions.
But, in non-experimentation, assumption (3) becomes, “assume that, during the study, no other variable or variables caused both independent and dependent variables, resulting in a spurious (i.e., non-causal) correlation between them;” a vastly more suspect assumption.}
{ As an example of how one might move toward Causal Statistics from Classical Statistics, consider the second error, made by those who claim (either explicitly or implicitly and either knowingly or unknowingly) to have discovered a causal connection. The second error is that the assumptions, upon which the causal conclusions are based, are virtually never stated or even known. In the RIF example, they could have validly concluded that B causes R, if they were willing to state the assumptions (1) that R did not cause B and (2) that no other variables caused both R and B. Under these assumptions, a researcher could make the valid causal inference that B causes R, but few would be deceived into accepting the conclusion because most people would be unwilling to accept one or both of the required assumptions and, if even one of the assumptions is incorrect, then the causal conclusions are highly likely to be incorrect.
This approach is an attempt to create a needed part of Causal Statistics by backing into it from Classical Statistics. The attempt is not improper, if executed correctly. But correct and complete execution along this path is very difficult, although not impossible. For this and other reasons, this is not the best approach, because (1) based on what I have observed over the last 40 years, not 1 in 100 researchers can reach a valid and complete causal inference in this manner; (2) the complete Causal Statistics paradigm would virtually never be developed in this way; and (3) no partial formulations would lead to a complete understanding of the nature, validity, accuracy, and weaknesses of the conclusions reached. In contrast, I have gone through the front door in developing and deriving Causal Statistics in my dissertation, which is presented on the Causal Statistics website, CausalStatistics.org
I began, as Euclid began with Geometry, by using a Meta language to state primitives, definitions, axioms, etc. From there, the whole of Causal Statistics was derived using both verbal and symbolic logic. Such an approach, allows complete understanding of the paradigm to anyone who is willing to put in the effort to follow the steps of its development, i.e., the derivation of Causal Statistics.
For a less rigorous, but adequate understanding of Causal Statistics -- i.e., an understanding adequate for a knowledgeable application of Causal Statistics to non-experimental research and utilization of the conclusions -- note the following.
Causal Statistics begins with a common sense, useful definition of “cause,” a non-trivial philosophical issue, given the numerous, inappropriate, and inadequate definitions forwarded by various philosophers and researchers over the past 3000 years. Additionally, Causal Statistics is based on three sets of assumptions. The first and most fundamental set postulates that we live in a causal universe, at least at the macro level, where cause and effect governs the behavior of all variables.
The second assumption set is for the purpose of isolating the system of variables considered by the specific research study. As an example of some of the possible assumptions in Set #2, note the assumptions discussed above, in conjunction with the RIF program. These assumptions were necessary to make the causal inference valid, although in no way do these assumptions make the causal conclusion correct, if one or more of these assumptions are incorrect. We can be confident that the causal conclusion is correct, only if the assumptions are correct.
The second assumption set need not totally rule out all influences from outside variables on variables inside the system of study. These assumptions need only prohibit outside influences that would replace just certain types of influences. To understand which set two assumptions are required and which are not, one must understand the components of association.
The association between two variables can have four primary sources; association due to (1) causal connection (e.g., B causes R and/or R causes B), (2) spurious correlation (e.g., S causes both R and B or S causes X and B and X causes R), (3) definitional overlay of internal variables (e.g., book ownership and school book ownership), and (4) statistical error. An observed association will often be a combination of all of these sources.
Association due to causal connection, source (1), is what researchers are looking for. Spurious correlation, source (2), is what researchers want to avoid and is what assumption set two needs to rule out. To be a little more precise, some spurious correlations are acceptable and others are not. It is not acceptable for any external variable(s) to cause a spurious correlation within an internal variable pair, if the variable pair, is the subject of a causal hypothesis test in the study. Such a spurious correlation must be eliminated by assumptions or by bringing external variables into the study. External sources of spurious correlation within any non-hypothesis variable pair are not a problem to the study and need not be eliminated.
Definitional overlap, source (3), is simply an error in research design and should be avoided. Statistical error, source (4), is inevitable in statistical studies and should be handled with the usual techniques of Classical Statistics. After all that’s what Classical Statistics is for and not for causal inference.
If the set two assumptions are properly designed and all else is handled correctly, the causal inferences drawn should be “valid.” But if these assumptions are not satisfied (i.e., false), the causal conclusion may be incorrect or at least the accuracy of the causal inferences will be diminished. The extent of diminution is usually related to the degree of error in the isolating assumptions.
The first two assumption sets are required only by Causal Statistics and not by Classical Statistics, but there is a third set required in both paradigms. This third set assumes no measurement error, no sampling bias, etc. Since the third assumption set impacts both Associative and Causal Statistics equally and our purpose here is to highlight the differences, we will simply consider the third set to be satisfied and not deal with these assumptions further.
In Classical Statistics, with the third set of assumptions satisfied, a positive result supports a non-causal hypothesis, e.g., an association, a mean value, etc., about the population, subject only to the error inherent in random sampling (i.e., statistical error). The probability of statistical error can be calculated and specified precisely. Such associative results are very close to simply reporting on what was observed, i.e., only slightly more inferential than reporting on what was observed.
In Causal Statistics, one or more causal hypothesizes are proposed; a study is designed; data are collected; the causal hypothesizes are either confirmed or not confirmed; subject (1) to all three assumption sets (although we have, for purposes of this discussion, stipulated that the third assumption set is satisfied.), (2) to a given definition of “cause,” (3) to calculated probabilities of statistical error, and (4) to the requirement that all assumptions be explicitly stated.
Causal connections are not directly observable and Causal Statistics requires more input to obtain its conclusions (i.e., the causal inferences). Associations are directly observable and Associative Statistics requires no additional inputs, except assumption set number three, (which was already considered satisfied) to obtain its conclusions, i.e., inferences to population associations. The additional [burden inherent] inputs required by Causal Statistics are a useful definition of “cause” and two additional sets of assumptions.}
The necessary assumptions can be grouped into three sets. The three assumptions sets could be broadly, but somewhat imperfectly (i.e., over simplified), labeled as follows: fundamental assumptions, isolating assumptions, and statistical assumptions.
VIII. B. 1. The First Assumptions Set
The first and most fundamental assumption set postulates that we live in a causal universe, where cause and effect governs the behavior of variables, at least at the macro level. These assumptions are presented in chapters 7, of the dissertation, below.
VIII. B. 2. Second Assumptions Set
The second assumption set is for the purpose of isolating and/or for limiting the interactive freedom of variables considered in a non-experimental study. These assumptions must be judiciously chosen in ways that will allow valid causal inferences.
VIII. B. 3. The Third Assumption Set
The third assumption set deals with the typical concerns of any statistical study: measurement error, sampling bias, etc. With regard to this set, there is no real difference between Causal Statistics and Classical Statistics. }
{There are four possible sources for the assertion that the causal connection from reading to book ownership is zero; 2), 3), 4) and/or 5) above. Let's take these possible sources one at a time.
The most likely source of our determination that R does not cause B is 2 ) above, by assumption and such assumption would have to be stated prominently along with the conclusions of the study so that any research consumer with no that the ultimate findings were based on this assumption and of course other assumptions, which would likewise have to be stated. The correctness of this assumption is quite doubtful, but that issue, the need for this assumption, and alternative ways of avoiding this assumption will be dealt with later.
Another source for this zero causal connection is 3) above, prior knowledge. Maybe a previous empirical study was performed in which it was determined that R does not cause B. As before, the utilization of this prior study in obtaining the conclusions of this research, must be stated along with the conclusions it contributed to.
In reading the above paragraph, one might be discomforted by the finding by the previous study that good reading performance does not cause the ownership of books. The comment might be made that you cannot prove a negative and R does not cause B is a negative finding. Actually, this comment is correct, but, at the same time, misleading.
It is true that it cannot be proven that he causal regression parameter for R causing B is exactly 0. Yet, it is also true that it cannot be proven that such regression parameter is exactly 2, or any other number. Stochastic or probabilistic error is always or potentially always present and such statistical error will lead to the regression parameter being only an estimate of the population parameter and distributed by some probability distribution around its expected value of zero or two or whatever number.
Hence, there is nothing unique about an estimated regression parameter equal to zero, except that humans have given that value a specific definition, namely, a negative finding. Similarly we could define a regression parameter of two to be a two-finding. Analogously we cannot prove a twofinding, although we could “prove” that the true population regression parameter fell between 1.9 and 2.1, with 95% Bayesian probability confidence and similarly for the confidence interval around zero.
Aside: this is not a math book; it is an understanding book.
[Aside: a zero cause could be arrived at because the putative effect occurs before the putative cause. On the other hand this is only strictly true for laws of the universe, i.e., micro causal laws and other causal connections close to that ilk. Macro causal laws would not be strictly limited in terms of time precedence. Maybe buying a house causes savings five years prior. This macro cause could result from the unmeasured anticipation and desire to buy a house in the future. In this case actually buying a house in the future could be a staring at for planning to buy a house in the past.
Looked at in another way a surrogate variable is not itself causal, even macro causal. If I bought houses for 100 people, the variable of home ownership today would not cause savings five years ago.]
Source 4), someone's prior theory, is another way to arrive at the belief that R does not cause B. Maybe some part or even all of a prior theory states that reading performance does not cause book ownership. This could then be a source for the zero causal connection and, like the assumption and prior study sources, the portion of the theory utilized in arriving at the results of this study must be stated along with the results.
The theory may be based on some other empirical study, a knowledgeable person's experience in the field, or nothing but a guess. Guess-theories could be considered very similar to assumptions in terms of their acceptability; theories based on empirical studies are similar to knowledge from previous studies; and theories based on personal experience falls somewhere in between. It is up to the researcher to present all necessary limiting information and up to the research consumer to evaluate these foundations as to their effect on the ultimate believability of or confidence in the conclusions of the study.
Another way to establish that there is no causal connection from R to B, is via source 5), time precedence, because a cause must occur before or simultaneous with its effect, except for Chuck Norris who's so tough he was born before his father. Just as an aside, he's also so tough that he sleeps with a pillow under his gun.
In the instant example, time precedence would not be relevant because reading performance and book ownership were presumably measured at the same time. Of course, one might argue that book purchasing decisions and ownership were both prior (say one year earlier) to the measured reading performance, hence reading performance could not have caused book buying or ownership.
The proper way to look at this is that reading performance is an operational measure of the theoretical variable, reading capability. The reading capability of a child is not a variable that generally fluctuates wildly over time. Hence, reading score today is an operational measure of reading capability today and reading capability today, controlled for age, is likely a good surrogate for reading capability one year ago. Therefore, reading capability a year ago could have been a cause of book purchasing decisions a year ago and in a more simplified, yet appropriate, approach, one could conclude that reading scores measured today could cause book buying decisions and book ownership.
Aside: On the other hand, causal time precedence is only strictly required for causal laws of the universe, i.e., micro causal laws. Macro causal laws would not be strictly limited in terms of time precedence. Maybe buying a house causes savings five years prior. This macro cause could result from the unmeasured anticipation and desire to buy a house in the future. In this case actually buying a house in the future could be a staring at for planning to buy a house in the past.
Looked at in another way a surrogate variable is not itself causal, even macro causal. If I bought houses for 100 people, the variable of home ownership today would not cause savings five years ago.}
[Book ownership (B) and reading performance (R) are correlated. There are many possible causal relationships which could produce the observed correlation. None of these causal connections can be observed; they can only be inferred from a combination of axioms, assumptions, previous bits of knowledge (e.g., the findings from a previous study that a one unit increase in socioeconomic status causes a 0.5 unit increase in the number of books owned), theories, and data and/or the results of data (like the correlation between B and R).
Figure: B and R and SES and X., with all causal connections and the correlation]
[X. [2)] A Three Variable Example of Causal Inference ]
Give researchers algorithm for applying Causal Statistics
Explain Hypothesis, testing vs. correlation matrices and the use of matrices to choose hypothesis.
Exemplify the 2 various research and associations, leading to the 3 variable research, leading to 10 variable research, etc. Using prior data. Someone else can figure out how or I’ll do it when I have time
Simultaneous experimental parameters partial out the considered correlation
A Ten Variable Example of Causal Inferences
A Ten Variable Example of Causal Inferences (A Ten Variable Follow-on Example of Causal Inference)
In the above example, one way to decrease the stringency of the required assumptions, would be to bring one or more of the relevant outside variables, inside. For example, collect data on B, R, and S (i.e., socioeconomic status of the family).
In this case, it would be perfectly reasonable to assume that B does not cause S and that R does not cause S. Further, we could assume that no outside variables causally affect B, R, or S in such a way as to change the correlations among any of the three considered variables and that any causal connection that do exist among the inside variables are linear.
The resulting possible causal connections can be represented by two simultaneous equations:
B = a + br + cs + d
R = g + hB + is + e
where a, b, c, g, h, and i are causal coefficients which can be determined by applying econometric estimation techniques to the studies data, and d and e are error terms. The error terms might be assumed to be normally distributed around a zero mean.
To use econometric estimation techniques, the system of equations must be “identified” (Google “econometric identification.”), analogous to having the same number of equations and unknowns in algebra. The above equation set is under identified. Therefore, econometric estimation cannot really occur until further assumptions are made and/or prior information (i.e., knowledge) is inserted, resulting in exact identification.
?? 3) Causal Statistics is not for the faint of Heart, nor the weak of Mind
Oh, so you think this sounds difficult, complicated, and renders even valid causal inferences highly suspect? I’m sorry but making valid causal inferences from non-experimental data is not for the faint of heart, nor the weak of mind.
Again, it is not Causal Statistics’ fault. Causal Statistics
[IX. [1)] A Two Variable Example of Causal Inference
Identification pb
Recursive equations
Endogenous
Exogenous, etc Dissertation, Chapter 13
Estimations
Parameter interpretations
Error handling
Structure and structural Delta
Start with general form of Causal Statistics
Recommend econometric sources and note their avoidance of “cause”
Simultaneous equation parameters partial out the considered correlation
Give researchers algorithm for applying Causal Statistics
Consider an empirical study in which variables B (i.e., number of book owned by the family) and R (i.e., reading capability of the child) are found to be correlated. If we assume (1) that no outside variables causally affect both B and R in such a way as to change their correlation, (2) that R does not cause B, and (3) that causal relationship are linear; one can validly conclude, subject to the usual statistical error, that B causes R with standardized strength equal to the correlation coefficient.
A “valid” causal inference is a causal inference which is a logically necessary result of the definitions stated, the assumptions made, and the data collected. In other words, a “valid” causal connection results from a causal inference arrived at by the proper application of Causal Statistics.
Note that a valid causal inference does not necessarily result in a correct causal connection. If one or more of the assumptions is incorrect, the causal connection will be incorrect. The greater the degree to which the assumptions are in error, generally, the greater the error in the causal connection drawn. This second assumption set can be very restrictive and questionable, yet, that is not the fault of Causal Statistics. It is, if you will, the fault of logic and the universe we exist in. Gravity can be inconvenient, if you want to fly, but the field of physics is not at fault. In fact physics can assist in overcoming the problem; same with Causal Statistics.]
{Click to go back to: Table of Contents, I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII, XIII, XIV, XV, XVI, XVII, XVIII}
IX. Causal Statistics: A Three Variable Example
{V. Classical and Bayesian Statistical Analyses of the Example Data (The Mathematics and Interpretation of Classical and Bayesian Association) xx
At the outset of this section, let me be clear that neither classical nor Bayesian statistics can produce valid causal inferences. "Cause" is not defined in either paradigm, like red is not defined in Euclidian geometry.
In this section, I will present the mathematics of associative statistics. The mathematics of causal statistics is not significantly different; it's the assumptions, the vocabulary [definitions, defined terms (like the concept of line or point), concepts and definitions not present in classical or Bayesian statistics], the input information, the understandings, the interpretations, and the results that are different.
Present a lot of mathematics of Association in general, e.g., nonlinear association, nonparametric Association, etc.
[Explain analysis of variance and analysis of covariance]
Along with the mathematics of associative statistics, I will present all of the assumptions associated with these mathematical/statistical forms, i.e., assumption set 3.
This is a chapter which presents the mathematics and interpretations of classical and Bayesian associative statistics. In the beginning of my research, I asked myself many times, "how do I know if a regression parameter or a correlation is causal or associative?"
Eventually, I came to half of the answer in a flash. I wasn't burning my brain to try to answer the question; I simply stumbled on to the question of fresh while dealing with another issue. At that moment half of the answer came to me in a stroke of intuition, but I had to spend some time thinking about it to be confident that my intuition was correct.
The first half of the answer is that a calculated regression parameter is nothing more than a representation of an observed Association. Causal connections cannot be observed; only associations can be observed. Therefore, the initial calculation of a correlation or regression parameter is an indication of observed linear association and indicates nothing about causation.
I arrived at the second half of the answer by a couple of flashes of intuition; no brain burning from just a brain that over time has acquired all the needed information and automatically brings information together to arrive at the correct intuition. It's like people who write music or lyrics or poetry sometimes say. It's like they are not creating the output at all; it's more like being channeled through them and they are writing it down. Sometimes they can't even write it as fast as is trying to slow out of their brain.
Your Anyway, the second half of the answer is that additional information; in the forms of assumptions, prior research results, etc.; must be added to the analytical system to decrease the number of possible causal mechanisms which could generate all or a portion of the observed correlation. When enough information has been added to reduce the possible causal mechanisms to one, the researcher can then infer that causal mechanism is responsible for some or all of the observed correlation, based on the input information and the observed data.
At that point, the correlation or regression parameter becomes at least partially causal. Any portion of the regression parameter which remains associative, might be attributable to some outside variable, x, which causes a portion of the correlation to be spurious.
[When it did come to me it came right out of the blue.]
[It's like opening the top of my head and throwing information in and then letting my brain grind around on it on its own time. Sometime; one day, two days, a few days; later the answer simply pops out.]}
[X. [2)] A Three Variable Example of Causal Inference
Give researchers algorithm for applying Causal Statistics
Explain Hypothesis, testing vs. correlation matrices and the use of matrices to choose hypothesis.
Exemplify the 2 various research and associations, leading to the 3 variable research, leading to 10 variable research, etc. Using prior data. Someone else can figure out how or I’ll do it when I have time
Simultaneous experimental parameters partial out the considered correlation ]
{Click to go back to: Table of Contents, I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII, XIII, XIV, XV, XVI, XVII, XVIII}
X. Causal Inference in Experimental Research
A. Non-Stochastic Causal Inference
in solving a problem, one should always start thinking about its simplest form (e.g., experimental causal inference) and solve that first. Then go on to the more complex problem (e.g. nonexperimental causal inference). In an experiment a correlation could be due to causal forces between the two variables or simply due to random error or Due to God' s intervention or...???
Present the proper algorithm and all assumptions required to draw an experimental causal inference.
1. Validity and Correctness
2. Causal Modeling and Theory Building
X. B. Statistical Causal Inference
1. Validity and Correctness
2. Causal Modeling and Theory Building
A part of the reason for the creping and slower progress in causal inference in the non-experimental sciences is that methodologists start, as did I, by considering causal inference in the non-experimental sciences, a highly complex environment for causal inference.
As a pedagogical device, I backed in the approach of beginning the presentation of causal inference in its simplest form or environment, to fabricate understanding, and then move on to more complex environments.
In retrospect, it is now obvious to me that this would have been the correct approach for my and others’ research into causal inference, i.e., understand and master the simplest form of causal inference first and then move on to the more complex environments. Try to walk, before trying to run. Unfortunately, no non-experimental research methodologist seems to have taken this “obvious” approach. “Obvious” in the sense that all ideas are obvious once someone tells you about it.
In the physical and biological sciences, experimentation is almost always possible. In general, if a variable can be repeatedly manipulated and another variable can be repeatedly observed to change, causal inferences will be intuitively drawn.
Say that a researcher wanted to study the effect of a force on the motion of a bowling ball. He could set up an experiment in which, on a series of runs, either a constant, horizontal force of 10 newtons (the metric unit of force) or no force is applied to a 5 kilogram ball. The motion of the ball is measured on each run and all else is held constant (i.e., controlled).
The experimenter would find that, on each run in which the force is applied, the ball accelerates horizontally at a constant rate of 2 meters/sec/sec so long as the 10 newton force is applied. When no force was applied, the acceleration would be zero.
In working with this data, a theoretician would likely develop the formula F=ma (10=5x2) to model the data from the experiment and then propose the formula as a theory to explain the relationship among the variables; force, mass, and acceleration.
Based on this experiment, the relationship inferred, F=ma, would be far from certain. But, if other, more varied experiments were performed, the equation would be found to hold up and confidence in it as a theory or ultimately a natural law would increase.
Yet, in experiments where the relative velocity between the bowling ball and the observer approaches the speed of light, a curious thing happens. The formula remains true, but the 10 Newton force results in an acceleration less than 2m/sec/sec and the mass of the bowling ball increases by a compensating amount, as phenomena explained by Einstein’s Theory of Relativity. This shows that gaining “knowledge” about the operation of the universe, especially at the extremes, is not as simple as it might first appear, but that my dear, is a story for another day.
[ IX. A.] [III.C.1. Non-Statistical Causal Inference from Experimental Data]
Looked at in one way, what was observed during the experiment was an associated (i.e., correlation) between force and acceleration. It is intuitive and easy for a researcher to make the small leap to the causal inference that the force caused the acceleration.
Speaking of words and “cause,” as noted previously, the logical foundations and development of Classical, Associative, and Bayesian Statistics do not mention, define, or incorporate “cause,” like “red” is not defined or incorporated in Geometry. Hence, any purported causal inference based on any one of the three statistical paradigms alone, would be invalid on its face. Technically, this is true even for their applications to experimental data for purposes of inferring causal connections. This is a surprising conclusion, given the great progress in the experimental sciences (e.g. Physics, Chemistry, and Biology), which has resulted from causal expectation. How can one explain this apparent contradiction?
Consider an experiment in which books were given to some children and not to others, all randomly selected. Suppose that that experiment discovered that; measured three years later, the treatment of group were superior readers. It would be easy and natural for a human brain to conclude that book ownership causes improvements in reading, but this conclusion would be technically invalid.
Cause is not a component of Associative Statistics, so this brain is making an intuitive jump and using Associative Statistics plus something else to draw the causal conclusion.
To make this causal conclusion logically tight (i.e. deductively complete), the valid experiment would have the define “cause”, (2) as that we live in micro causal universe, (3) as that no outside variables are causing both the experiment to manipulate ownership and the treatment group to read better, and (4) as that the correlation between ownership and reading is not a result of either random or measurement error.
Note that we have chosen these assumptions quite judiciously in such a way that, if they are true, there is one and only one explanation for the observed data. That explanation is that bank ownership causes improved reading in children. Therefore, if the definition and assumptions had been explicitly stated, the causal conclusion that that ownership causes improved reading would be valid and if the Associative Statistics are all true, the conclusion should be correct.
For 100’s of years, experiments have made causal inferences from experimental data without defining “cause” and with no reference to the required assumptions. Technically, this is improper Causal Statistics and is not valid. But, looked at it in another way, these required assumptions generally are acceptable to researchers that they don’t bother to state them, considering the assumptions to be implicit and the causal inferences, therefore, valid. This reasoning generally comports with observed behavior in the experimental sciences and, therefore, is probably the best explanation of what actually happened.
It’s a small leak, with generally acceptable and accepted assumptions. Thats why the experimental sciences have been so successful with their causal conclusions.
The non-experimental sciences are dealing with another kettle of fish. Causal inferences, from non-experimental data using Associative Statistics require a huge and almost always incorrect leap of intuition. This is the reason for the relative lack of success in general in causal theory building in particular in the social sciences, epidemiology, etc.
The essence of this difference in causal inference between experimental and non-experimental studies resides in one factor, the vastly different nature of assumption (3), above, between experimental vs. non-experimental research.
In experimentation, assumption (3) simply asserts that nothing cause both the experimenter to manipulate the independent variable (ownership) and the change in the dependent variable (reading scores), resulting in a non-casual correlation between the two variables; an eminently reasonable and acceptable assumption.
But, in non-experimentation, assumption (3) becomes “assume” that, no outside (i.e. unconsidered) variable(s) caused both independent and dependent (reading scores) variable, resulting in spurious (i.e., non causal) correlation between them.
It is difficult to imagine that, the correctness, or leak thereof, are little, or even large, assumptions could account for the large difference in success between the physical and social sciences. Actually, the mobility to draw valid causal inferences in the social sciences doesn’t account for all of the difference in success, but I would say at least 50%.
Of course, non-experimental researchers and research consumers recognize this and attempt to draw causal inferences with the tools they have. Unfortunately, the tools of Classical, Bayesian, and Associative Statistics are so inadequate to this task in non-experimental research, that most such causal conclusions are incorrect and therefore not only useless, but counter productive.
A select, very few such researchers also recognize the intuitive leak they are required to make to get from non-experimental data and Associative Statistics to causal inferences and attempt to fill in the intuitive gap in one way or another. Such researchers are attempting to go beyond the standard statistical paradigms, to a new statistical realm which I call Causal Statistics. A few such researchers and methodologists have moved some distance into that new statistical paradigm, but not many and not far.
Note that researchers who attempt to draw causal conclusions from experimental data using Classical Statistics are making only a small leap of intuition beyond the tenets of the field. This is why the physical (experimental) sciences have not generally encountered significant problems in drawing causal conclusions. On the other hand, the social and other non-experimental sciences have been largely paralyzed by their difficulties in making valid and correct causal inferences.
As an example of how one might move toward Causal Statistics from Classical Statistics, consider the second error, made by those who claim (either explicitly or implicitly and either knowingly or unknowingly) to have discovered a causal connection; i.e. the assumptions, upon which causal conclusions much necessarily be based, are virtually never stated or even known.
In the RIF example, they could have validly concluded that B causes reading scores, R, if they were willing to state the assumptions (1) that R did not cause B and (2) that no other variables caused both R and B. Under these assumptions, a researcher could make the valid, but likely incorrect causal inference that B causes R. if these assumptions are stated, few would be deceived into accepting the conclusion that B causes R because most people would be unwilling to accept one or both of the required assumptions, if even one of the assumptions is incorrect, then the causal conclusions are highly likely to be incorrect, at least in terms of these magnitude.
This approach would be an attempt on the part of a very few sophisticated researchers in the field, to create a needed part of Causal Statistics by backing into it from Associative Statistics. The attempt is not improper, if executed correctly. But correct and complete execution along this path is very difficult, although not impossible. For this and other reasons, this is not the best approach, because (1) based on what I have observed over the last 45 years, not 1 in 1000 researchers can reach a valid and complete causal inference in this manner; (2) the complete Causal Statistics paradigm would virtually never be developed in this way; and (3) no partial formulations would lead to a complete understanding of the nature, validity, accuracy, sensitivity, and weaknesses of the conclusions reached.
In contrast, I have gone through the front door in developing and deriving Causal Statistics in my dissertation, which is presented on the Causal Statistics website, CausalStatistics.org
I began, as Euclid began with Geometry, by using a Meta language to state primitives, definitions, axioms, etc. From there, the whole of Causal Statistics was derived using both verbal and symbolic logic. Such an approach, allows complete understanding of the paradigm to anyone who is willing to put in the effort to follow the steps of its development, i.e., the derivation of Causal Statistics.
For a less rigorous understanding of Causal Statistics, but adequate for a knowledgeable application to non-experimental research and utilization of the conclusions, the following should suffice.
Causal Statistics begins with a common sense, useful definition of “cause,” a non-trivial philosophical issue, given the numerous, inappropriate, and inadequate definitions forwarded by various philosophers and researchers over the past 3000 years.
Additionally, Causal Statistics is based on three sets of assumptions. The first and most fundamental set postulates that we live in a causal universe, at least at the macro level, where cause and effect governs the behavior of all variables.
The second assumption set is for the purpose of isolating the variables considered by the specific research study. As an example of the types of assumptions in Set #2, note the two assumptions presented above, in conjunction with the RIF program. These assumptions were necessary to make the causal inference i.e. , that book ownership causes improved reading; valid. Nevertheless, these assumptions in no way make the causal conclusion correct, if one or more of these assumptions are incorrect. We can be confident that the causal conclusion is correct, only if the assumptions are correct.
The second assumption set need not totally rule out all influences from outside variables on variables inside the system of study. These assumptions need only prohibit outside influences that would exert just certain types of influences. To understand which set two assumptions are required and which are not, one must understand the components of association.
The association between two variables in a study have the major sources; association due to (1) causal connection (e.g., B causes R and/or R causes B), (2) spurious correlation (e.g., S, socioeconomic status, causes both R and B or S causes X, any outside variable, and B and X causes R), and (3) definitional overlay of internal variables (e.g., book ownership and school book ownership), and (4) study error (e.g. measurement bias) statistical error. An observed association will result from one or a combination of all of these sources.
Associations due to causal connection, i.e., source (1), are what researchers are looking for. Spurious correlation, source (2), are what researchers want to avoid and is what assumption set two needs to rule out.
To be a little more precise, some spurious correlations are acceptable and others are not. It is not acceptable for any external variable(s) to cause a spurious correlation within an internal variable pair, if the variable pair is the subject of a causal hypothesis test in the study. Such a spurious correlation must be eliminated by assumption(s) or by bringing such external variable(s) into the study to make the causal conclusion valid. Yet, if the assumption is erroneous, the causal conclusion will be similarly erroneous and the problem external variable(s) must be brought into the study. External sources of spurious correlation within any non-hypothesis variable pair are not a problem to the study and need not be eliminated.
Definitional overlap, source (3), is simply an error in research design and should be avoided. Study error, source (4), is no more likely in causal studies than any other experimental research and should be handled with the usual techniques.
If the set two assumptions are properly designed and all else in the research is handled correctly, the causal inferences drawn should be “valid.” But if these assumptions are not satisfied (i.e., false), the causal conclusion may be incorrect or at least the accuracy of the causal inferences will be diminished. The extent of diminution is usually related to the degree of error in the isolating assumptions.
The first two assumption sets are required only by Causal Statistics and not by other types of Statistics, but there is a third potential assumption set required in all paradigms. This third set assumes no measurement error, no sampling bias, etc. Since the third assumption set impacts all forms of Statistics equally and our primary purpose here is to highlight the differences, we will simply consider the third set to be satisfied and not deal with these assumptions further.
In the standard statistical paradigms, with the third set of assumptions satisfied, a positive result supports a non-causal hypothesis, e.g., an association, a mean value, etc., about the population, subject only to the error inherent in random sampling (i.e., statistical error). The probability of statistical error can be calculated and specified precisely. Such standard statistical results are very close to simply reporting on what was observed, i.e., only slightly more inferential than reporting on what was observed.
In Causal Statistics, one or more causal hypothesizes are proposed; a study is designed; data are collected; and the causal hypothesizes are either confirmed or not confirmed; subject (1) to all three assumption sets (although we have, for purposes of this discussion, stipulated that the third assumption set is satisfied.), (2) to a given definition of “cause,” (3) to calculated probabilities of statistical error, and (4) to the requirement that all assumptions be explicitly stated.
Note that causal conclusions are not directly observable and therefore Causal Statistics requires more input to obtain its results (i.e., the causal inferences). The standard statistical paradigm require only data and no additional input, except assumption set number three (which was already considered satisfied), to obtain its conclusions, i.e., inferences to population associations. The additional [burden inherent] inputs required by Causal Statistics are a useful definition of “cause” and two additional sets of assumptions.
IX.A.1. [ III.C.1.a. Validity and Correctness]
See[ [IV.B]]
“Intuitive and easy,” yes, but valid? No and yes, in that order. What about “correct?” For such an experiment, very likely.
Technically, the causal inference that force causes accelerations draws from the experiment, is not valid, because one cannot logically/deductively arrive at the causal inference using only the data from the experiment, as the researcher seems to have done. The additional inputs needed are an appropriate definition of “cause“ (see …) and a set of assumptions about the causal nature of the universe and the conditions extant during the experiment.
Specifically, the required assumptions for the experiment are as follows:
(1) We live in a causal universe (This means that the behavior of our universe is not simply an innumerable series of random events that only appear to be causal and that no greater power is intervening and making the behavior appear causal.)
(2) The operation (i.e., causal and any other “laws”) of the universe does not change over time,
(3) No variable caused both the experimenter to exert the force and the ball to, independently, accelerate,
(4) None of the correlation between force and acceleration is the result of random fluctuation in one or both of these variables. {Make this assumption or accept 99.5 confidence in the causal connection or causal inference.}
(5) the observed accelerations are not due to measurement error.
What a result! We did not use intuition to leap across a chasm of unknowns, to the conclusion that force causes acceleration. We used stated assumptions to build a plank bridge across the gorge and walked across to deductive causal inference in the way that Euclid started with definitions and assumptions and deductively concluded (i.e., prove, derived) that all Triangles contain 180 degrees. If the assumptions are true, the conclusion is true. [It must be noted that these assumptions (planks) are always suspect and, if even one assumption is not true, there is a real possibility of falling to the bottom of the gorge.]
Note that, in the non-experimental sciences the situation with regard to drawing of valid causal inferences is similar in form, but vastly different in confidence. The required assumptions are similar, but the analogue to as. (3) is for more suspect [assumptions (4) and (5) are less true and control is less].
For 100’s of years, experimenters have made causal inferences from experimental data with no reference to the required assumptions. Technically, these are improper and are not valid causal inferences. But, looked at it in another way, these required assumptions so are generally acceptable to researchers that they don’t bother to state them, considering the assumptions implicit and the causal inferences, therefore, valid. This reasoning generally comports with observed behavior in the experimental sciences and, therefore, is probably the best explanation of what actually happened.
Now, what about the question as to weather or not the inferred causal connection is “correct?” If one is 99% confident (Bayesian probability) that the assumptions are correct, then he can be at least 99% confident that the causal inference is correct. I say “at least” because there is a possibility that an assumption may be false and yet the conclusion (causal connection) be true or correct. For the person in the about example, the causal inference is valid, but he is not certain that the causal assertion is correct, but almost, i.e., greater than or equal to 99%.
IXA2 [ III.C.1.b. Causal Modeling/Theory Building ]
A causal theoretician, looking at these experimental results, would likely conclude that force causes acceleration in an amount equal to F/m or according to the equation, a=F/m.
IXB [III.C.2. Statistical Causal inference]
In the above experiment, there is no need for statistical analysis because there is no significant random component or random error in the experiment.
The only relevant variables which vary, from run to run, are the force and the acceleration, which are the variables of interest in the study. All other relevant variables (e.g., mass of the bowling ball, wind speed, direction of the force) are controlled to be unchanged from run to run. Also, there was no significant measurement error.
In experiments with random error, the greater the stochastic component relative to the size of the effect, the greater the need for statistical analysis. Consider the above experiment, but designed by a researcher who failed to control some of the relevant variables, like mass.
Say that, due to lack of knowledge of the subject, the researcher used a different bowling ball, randomly selected, on each run and hence did not control for the different masses. The force would lead to various measured accelerations, rather than the identical accelerations observed in the first experimental design.
If on a given run the mass was 4 kg, the measure acceleration would be 2.5 m/sec (a= F/m = 10/4= 2.5 m/sec). In the original experiment, the mass was 5 kg and the measured acceleration was 2 m/sec. Hence, the new, measured acceleration contains and error term equal to .5 m/sec (a=2.5= 2+e, e=.5 m/sec.)
Say that the mass, for the next run, is 6 kg. The measured acceleration would be 1.67 m/sec (a=F/m =10/6 = 1.67), which would contain an error term equal to -.33 m/sec (a= 1.67= 2+e, e= -.33).
Note that the distribution of masses, from which the balls are randomly selected, could be used to calculate the distribution of the error term.
As an aside, there is pedagogical advantage to this procedure. Advanced Statistics classes are taught starting from the error terms, without looking to the origin of the error terms. Starting form the source of the error and then determining what the error should look like, is a good way to teach someone how to go the opposite the direction, i.e., error distribution to source.
Note that, in an experiment where the relative magnitude of the stochastic processes are large, there is some likelihood that even though the sample correlation is positive, the true correlation might be zero. This would require that a hypothesis test be done to determine if the experimental data yields a sample correlation between force and acceleration significantly different from zero.
If a Bayesian hypothesis test results in a probability of 95% that the true correlation is greater than zero, the researcher could validly conclude that force is a positive cause of acceleration with 95% confidence, based on the previously stated assumptions.
“Valid” does not mean “correct” or “certain.” It means the process (e.g., for making causal inferences) has been followed properly. A Classical Statistical inference, from a random sample of people, that the mean population age in the U.S. is greater than 35, may or may not be correct, but, if all statistical procedures were followed, the inference is “valid.”
If the research consumer believed with 90% confidence that the assumptions were true, his confidence that this causal connection is correct would be .95 x .90 = .855 or 85.5%
The apparently random error would increase, if the number of uncontrolled items increased, e.g., the experimenter did not control the direction of the force. In the first effort, where the force was horizontal, gravity did not affect the measure acceleration. But, in an experiment in which direction is uncontrolled and also not controlled for analytically, more stochastic error is introduced. [If the choice of directive on each run is randomly selected the error in measured acceleration will also be random. But, if the direction is selected systematically, the error term will also be systematic. For example, if the direction of force is always more or less down, ranging form 3:00 to 9:00, the effect of gravity will always be greater than or equal to zero. Therefore, the error in measured acceleration due to gravity will always be greater or equal to zero. This will lead to a positive bias in the measured acceleration, with the error distributed around a positive number, rather then zero.
IXB1 [? III. C. 2. a. Validity and Correctness]
IXB2 [III. C. 2. b. Modeling/Theory Building ]
If the researchers or theoretician wished to calculate the strength of the causal connection, i.e., the amount of change in acceleration caused by a unit change in force, he/she could use Classical Statistics in the form of regressive analysis: a= c_{1} + c_{2 }(F/m) + e, where c_{1} and c_{2} are regression parameters, estimated form the experimental data.
If all of the error assumptions of regression analysis are satisfied, c_{1}, should be close to zero and c_{2 }should be close to one, yielding the following equation.:
a @ F/m +e
which is consistent with the equation developed from the zero error experiment, namely a = F/m.
IXC [III.C.3. Conclusions ]
Causal inference from experiments, even highly stochastic experiments, are much more difficult and much more complicated. These difficulties have lead to philosophical, definitional, and validity concerns that have permeated work with and the understanding of causality in non-experimental research.
The essence of this difference in causal inference between experimental and non-experimental studies, is (1) the difference in assumption (3), above, and (2) the usually diminished ability of the researcher to control (e.g., hold constant) other variables.
In experimentation, assumption (3) simply asserts that nothing causes both the experimenter to manipulate the independent variable (force) and the change in the dependant variable (acceleration), resulting in a non-causal correlation between the two variables; an eminently reasonable and acceptable assumption.
But, in non-experimentation, assumption (3) becomes, “assume that, during the study, no other variable or variables caused both independent and dependent variables, resulting in a spurious (i.e., non-causal) correlation between them;” a vastly more suspect assumption.
In the first experiment, above, the research was perfectly controlled. The only two relevant variables that changed were force and acceleration. In non-experimental research….
In fact, causal inferences are generally so easy to draw from experiments that, most of the time, no or only minor thought is required and, in stochastic experimental, only rudimentary statistical analysis is required. In such cases, Classical Statistics has generally been use to assist in making these causal inferences from experimental data, but Classical Statistics alone would not be sufficient for drawings valid causal inferences from even experimental data because “causality” is not defined in or considered by Classical Statistics.
Therefore, something additional is necessary. It turns out that that that extra something is human’s natural, innate intuition for making causal inferences. It is so natural for us that we don’t even realize that we are making a leap of intuition to causality, something, as Hume said, that we cannot sense directly, but can only infer.
But, such intuition cannot be a logical basis for valid causal inference. Hence, as was shown earlier, valid causal inferences from experimental data must additionally be based on a generally acceptable set of assumptions.
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XI. Causal Inference in Non-experimental Research
In non-experimental research, everything changes. Causal inference is extremely difficult, leading to philosophical, definitional, and validity concerns that permeate work with and the understanding of causality in non-experimental research.
In the Social Sciences and in Epidemiology, experimentation is seldom feasible. In these and other non-experimental sciences, the situation with regarding to drawing causal inferences is vastly different. It is much more difficult and much more complicated. These difficulties lead to philosophical, definitional, and validity concerns that permeate work with and the understanding of causality in non-experimental research.
The essence of this difference in causal inference between experimental and non-experimental data is (1) the difference in assumption (3), above, and (2) the usually diminished ability of the researcher to control (e.g., hold constant) other variables.
In experimentation, assumption (3) simply asserts that nothing causes both the experimenter to manipulate the independent variable (force) and the change in the dependant variable (acceleration), resulting in a non-causal correlation between the two variables; and eminently reasonable and acceptable assumptions.
But, in non-experimentation, assumption (3) becomes, “assume that, during the study, no other variable or variables caused both independent and dependent variables, resulting in a spurious (i.e., non-causal) correlation between them;” a vastly more suspect assumption.
In the first experiment, above, the research was perfectly controlled. The only two relevant variables that changed were force and acceleration. In non-experimental research….
No valid causal conclusion can be drawn from this study by the use of Classical Statistics because the only way to make valid causal inferences from this non-experimental study is to use Causal Statistics, either explicitly or implicitly.
A management researcher, unsatisfied with the problems inherit in non-experimental studies, could avail himself of the option to perform an experimental study, likely in a lab and using students. Such a study would be experimental and allow for valid causal inferences, but the artificial, over simplified environment would greatly limit the generalizability(sp?) of any causal findings. This is often the dilemma in the non-experimental sciences, (1) to experiment in an artificial environment and obtain causal inferences of questionable generalizability or (2) to carry out a non-experimental study in out carry a rich environment and relinquish the possibility of reaching valid causal conclusions.
Enter Causal Statistics. The proper use of Causal Statistics can distinctly diminish these problems for non-experimental researchers. I developed Causal Statistics 40 years ago, spend 10 years, when I wasn't teaching, trying to push the idea and to get funding, without notable success, and eventually went on to other things.
Now, I’m back, with a fresh look and new insights and approaches. Sad to say, 40 years ago, I thought people, even highly educated people, were a whole lot smarter than they turned out to be. Also, I now know that 99% of academics, funders, etc. look at a new, eclectic, and revolutionary idea to find what might be wrong with it and not how important it would be, if successful, and not how any apparent problems could be overcome to make it successful.
I now know, even highly educated, smart people have to be lead by the hand through the whole continent of a huge, revolutionary idea and through solutions to all the potential problems.
Consider the 50 year fight to “prove” that tobacco use causes lung cancer. If a valid causal inquiring system had been used, rather than Associative Statistics, it shouldn’t have taken more than 10 years, even with the disinformation campaigns mounted by tobacco companies. How many lives would have been saved? The ignorance of and/or the act of ignoring Causal Statistics are not a meaningless condition, equivalent to the argument of how many angles can dance on the of a pen, but a scientific/methodological gaf of high gravity, with great human and financial costs.
A paragon of the problems flowing from the application of Associative Statistics, and the non-use of Causal Statistics to analyze non-experimental data, is a program called Reading is Fundamental (RIF) which has been in existence for 41 years and is “the nations largest children’s literacy organization.” The original research found a correlation between children’s reading performance (R) and the number of he/she owned (B).
RIF was formed to increase book ownership so Johnny would read better, based on the invalid inference, not explicitly stated, that B causes R.
How did the founder of RIF know that correlation between B and R wasn’t due to the fact that R caused B? Looking a little further a field, how did they know that socioeconomic level of the child (S) didn’t cause both B and R, leading to a spurious (i.e., non-causal) correlation between B and R?
40 years, 300 million books distributed, and millions of people duped by bad research, bad scientists, bad research consumers, and bad statistical inference.
RIF exemplifies two errors in causal inference that are common in non-experimental research. The first error is that causal inferences are often drawn, both explicitly and implicitly, by scientists who almost as often deny, cover up, and/or are unaware that, in their results, causal connections between variables are either stated or implied.
The RIF error was accomplished by (1) not specifically stating that B causes R, but saying that kids who own books are better readers, misleading the majority of human minds to jump to their own invalid causal inferences, and (2) then proceeding to develop a program and establish an organization, as if it had been proved that B causes R.
Other researchers hide the fact that they are making causal inferences (typically in an invalid manor) by not using the word “cause,” but by using synonyms like “yields,” “results in,” “produces,” “brings forth,” “brings out,” “creates,” “effectuates,” “elicits,” “is due to,” “generates,” “induces,” “leads to,” “makes,” and more.
The second error made, by those who claim (either explicitly or implicitly and either knowingly or unknowingly) to have discovered a causal connection, is that the assumptions, upon which the causal conclusions are based, are virtually never stated or even known. In the RIF example, they could validly conclude that B causes R if they were willing to state the assumptions (1) that R did not cause B and (2) that no other variables caused both R and B. Under these assumptions, a researcher could make the valid causal inference that B causes P, but few would be deceived into accepting the conclusion because most people would be unwilling to accept one or both of the required assumptions and, of course, if even one of the assumptions is invalid, then the conclusions are invalid.
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XII. A Ten Variable Example of Causal Inferences (A Ten Variable Follow-on Example of Causal Inference)
In the above example, one way to decrease the stringency of the required assumptions, would be to bring one or more of the relevant outside variables, inside. For example, collect data on B, R, and S (i.e., socioeconomic status of the family).
In this case, it would be perfectly reasonable to assume that B does not cause S and that R does not cause S. Further, we could assume that no outside variables causally affect B, R, or S in such a way as to change the correlations among any of the three considered variables and that any causal connection that do exist among the inside variables are linear.
The resulting possible causal connections can be represented by two simultaneous equations:
B = a + br + cs + d
R = g + hB + is + e
where a, b, c, g, h, and i are causal coefficients which can be determined by applying econometric estimation techniques to the studies data, and d and e are error terms. The error terms might be assumed to be normally distributed around a zero mean.
To use econometric estimation techniques, the system of equations must be “identified” (Google “econometric identification.”), analogous to having the same number of equations and unknowns in algebra. The above equation set is under identified. Therefore, econometric estimation cannot really occur until further assumptions are made and/or prior information (i.e., knowledge) is inserted, resulting in exact identification.
?? 3) Causal Statistics is not for the faint of Heart, nor the weak of Mind
Oh, so you think this sounds difficult, complicated, and renders even valid causal inferences highly suspect? I’m sorry but making valid causal inferences from non-experimental data is not for the faint of heart, nor the weak of mind.
Again, it is not Causal Statistics’ fault. Causal Statistics is an optimized system, doing the best it can with the logical and physical universe it was given. Maybe some parallel universe would make Causal Statistics as simple to apply as Classical Statistics, but getting to that universe would likely be even more difficult than properly applying Causal Statistics.
I dare say, you have probably never before seen non-experimental causal inferences drawn in this way. What do you think this means? I would suggest that you have probably never before seen valid causal inferences drawn.
This is not to say that no correct, non-experimental causal connections have been established in the past 100 years. But, I would say that the number that are as accurate as would have been possible with the proper application of Causal Statistics, is far less than 1% and the number of non-experimental causal inferences that are correct to any level of accuracy is probably less than 10%.
Further, our efficiency in determining causal connections has been dismal. Consider the 50 year debate over the causal connection between smoking and lung cancer.
Now, it should be clear why I have complained of the vast financial and intellectual waste in the non-experimental sciences over the past 100 years.
As you may or may not remember, we were discussing the second assumption set. Now, on to the third.
?? C. Statistics
Now, back to the three disparate fields upon which Causal Statistics is based. Thirdly, the final formulation of Causal Statistics is statistical, in that the results are subject
to calculatedly [ ] sampling error, just like the results from Classical and Bayesian Statistical studies.}
(Add parts of WP3)
?? 3) Causal Statistics is not for the faint of Heart, nor the weak of Mind
?? c. The Third Assumption Set
?? C. Statistics
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XIII. How does Causal Statistics Fit into the field of Statistics, into the Social Sciences, and into Epistemology
A. The Relationship among the Three Statistical Paradigms
A. How does Causal Statistics fit into the Pantheon (Mosaic) of other Statistical Paradigms?
B. [A.] Understanding the Nature of Mathematical and scientific “Knowledge.”
?? B. Where Does the Dissertation Fit in?
A. [C.] The Logical Constructs within Statistics and their Relations to each other
XII. [V.] A. The Relationship among the Three Statistical Paradigms
Classical Statistics, Bayesian Statistics, and Causal Statistics; like virtually every major mathematical discipline; are a/d constructs. These three statistical paradigms are similar to, but different from, each other; akin to the way that Euclidian and the various non-Euclidian Geometry’s are similar to, but distinct from, each other.
Classical Statistics is concerned with making inferences from sample statistics to population statistics; like means, correlations, and standard deviations. Causality is not considered by Classical Statistics; nor is “cause” or any of its synonyms within the vocabulary of Classical Statistics. Therefore, any consideration of causal inference is outside the domain (i.e., the area of applicability) of Classical Statistics. Researchers, who attempt to use Classical Statistics alone to draw valid causal inferences from non-experimental data, as many have done, are bound to fail.
Causality and causal inference is at the heart of the domain of Causal Statistics and it is the only complete inquiring system designed specifically for making causal inferences.
At this point the question often arises, “Which one of the three is correct?” The answer is that each is correct or useful with in its own domain, like an individual’s depression is an issue for psychology, but not for sociology or economics. Causal Statistics is absorbed with making valid causal inferences and Classical Statistics is totally unconcerned with causality in any way.
Classical Statistics and Causal Statistics are like Physics and Chemistry or like Sociology and Psychology. The pairs are not in conflict with each other; they deal mostly with different issues; and each is applicable and useful within its own domain. Classical Statistics and Causal Statistics are not different pieces of the same thing. They are different things; different definitions, different assumptions, different concepts, different interpretations, different domains; with some similarities in mathematical form.
So, it’s not a question of which one is right or which one is wrong. Both Classical Statistics and Causal Statistics are right and correct and usefully and usable in their own domains. It’s when scientists use one in a domain it’s not suited for, that the validity of their research suffers, like using Classical Statistics on non-experimental data, in an attempt to draw causal inferences.
On the other hand, it would not be improper to use both statistical paradigms together, if each addresses itself only to elements within its own domain.
For example, a researcher might use Causal Statistics to extract a causal coefficient from non-experimental data and utilize Classical Statistics to infer from the sample causal coefficient to the population causal coefficient: Alternatively, one could use Bayesian Statistics rather than Classical Statistics and calculate a Bayesian confidence interval for the causal coefficient.
Aside: Ultimately, these are simply two different paradigms, based on different definitions of probability. So who is right? The answer is that neither of these logical constructs is absolutely right or wrong. The appropriate question is, “Which one is most useful and in what situations, when exposed to empirical applications (i.e., reality)?” Both Euclidian and some Non-Euclidian Geometries work well in our everyday experience and with Newtonian Mechanics, but Euclidian Geometry breaks down when applied to Einstein’s Relativistic universe. There, it turns out, that one of the Non-Euclidian Geometries works best for both. It turned out that Euclidian Geometry is just a very good approximation in our everyday experience, but ultimately wrong.
Carrying things a little further, even Relativity breaks down when applied to particles at the atomic scale and the quantum mechanics paradigm takes over. Yet, neither can explain the behavior of matter at the center of a black hole, a place somewhat removed from everyday experience.
?? Aside: Are the assumptions true?
Are the def’s true? True is not relevant, useful is better.
?? Aside: Plane Euclidean Geometry. 3D by solid Geometry. Extension because no conflict. Thought to explain the world Non-Euclidean Geometry.
?? Aside: Causal Statistics could use Classical Statistics or Bayesian Statistics for handling the statistical element of Causal Statistic, like hypothesis Test, Type I and II error, confidence intervals, etc. I will use Classical Statistics for that because it is much more familiar, although a little less meaningful.
WP3
XII. A. [VII.] How does Causal Statistics fit into the Pantheon (Mosaic) of other Statistical Paradigms?
To understand the importance of the fact that Causal Statistics and Classical Statistics are different axiomatic/deductive constructs, let us step back for a moment and look more broadly at the fields within the realm of statistics. For a discussion of how Causal Statistics and Associative Statistics are related and how the aforementioned causal inquiring system is founded on these three, apparently dispirit, disciplines, see Working Papers #3 and #4, below.
XII. B. [A.] Understanding the Nature of Mathematical and scientific “Knowledge.”
?? B. Where Does the Dissertation Fit in?
In science and math, the big, encompassing theories and mostly logical, deduct paradigms; like Causal Statistics in my dissertation, like Euclid derived Geometry, like Einstein derived the Theory of Relativity, like Whitehead and Russel derived Arithmetic and Algebra in Principia Mathematica, and others to derive Classical and Bayesian Statistics.
The interested reader might ask, “So what?” and the uninterested reader has probably already moved on to taking out the trash or to a porn site. Anyway, for you that me left, back to, “So what?”
Consider Einstein’s contribution with his derivation of Relativity. Much of the mathematical formulation of Special Relativity, the Lorentz Equations, were already known from experimentation. But nobody understood why these equations were the way they were; i.e., why, for fast moving objects, their length shortened; their mass increased; and time slowed down. So the equations couldn’t be applied with understanding. Einstein, rather judiciously, formulated a set of assumptions (e.g., the measured velocity of light is constant, no matter what the velocity of the observer) about the nature of the universe and derived the Lorentz Equations and then went beyond the empirically known, to derive E=mc².
I did a similar thing in deriving a causal inquiring paradigm, which I call Causal Statistics. The application of the Causal Statistics paradigm to non-experimental data can now enable researcher to draw valid causal inferences with complete understanding of the basic definitions utilized, the assumptions made, the use of prior information, and the proper interpretation of causal coefficients. See Working Papers #___ for a simple, but complete, application oriented formulation of Causal Statistics.
XII. A. [C.] The Logical Constructs within Statistics and their Relations to each other
Some will still argue that Causal Statistics conflicts with Classical Statistics and, hence, Causal Statistics must be wrong or that Causal Statistics is an invalid use of Statistics. My response is that Causal Statistics deals with causality and Classical Statistics does not. So they are not in conflict for the same reason that chemistry is not in conflict with nuclear physics and sociology is not in conflict with psychology. They are different paradigms dealing with different aspects of the universe. Classical Statistics and Causal Statistics are not different pieces of the same thing. They are different things (different definitions, different assumptions, different interpretations, different domains) with similarity in mathematical form.
Concerning the certainty that people will call Causal Statistics an invalid statistical technique: it is a logical construct, like Relativity or Classical Statistics are logical constructs. Therefore, it is not per se valid, because it is internally consistent. Yet, the real proof of the pudding for any logical construct, like Geometry is in the eating. When applied to the real world, are the results useful? I believe that Causal Statistics will eventually be proved so useful that it will revolutionize non-experimental research.]??
?? Aside: Objective: to give all people all understandings necessary to understand context, nature, etc of Causal Statistics and to apply Causal Statistics.
??[IV.]
?? VIII. How does Dissertation, fit into the field of Statistics
Causal Statistics is axiomatic/deductive constructs like virtually all other math. Dissertation derives Causal Statistics like contribution of Einstein.] ??
(No??) [IX. The Importance of Causal Statistics]
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XIV. The Design of Non-experimental Causal Studies and Causal Study Sequences
XV. History of Causal Philosophy and Causal Inference, as Seen from the High Ground
A. Conditionals, Counterfactuals, etc.
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XVI. My Post Dissertation Attempts to do Causal Statistics and Obtain Funding
XVII. Why is it Taking so Long? And how much longer will it take?
Dissertation in Library of Congress and University of California Berkeley Graduate Library
Best year of my life. I learned so much out of my own head, I could almost become a rationalist.
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XVIII. Where do we go from Here?
Dissertation, Chapters 14, 1, or 2
A. Get researchers using Causal Statistics
1. Develop a simple presentation of the Causal Statistics paradigm.
2. Answer all objections.
3. Publicize.
4. Free application consulting.
5. Book? With basics and examples
6. Funding: only after it’s no longer innovated and is acceptable. When it’s too late, there will then be adequate money.
Old Table of Contents
N I. In a Nutshell, What is Causal Statistics and How Important Is It?
N II. Origin, Background, Setting, and Context of Causal Statistics
N III. The Needs for Non-Causal and Causal Inference
N IV. Empirical Research, Associative and Causal (An Example)
A. Nonexperimental Research, Associative and Causal (An Example)
B. Experimental Research, Associative and Causal (An Example)
N IV.1. Classical and Bayesian Statistical Analyses of the Example Data (The Mathematics and Interpretation of Classical and Bayesian Association)
Present a lot of mathematics of Association in general, e.g., nonlinear association, nonparametric Association, etc.
[Explain analysis of variance and analysis of covariance]
This is a chapter which presents the mathematics and interpretations of classical and Bayesian associative statistics. In the beginning of my research, I asked myself many times, "how do I know if a regression parameter or a correlation is causal or associative?"
Eventually, I came to half of the answer in a flash. I wasn't burning my brain to try to answer the question; I simply stumbled on to the question of fresh while dealing with another issue. At that moment half of the answer came to me in a stroke of intuition, but I had to spend some time thinking about it to be confident that my intuition was correct.
The first half of the answer is that a calculated regression parameter is nothing more than a representation of an observed Association. Causal connections cannot be observed; only associations can be observed. Therefore, the initial calculation of a correlation or regression parameter is an indication of observed linear association and indicates nothing about causation.
I arrived at the second half of the answer by a couple of flashes of intuition; no brain burning from just a brain that over time has acquired all the needed information and automatically brings information together to arrive at the correct intuition. It's like people who write music or lyrics or poetry sometimes say. It's like they are not creating the output at all; it's more like being channeled through them and they are writing it down. Sometimes they can't even write it as fast as is trying to slow out of their brain.
Anyway, the second half of the answer is that additional information; in the forms of assumptions, prior research results, etc.; must be added to the analytical system to decrease the number of possible causal mechanisms which could generate all or a portion of the observed correlation. When enough information has been added to reduce the possible causal mechanisms to one, the researcher can then infer that causal mechanism is responsible for some or all of the observed correlation, based on the input information and the observed data.
At that point, the correlation or regression parameter becomes at least partially causal. Any portion of the regression parameter which remains associative, might be attributable to some outside variable, x, which causes a portion of the correlation to be spurious.
[When it did come to me it came right out of the blue.]
[It's like opening the top of my head and throwing information in and then letting my brain grind around on it on its own time. Sometime; one day, two days, a few days; later the answer simply pops out.]
N V. Foundations and Derivation of Causal Statistics
A. Epistemology
1. The Empiricists
2. The Rationalists
3. The Eclectics
4. Observability
5. Un-observability (Metaphysics)
B. Philosophy of Causality and Definition of Cause
C. Physics and Metaphysics
2.Explain Quantum Mechanics
Does a single electron, going through two grates of a diffraction grating, break into quanta or as the breakup continuous?
If the universe is stochastic, causal " laws" still operate between fundamental particles and their stochastic causal mechanisms determine the apparently random behavior of larger (composite) particles, e.g., electrons or protons.
1. Fundamental Particles
3. Etc.
D.. Deductive Logic
The beginning: definitions, postulates, axioms, etc.
Analogous to Euclid, Einstein, Whitehead and Russell, Probability Theory, etc.
E. Research Methodology
F. The Derivation of Causal Statistics
Micro Cause, micro variable, discrete causes (quantum mechanics), macro variable, macro causal chain, error term, general form of causal statistics
G. The Assumptions and the General Model of Causal Statistics
1. The First Assumption Set
2. The Second Assumption Set
3. The Third Assumption Set
H. Mathematical statistics (The General Form of Causal Statistics)
I. Inductive Logic in Causal Statistics
N VI. Causal Statistics: A Two Variable Example
N VII. Causal Statistics: a Three Variable Example
N VIII. Causal Inference in Experimental Research
A. Non-Stochastic Causal Inference
1. Validity and Correctness
2. Causal Modeling and Theory Building
B. Statistical Causal Inference
1. Validity and Correctness
2. Causal Modeling and Theory Building
N VIII. Causal Inference in Nonexperimental Research
N IX. Ten Variable Example of Causal Inferences (A Ten Variable Follow-on Example of Causal Inference)
?? 3) Causal Statistics is not for the faint of Heart, nor the weak of Mind
?? c. The Third Assumption Set
?? C. Statistics
N X. How does Causal Statistics Fit into the field of Statistics (the pantheon of statistical paradigms), into the Social Sciences, and into Epistemology
A. The Relationship among the Three Statistical Paradigms
A. How does Causal Statistics fit into the Pantheon (Mosaic) of other Statistical Paradigms?
B. [A.] Understanding the Nature of Mathematical and scientific “Knowledge.”
?? B. Where Does the Dissertation Fit in?
A. [C.] The Logical Constructs within Statistics and their Relations to each other
N XI. The Design of Nonexperimental Causal Studies and Causal Study Sequences
[c13]XIII. Why is it Taking so Long? And how much longer will it take?
[c14]XIV. Where do we go from Here?
XIV. A. Get researchers using Causal Statistics
1. Develop a simple presentation of the Causal Statistics paradigm.
2. Answer all objections.
3. Publicize.
4. Free application consulting.
5. Book? With basics and examples
6. Funding: only after it’s no longer innovated and is acceptable. When it’s too late, there will then be adequate money.
An Exposition toward the Ultimate Goal of this Web site, i.e., the Extraction, Formulation, Explanation, and Exemplification of Causal Statistics
Rough Draft
A critical introduction to the methods used to collect data in social science: surveys, archival research, experiments, and participant observation. Evaluates "facts and findings" by understanding the strengths and weaknesses of the methods that produce them. Case based.
Make statistical def of cause
Simi exp’al, quasi exp’al, etc
Who should be able to understand this book
CAUSAL INFERENCE VIA CAUSAL STATISTICS
The remainder of this web
page is dedicated to accomplishing the goal of making a sea change in the
way non-experimental scientists conduct their research. Specifically,
the goal is
it is desired
that social scientists, epidemiologists, and other non-experimental
researchers--when appropriate--utilize Causal Statistics in the design,
conduct, analysis, and reporting of their empirical research. The more
precise purpose of this Exposé is to tackel Objective 4, above, i.e., to
present Causal Statistics in an easily
accessableaccessible
(intellectually) way,
(even
multiple ways).
This article
draws
takes the
extractions from the dissertation and presents a minimal,--
yet sufficient,--
formulation of Causal Statistics, along with examples of its
application in non-experimental research. Additionally, the presentation is
interspersed with background information (necessary for understanding),
answers to potential objections, explanations of how the various components
being presented relate to each other, and delineations of how these
components fit into their philosophical, logical, and statistical
environments and into the overall mosaic of human knowledge.
This presentation will proceed via the following sections:
0.1 In a Nutshell, What Is Causal Statistics and How Important Is It?
“Causal Statistics is a mathematical inquiring system which enables empirical researchers to draw causal inferences from non-experimental data, based upon the minimum required assumptions, explicitly stated.” 1968
"Causal Statistics is the only causal inquiring system which is a deductive mathematical construct, in the sense that Euclidian geometry is an axiomatic, deductive, logical construct. It's derivation is founded in causal philosophy, physics, epistemology, symbolic logic, statistics, and non-experimental research methodology." 1976
“The development and utilization of Causal Statistics will eventually be as important to the non-experimental sciences as the codification and utilization of the scientific method was to the physical (i.e., experimental) sciences." 1969
“100 years from now, research results and theories in the non-experimental sciences will consist mostly of large arrays of variables, connected by multi-equation causal models, inferred from a single large or a compounded succession of smaller applications of Causal Statistics in empirical research studies.” 2007
I. Origin, Background, Setting, and Context
In 1967, I was at the University of California (Berkeley), finishing an M.S. in Nuclear Engineering and beginning a Ph.D. in Business Administration. The dissertation research plan was to carry out a non-experimental management study, in an attempt to determine what managerial behaviors caused increased productivity of scientists and engineers engaged in physical sciences research and development.
. [causal inferences were difficult to impossible to draw from non-experimental studies.] [ I had expected a completely understandable, logically tight causal inquiring system to be one of the ]
As work on the dissertation progressed, I was amazed to discover that there was no well-defined technique which social scientists used to make causal inferences. My sojourn in the physical sciences led me to think that such an important and needed component of research methodology would be a standard arrow in the quiver of all social science researchers and, if it wasn't, the whole field of social science would be moving heaven and earth to discover and/or develop such an arrow. [As a recent convert from the physical sciences,] [as a recent refugee from the physical sciences,] [What seemed even more amazing to me was that no one seemed to be working on the development of an understandable, logically tight causal inquiring system.]
Well, there was no such standard arrow. As I would discover later, there were ongoing efforts on the parts of some nonexperimental research methodologists to develop causal inquiring systems, but the frenzied, almost frantic, intensity which I expected was not observable. Most researchers and research consumers seemed to accept the limitations on drawing causal inferences as a given, as part of the nature of the field. Many even felt that any discussion of causality was somehow inappropriate and even unscientific. Even the social scientists and others who wished to have a complete, understandable, and well-founded causal inquiring system generally felt that such a thing was impossible. [There any significant pressure or effort to develop one. So I set about finding out what work had been done in causal inference. ]
After some time I found that the available causal inquiring systems broke down into three, more or less distinct, mathematical forms, all forms of classical statistics. All three were attempts to stretch classical statistics in a way that causal inferences could be produced. One of the problems with these attempts was that they were basically intuitive and not based on logical derivation or deduction, resulting in an understandable, logically tight causal inquiring system.
"Cause" is not a term defined in classical statistics; therefore, causal inferences cannot be established from classical statistics. Each of the three forms of causal inquiring system utilized a slightly different mathematical form from classical statistics. Each then gave an intuitive, handwaving argument to arrive at its causal inquiring system, utilizing the initial mathematical form. No wonder the systems, assumptions, definitions, etc. were not completely understandable; they were never actually stated nor was any continuous logical argument to derived them from classical statistics ever made.
So what you had was the domain of classical statistics and then three, somewhat overlapping causal inquiring systems some distance away, in the logical space, from classical statistics and no real, logical, deductive connection between classical statistics and the causal inquiring systems.
This is not to impugn the work done by these authors. Even to see the problem put them far ahead of 90% of the other professionals in the field. Then, to develop a causal inquiring system, no matter how intuitive, was godlike compared to everyone else. The point that I am making is only that these causal inquiring systems were not derived analytically, but intuitively, and therefore could not be applied with complete understanding or confidence.
What I did was (1) in a sense, to fill that opening (Actually, I started from scratch and fill the space from zero to the general form of causal statistics with definitions assumptions the deck of logic etc.) with deductive logic; (2) derive a general, rather than specific, form for causal inquiry; (3) generate all the required definitions; (4) produce all the required assumptions; and (5) explain all of the above in a manner hopefully understandable to a careful reader.
Now, in this book, I am engaged in simplifying and extracting all of the above from the dissertation; adding some explanations, global and epistemological information; and presenting a complete, understandable, and well-founded algorithm (which I call causal statistics) for making causal inferences from nonexperimental studies.
{2,5 Extant Causal Inquiring Systems
2.5.1 Summary
At present, there are three, more-or-less distinct,
causal inquiring systems. They are path analysis,
econometrics, and the Simon-Blalock approach. In actual
fact they are virtually identical to one another.
2.5.2 Path Analysis
Path analysis was introduced in a phenomenally
innovative paper by Sewall Wright* in 1921. Since that
*Wrlght, Sewall: "Correlation and Causation,"
J. of Agricultural Research, Vol. 20, 1921, pp.
557-85.
time additional innovations and, also, acceptance have
been amazingly slow. Path analysis considered only oneway
causation until 1954 when John Tukey** introduced
**Tnkey, John Wilder: "Causation, Regression, and
Path Analysis," in Oscar Keinpthorne, T. A.
Bancroft, J. W. Gowen, and J. L. Lush, eds.,
Statistics and Mathematics in Biology, Ames: Iowa
State College Press, 1954, pp. 35-66
two-way path analysis. This is the only innovation in
path analysis of major importance since 1921.
Basically, path analysis is a linear regression or
simultaneous linear regression technique in which the
coefficients are causal, assuming that the basic
assumptions of the model are valid. These coefficients
cj’s are called path regression coefficients. See
WrIght* 1960 for a summary of path analysis.
*Wright, Sewall: "Path Coefficients and Path
Regressions: Alternative or Complementary
Concepts?" Biometrics, Vol. 16, 1960, pp. 189-202.
2.5.3 Econometrics
Econometrics employs regression and simultaneous
equation models. It is far more advanced mathematically
than path analysis, but there are comparatively few
papers in the field which consider the causal implica
tions of these mathematical techniques.
Econometricians try to avoid the word "cause"
because of’ their misinterpretation of Humian philosophy
on the subject. Due to their avoidance of this word,
econometricians have failed to consider sufficiently a
many of the causal implications and proertIes of
econometrics and b many of the problems and benefits
connected with causal prediction.
Two good econometric references are Johnston** and
Goldberger***.
**Johnston, J.: Econometric Methods. New York,
McGraw-Hill, 1963.
***Goldberger, Arthur S. z Econometric Theory. New
York, John Wiley & Sons, 1964.
2.5.4 The Simon-.Bialock Approach
The Simon-Blalock approach began with a 1954 paper
by Herbert Simon*. This paper served as the foundation
*Simon, Herbert A.: "Spurious Correlation: A
Causal Interpretation," J. of the American Statis
tical Association, Vol. 9, 19514, pp. 467-47.
for a great deal of later work by Hubert Blalock.
Basically, this approach gives causal Interpreta
tion to some of the more elementary f’ormalizations of
econometrics. An exception is Blalock** 1969 in which
**Blalock, Hubert N., Jr.: Theory Construction:
From Verbal to Mathematical Formulati, Englewood
Cliffs, Prentice-Hall, 1969.
he gives preliminary consideration to some simple
differential equation models.}
mathematical treatise on the subject was published by Sewall Wright, with his innovative, 1921 article, “Correlation and Causation”, in J. of Agricultural. Research, Vol. 20, 1921, pp. 557-85. Dr. Wright presented a regression analysis approach to causal inference. That should have gotten the ball rolling, but amazingly it didn't.
33 years later, John Tukey published “Causation, Regression, and Path Analysis,” in Oscar Kempthorne, T. A. Bancroft, J. W. Gowen, and J.L. Lush, eds., Statistics and Mathematics in Biology Ames: Iowa State College Press, 1954, pp. 35-66 and Herbert Simon published “Spurious Correlation: A Causal Interpretation,” in J. of the American Statistical Association, Vol. 49, 1954, pp. 467-479.
These and other causal inquiring systems were incomplete and their foundations in philosophy and axiomatic logic not established. Nevertheless, these insightful initial efforts should have triggered a tidal wave of research into causal inquiring systems and their foundations.
Yet, there was no more than a diminishing ripple on the ponds of non-experimental research, statistics, and research methodology.
Econometrics, with its multi-equation and sometimes recursive forms, was mathematically superior to, i.e., more general than, the other forms of the day for making causal inferences. Econometrics employs regression and simultaneous equation models. It is more advanced mathematically than path analysis, but the number of papers in that field which consider the causal implications of these mathematical techniques is small.
Econometricians try to avoid, the word “cause” because of their misguided belief that Hume put a stake in the heart of causality. Due to their avoidance of this word, econometricians have failed to consider sufficiently (a) many of the causal implications and properties of econometrics and (b) many of the benefits that could be gleaned from facing cause inference from econometric analysis honestly and straight forwardly.
In 1968, I attempted to utilize the aforementioned systems to draw causal inferences in my nonexperimental management study, but I found that I could not apply these systems for causal inference with complete understanding, insight, or confidence. For example, many of the assumptions implicit in the various systems were unknown, the nature of statistical causal connections was not understood, etc.
13 years after Tukey’s and Simon’s first articles, I stumbled into the field of causal inquiring systems because of a desire to do good empirical research, rather then me-too research, with inferior and inappropriate statistical tools.
Recognizing that causal results were, by far, the most desired and that most research in the social sciences, epidemiology, management, etc. was non-experimental; I realized that the development of a non-experimental causal inquiring system, which could be applied with complete understanding and confidence, was of transcendent importance. Hence, I discontinued the original empirical study and turned my efforts to the development of a definitive causal inquiring system.
This statement rolls off the tongue very easily, but what kind of ego maniac would so cavalierly set about solving one of the most important problems in philosophy and THE most important and difficult problem in the social sciences, epidemiology, and statistics: first the problem of causality--a concept challenged with only a modicum of success by philosophers for over 3000 years--and second, the problem of non-experimental causal inference, a task never adequately handled by social scientists, epidemiologists, nor statisticians? Answer: a young graduate student who didn’t know better than to believe that he could ultimately solve any problem that had a rational solution.
{ In 1967-8 I suspected that the problem of causal inference must be pretty difficult, in that it was the most important problem in the field and it hadn't been solved. Even so I didn't realize how extremely hard it was; I spent at least two man years posing questions to myself, thinking, doing library research, and writing down the answers.
In another sense, this was an easy project for me because of the sheer joy of attacking such difficult and such interesting unsolved problems and, over time, seeing them yield to the relentless assault of pure reason. . I had no idea how much I could learn out of my own head by just thinking.
The concept of a theoretical dissertation was appealing to me from the first time I heard that such a thing was possible, sitting on the football/softball intramural field at Berkeley and talking to a Ph.D. candidate in nuclear engineering who was doing a theoretical dissertation. It turned out to be all I imagined and more.
I attacked the problem with all the confidence of one who didn't know better.}
Aside: I would note that any researcher, who doesn't believe he can solve his research problem, is probably right. He probably couldn't solve it. Einstein, Enrico Fermi, and R. A. Fisher certainly attacked their respective, important problems with the belief that they could solve them.
{I studied causal philosophy from Plato to Hume.}
After trying many approaches to the development of a complete causal inquiring system, I eventually did as Euclid did for Geometry and Einstein did for Relativity and no one had or has done for causality. I went back to basics and--beginning with axioms, definitions, primitives, etc., about the nature of the universe and the nature of empirical research--performed a logical derivation of the general formulation of a causal inquiring system, I called Causal Statistics. This research was reported in my Ph.D. dissertation, entitled Foundation of Mathematical Epistemology: A Derivation of Causal Statistics, published in 1972.
During the time I was working on the dissertation and after its completion -- when I was looking for funding to carry the research to the next stage--I was amazed at the negative reactions of many ostensibly intelligent people (professors, funders, etc.) to my research. Over 40 years, I have heard it all. It couldn't be done; it needn't be done; Hume has already considered it and proved that it couldn't and needn't; it should be done, but someone else should fund its further development (e.g., it's too interdisciplinary, too different, too eclectic, too innovative, or conflicting with accepted paradigms to be funded here or evidently anywhere); it's too risky (i.e., it might fail) for results oriented departments and especially for the untenured; etc.; etc.; etc.
There were about as many different, negative criticisms as there are people reacting and, in the final analysis, almost none of the negative opinions held any water. If there had been some convergence of opinion about what was wrong with my research, the critiques would have been more believable.
Nevertheless, I analyzed every criticism and modified my results for those very few with merit. Even so, such corrections almost never satisfied the doubter whose critique was accepted and corrected. These critics would then come up with some new, off-the-wall criticism. I called these critiques the “yes-buters.” "Yes, but what about this new criticism?"
Over the years, I've had many different thoughts about why people with the right educational background, have produced so many negative and incorrect reactions. I have come to several different conclusions: (1) Many just can't think out of the box, even when led by the hand. (2) Many fear, or just naturally resist, the new and different, i.e., anything more than 1 millimeter deviant from what they were taught at their professor’s knee. (3) Most don't have the breadth of mind to comprehend the whole of a large and complicated project (what I call a mega project) all at one time. They can only see distinct pieces. (4) A surprisingly large number just don’t have the mental where-with-all to even make the aforementioned three errors, e.g., "We like the way we've been doing statistics previously."
{In all fairness, might some of the blame for the retarded acceptance of causal statistics properly be placed at my door?
Could I have explain things better? With regard to the dissertation itself, I think things were explained quite well. But concerning the relationship of causal statistics to other statistical paradigms, my early explanations were lack.
Could I have published in journals to get the word out there? Maybe, but there were no journals in any way related to or interested in causal inference. Getting something in a journal which the publisher and referees feel are off-topic, is difficult in the best of circumstances. Doing it with a paper on causal inference, a topic of anathema to most academics, was multipally harder, apropos my experience with funding applications.
Would it have been better to apply causal statistics to nonexperimental research data and use the resulting report as an example of the benefits of causal statistics? I did exactly that with a 10 variable analysis of DDT, DDE, hypertension, pesticides, etc. on the data of a nationwide EPA study. They had collected data for about 10 years and, in all that time, had been incapable of sorting out the causal connections. I did exactly that in about two months and wrote a research report. The local scientists were happy, but those in Washington couldn't be less interested. Again, I was amazed. Is all the world mad or just most of it?}
As a consequence, we are 40 years down the road and Causal Statistics is still a Ph. D. dissertation and generally unavailable to non-experimental researchers: 40 years of lost time, billions of wasted dollars in the non-experimental sciences, untold waste in competent human resources--i.e., social science and epidemiological researchers getting 10% of what they could get from their research efforts--and many lives lost (e.g., from lung and other environmental cancers) and destroyed (e.g., due to the dearth of good causal research and theories to correct criminal, social, and health problems).
If I had been able to spend a substantial portion of the last 35 years carrying out the beyond-dissertation steps in the Causal Statistics project, I believe that today Causal Statistics would be in broad usage. But, for the lack of less funding than that required for one government secretary, surprisingly little progress toward codifying an understandable, complete, and algorithmic arrow for causal inference has been made in the last 35 years. Such collective stupidity on the part of funders and funding agencies is beyond belief.
At least one other explanation is possible, maybe I'm insane. The mental case is always the last to know. Well, then my dissertation chairman, Professor C. West Churchman and the dissertation committee must also have been insane. Further, if I was insane then, I still am. The research makes just as much sense to me now as it did in 1972.
As I said before, the dissertation derived the general form of Causal Statistics. Originally, I had planned to get research funding to extract an application oriented formulation of Causal Statistics from the dissertation. When the funding didn't materialize, I went on to other things, with the belief that some extremely analytic and dedicated researcher would either do the extraction or develop his/her own application oriented formulation of Causal Statistics.
About two years ago, as I started moving toward retirement, I looked at the developments in the field over the past 35 years and found that a great deal had been written about causal inference and causal theory construction. Virtually all attempts were laudable and most were correct, as far as they went. These writers were obviously people who were (1) sharp enough to see the need and (2) insightful enough to say something meaningful and correct about the subject. Nevertheless, they were about where I was in 1968, a year after I began researching the subject and before realizing the need to return to fundamentals and take the deductive approach.
A few methodologists--like Pearl, Rubin, and others--have gone further, but no one has even come close to the total package, i.e., an understandable, complete, algorithmic causal inquiring system for nonexperimental research. Neither has anyone taken the deductive approach, which I believe to be the optimal. My research is still the only effort to do what Euclid and Einstein did; i.e., go backwards, down to basics, start from definitions, axioms, primitives, etc. and derive the whole field. This approach gives a logical foundation to causal inference and allows a complete understanding of the inferential process, just as Euclid's and Einstein's deductive approaches to Geometry and Relativity, respectively, were the only routes to real understanding in their fields. Their derivations effectively ended controversies concerning origins in their fields and their colleagues move on to research in other aspects of their fields.
Not all deductive constructs are as earth shattering as Euclid's and Einstein's. Alfred North Whitehead and Bertrand Russell used symbolic logic to derive arithmetic and algebra in their book, Principia Mathematica. Few people who use arithmetic or algebra refer to or even know about their work.
In one sense, the derivation of causal statistics is of greater utility to its field than the work of Whitehead and Russell was to their field. Euclid's deductive construct was earth shattering in terms of its importance to deductive and symbolic logic, but less so to the field of plane geometry. Interestingly, Euclid's derivation of plane geometry was more important for its contribution to the development of non-Euclidian geometries then for Euclidian geometry.
As everyone knows, Einstein's derivation of relativity was and is unequaled in its effect on the field of physics and in popular culture (the public consciousness). At the turn of the 20th century, physics was where nonexperimental causal inference was in 1968. Most of the mathematical formulations have been developed, but they could not be applied with confidence or understanding because no one understood why the formulas were as they were. Einstein's logical construct, relativity, derived the Lorentz equations and specified the assumptions and concepts on which they were based. After Einstein published his work on relativity, physicists progressively moved toward the accept that's of relativity and its assumptions as the intellectual foundations for the Lorentz equations and moved on to other problems in physics.
In the above sense, the derivation of causal statistics is not comparable in importance to the derivations of Euclid and Einstein. Yet, in the effect on human lives in the long run, the derivation of causal statistics will arguably be more important than any of the others. If causal statistics were used in all appropriate nonexperimental research, the increase in medical, social, etc. knowledge would be so great and the application of this knowledge so earth shattering in its effects that, in that sense, the derivation of causal statistics would surpass the others.
Now you should have a ballpark understanding of the content of the 1972 dissertation and where the field of causal inquiring systems is today. About two years ago I started a website called causalstatistics.org. The initial purpose of the web site was to make my dissertation easily accessible. A downloadable, selectable, and searchable copy of the dissertation is presented on that website.
The dissertation presents Causal Statistics at a level that extremely analytic and dedicated researchers could apply the paradigm in non-experimental research and obtain valid causal inferences. Nevertheless, greater simplification is necessary for the vast majority of social science researchers to utilize Causal Statistics with complete understanding and confidence.
{ The remainder of this web page is dedicated to accomplishing the goal of making a sea change in the way non-experimental scientists conduct their research. Specifically, the goal is that social scientists, epidemiologists, and other non-experimental researchers--when appropriate--utilize Causal Statistics in the design, conduct, analysis, and reporting of their empirical research. The more precise purpose of this Exposé is to tackle Objective 4, above, i.e., to present Causal Statistics in an easily accessible (intellectually) way, even multiple ways.
This article draws extractions from the dissertation and presents a minimal, yet sufficient, formulation of Causal Statistics, along with examples of its application in non-experimental research. Additionally, the presentation is interspersed with background information (necessary for understanding), answers to potential objections, explanations of how the various components being presented relate to each other, and delineations of how these components fit into their philosophical, logical, and statistical environments and into the overall mosaic of human knowledge.}
But, as long as I'm at it, I might as well take the opportunity to handle all other relevant issues that I can think of. Hence, the presentation will be interspersed with background information (necessary for understanding), answers to potential objections, explanations of how the various components being presented relate to each other, and delineations of how these components fit into their philosophical, logical, and statistical environments and into the overall mosaic of human knowledge. (Moved this paragraph to preface?)
[the purpose of this book is to design an algorithm for properly applying causal statistics in non-experimental research so that, after reading this book, any non-experimental researcher will be able to apply causal statistics to their work correctly and with understanding. In furtherance of that purpose, the book should answer any questions and handle any problems that the researcher is
Hence, as my work has progressed, my ultimate goal has become more far-reaching. The goal has evolved toward making a sea change in the way non-experimental scientists conduct their research. Specifically, it is desired that social scientists, epidemiologists, and other non-experimental researchers, when appropriate, utilize Causal Statistics in the design, conduct, analysis, and reporting of their empirical research.
In an effort to accomplish this goal, I have established five objectives (“Objectives” are steps on the path toward accomplishing the overall goal.):
1. To make the dissertation readily available to all (accomplished via the presentation of the dissertation at causalstatistics.org),
2. To extract from the dissertation portions that are, in sum, necessary and sufficient for formulating a physics/logically/epistemology/ statistically/research methodology/philosophically based causal inquiring system,
3. To utilize these extractions to formulate Causal Statistics in a complete, coherent, and interrelated (i.e. with consideration of how Causal Statistics related to its epistemological environment) form,
4. To present this formulation of Causal Statistics in an easily accessible (intellectually) way (even multiple ways) to present and future research methodologists, to the researchers themselves, and to research consumers. (The initial presentation will be accomplished through the development of this book),
5. To challenge non-experimental scientists and research methodologists to do the hard work to study, understand, analyze, critique, extend, and apply Causal Statistics
[GOAL:] The initial impetus for this book is to extract from the dissertation the elements necessary for a minimal, yet sufficient and usable, formulation of Causal Statistics and present it herein, along with examples of its application in non-experimental research.
{But, as long as I'm at it, I might as well take the opportunity to handle all other relevant issues that I can think of. Hence, the presentation will be interspersed with background information (necessary for understanding), answers to potential objections, explanations of how the various components being presented relate to each other, and delineations of how these components fit into their philosophical, logical, and statistical environments and into the overall mosaic of human knowledge.
[the purpose of this book is to design an algorithm for properly applying causal statistics in non-experimental research so that, after reading this book, any non-experimental researcher will be able to apply causal statistics to their work correctly and with understanding. In furtherance of that purpose, the book should answer any questions and handle any problems that the researcher is likely to encounter in either application or understanding.
In a]}
Take your recent example. A researcher found a correlation between lack of sleep and poor grades, on the part of children. The newscaster said, “You should make sure your children get lots of sleep so they will get better grades."
Is it possible that parents who don't care enough to make sure that their children get sufficient sleep also don't care to push or to help them make better grades? Or could it be a spurious correlation due to a genetic factor?
What about correlations between book ownership and reading scores of children? Or between....
What about the causes and the effects of mental illness? What about the environmental causes cancer? What about the effects, of lax law enforcement in low income areas and on young people, on the future criminal behavior of these young people. Etc., etc., etc.
I have virtually never seen a non-experimental study reported in the media correctly. The original research report almost never uses the word cause, but the media people almost invariably give an inappropriate causal interpretation, even if the word cause is not used by them either.
I.1 Causal Statistics: A Two Variable Example
Causal statistics, unlike any other causal inquiring system, is a deductive system. See preface.
I.2 Causal Statistics: A Three Variable Example
II. The Need for a Complete, Non-Experimental Causal Inquiring System (Put in I and/or III)
Dissertation in Library of Congress and University of California Berkeley Graduate Library
WP2
WP6 Paragraph 1-4
WP7
A government planner, as a consumer of social science research, might want to predict recidivism rates as a function of the hours per week an inmate spends in self-improvement and educational programs, in order to plan for future prison space requirements. For a non-experimental study to produce the information that the government planner needs, the researcher need only obtain the association (i.e., correlation) between prison program participation and recidivism (a dichotomous variable) or the amount of time prior to return (a continuous variable). In either event, the data analysis would require only a straight forward application of Associative Statistics, a subset of Classical Statistics, and no information about causation is needed.
From the correlation or, more desirably, the regression results, government planners could predict recidivism rates and, from that and other information, prison space requirements.
But more importantly, government planners, counselors, administrators, etc. would like to know how to intervene to reduce the recidivism rate. Such manipulation would require knowledge of the causes, both positive and negative, of recidivism rate for one or more of the variables which government can control.
The demand by research consumers for prediction, without intervention, is miniscule compared to the demand (generally, unsatisfied in the Social Sciences) for research results enabling prediction, with intervention or manipulation, i.e., causal results. Accurate causal theories enable us to control the future rather than just forecast it.
Unfortunately, no complete or totally understandable research tool is available or drawing valid causal inferences from non-experimental data. The dominant, almost exclusive, inquiring tool is Classical Statistics, which yields correlations and/or regression coefficients. And, as basic statistics texts will tell you, if they deal with the subject at all, correlation does not imply causation. “Cause” is not even a term in the vocabulary of Classical Statistics; therefore, Classical Statistics alone could not draw valid causal inferences. Yet, the need for causal results and theories is so great that many researchers and research consumers have incorrectly purported to do just that, utilize Classical Statistics to draw valid causal inferences from non-experimental data.
Now, stop a moment to let this lunacy sink in. Causal inferences are the most desired and needed conclusions in the non-experimental sciences. But, there is no complete or logically consistent inquiring tool available to draw the needed valid causal inferences from non-experimental data. [Yet, little funding or effort is being expended to remedy this gross and debilitating short coming. Is such mass stupidity possible? Regretfully Dorothy, it is.]
Presented with these facts, any intelligent person would imagine that a tremendous amount of research money and effort would have been devoted to the development of a complete, understandable, and valid causal inquiring system. But this intelligent person would be, not only wrong, but grossly wrong. No significant money and only slightly more effort has been expended on the methodological problem of causal inference.
Aside: One might argue to the contrary, in that a significant portion of my dissertation was supported by the National Aeronautics and Space Administration through the Space Sciences Laboratory at the University of California (Berkeley). While true, and even though I had worked at the Manned Spacecraft Center (now the Johnson Space Center) for NASA in the late 60's, in the Theoretical Physics Branch analyzing the Bremsstrahlung and other radiation hazards to Apollo astronauts, it was not NASA's intent to fund me to develop Causal Statistics. The original funding was for an empirical study to determine the causes (both positive and negative) of productivity in scientific research and development.
Since the study was non-experimental, causation was difficult to establish. I attempted to use the causal inquiring systems available at that time, but could not apply them with understand, insight, or confidence--e.g., the assumptions implicit in the various systems were unknown, there were no proofs of the appropriateness of such systems, and it was unclear how to disentangle causal components from associative components in coefficients (e.g., regression coefficients) inferred.
I then turned my efforts to the development of a causal inquiring system which could overcome the aforementioned problems. As this work proceeded, I saw a far-reaching importance of this line of statistical research. Its significance dwarfed that of the original R & D study. For this reason the R & D study was discontinued, with the causal statistics project taking its place.
At this point, NASA would likely have cut my funding. But Professor C. West Churchman--the head of Berkeley's Space Sciences Laboratory, my dissertation chairman, and all around intellect and great human being, renewed my funding, probably without saying a word to NASA.
Before completing the dissertation I left Berkeley to teach for a year at the University of Hawaii. In the next year, I completed the dissertation funded by unemployment and food stamps.
As a further aside, I would have gladly continued my research at that time had any government or private organization been willing to fund me at the level of my unemployment and food stamps. I searched extensively, but no one was willing: 40 years of lost time, billions of wasted dollars in the non-experimental sciences, and many lives lost (e.g., from lung and other cancers) and destroyed (e.g., due to the dearth of good causal research and theories to correct criminal and social problems). All for the lack of less funding than that required for one welfare client. Such collective stupidity is beyond belief... [unless one understands that the intelligence of an organization is generally inversely proportional to the number of people in it.]
To me, this state of affairs almost defies explanation, but not quite. The reason for “not quite,” is that I have endured 40 years of ostensibly intelligent people telling me it couldn't be done; it needn’t be done; Hume has already considered it and proved that it couldn’t and needn’t; it should be done but someone else should fund it (i.e., it’s too interdisciplinary, eclectic, different, innovative, or conflicting with accepted paradigms to be funded here or evidently anywhere); it’s too risky (i.e., it might fail) for results oriented departments, especially for the untenured; etc.; etc.; etc. The need for a usable, complete, understandable, and valid causal inquiring system is beyond question to clear thinkers, those one in every million or so. “I think, therefore I see the need.” This is the need which Causal Statistics is designed to fill.
It is my belief that, 100 years from now, research results and theories in the non-experimental sciences will consist of large arrays of variables, connected by multi-equation causal models, inferred from a single large or a succession of smaller, Causal Statistics based, empirical research study(s) This, of course, assumes that I am able to complete my work on the objectives (stated above) of this web site.
Should I die before completing this work and be unable to fund the effort in my will, I estimate it would be in the range of 300 years before this type of advanced, non-experimental research and theory construction will be common place. This conclusion is drawn inductively from my observations of the glacial progress in the field of causal inference in the last 100 years.
A 200 year delay sounds pretty extreme, especially considering that, in recent years, a great deal has been written about causal inference and causal theory construction, as one can see by Googling “causal inference.” Virtually all attempts are laudable and most are correct, as far as they go.
These writers are people who are (1) sharp enough to see the need and (2) insightful enough to say something meaningful and correct about the subject. This is where I was in 1967, before realizing the need to return to fundamentals.
My research in causal inference methodology is the only effort to do what Euclid and Einstein did, i.e., go backwards, down to basics, and start from definitions, assumptions, etc. and derive the whole field; giving a logical foundation and allowing a complete understanding of causal inference. Further, I see not even a tendency, on the part of any writer in the field, to so much as glanced in the direction of the deductive approach to causal inference, which is the only real route to a complete and understandable causal inquiring system for the non-experimental sciences, just as Euclid’s and Einstein’s deductive approaches to Geometry and Relativity, respectively, were the only routes to real understanding in their fields.
Whenever this advanced research and theory building paradigm does enter common research usage, that research methodology and its foundation, i.e., Causal Statistics, will be generally considered to be the most important developments ever, for progress in the non-experimental sciences.
III. Causality and the Empirical Research Environment
Prior to a detailed presentation of Causal Statistics, it is necessary to understand the empirical research environment which gives rise to the need for Causal Statistics and to see how Causal Statistics fits into it.
The empirical research environment includes not only the nature and types of empirical research, but also the methodological and analytical tools available, the types of findings desired, and the uses to which research findings are put.
Aside: It is my tendency to present a subject by starting at the beginning and proceeding in chronological/logical order to the end/conclusion, Q.E.D. Although logical, this approach is pedagogically problematic. The problem is that, at best, the reader usually knows only generally where he is going, but not precisely. Hence, subtleties in the step-by-step presentation are not appreciated when they are read, and, as a result, not committed to memory and/or not place in the proper juxtaposition to other elements for reaching a well-reasoned, logically tight conclusion. Hence, you might consider reading the concluding chapter and then return to chapter II with a comprehension of exactly where you are going.
III. A. Prediction, without Intervention
As indicated by the government planner example at the beginning of Section I.B., non-experimental research, to enable research consumers to make forecasters, without manipulation of any of the variables, requires no knowledge of causal connections between the variables and is comparatively simple. All the scientist needs to do is use Classical Statistics to obtain correlations or regression coefficients, pretty much a straight forward calculation within the domain of Classical Statistics.
A research consumer might desire a confident interval or a hypothesis test on the results, which is only a little more complicated calculation with a Classical Statistics computer algorithm.
For prediction, without intervention causal information is not needed and, therefore, neither is Causal Statistics.
[Prediction, without intervention is in small demand. Prediction, without intervention is not so simple in a game against a thinking opponent or if the structure has changed.]
III. B. Prediction, with Intervention
As invented in Section I.B., the demand by research consumers and theoreticians for results to support prediction, without intervention, is miniscule compared to the demand for support of prediction, with intervention. To support prediction, with intervention, these results must be causal connectors and presently these are no available tool for drawing valid causal connections from non- experimental date.
It is beyond incredulity that, in the 150 year history of the modern social sciences, so little money, thinking, and work have been devoted to the development of methodological tools and techniques for making valid causal inferences. This is an indictment of funders, methodologists, and methodology developers.
In the absence of such tools and techniques, Social Science researchers have been confronted with a classic dilemma. Associative results lead to unsatisfied and unhappy theoreticians and practitioners and invalid causal inferences lead to failure in the long run. Researchers who take the latter path, i.e., draw invalid causal inferences, are usually either unscrupulous or naive in the ways of causal inference. The ultimate result of such invalid causal conclusions is incorrect theories and failed interventions and programs.
As a result of this unhappy, no-win dilemma, social scientists have largely concerned themselves with definitions, concepts, associative theories, analytically developed theories or models (like in Economics), controversial experiments, case studies, and questionable causal studies and theories (usually based on improperly drawn and highly suspect causal inferences). All of these, except the invalid causal inferences, are valuable. But, if the social scientists were able to supplement their formulations with validly drawn causal connections and large multi-variable causal theories, the overall power and value of the Social Sciences could more then doubled.
This is the contribution which Causal Statistics is poised to make.
[?The dominant mode of theory building in the physical and biological sciences is based on “knowledge” of the cause and effect relationships among variables. The social and other non-experimental sciences would like to do the same, but generally can’t, because of the difficulty, in obtaining valid causal inferences from non- experimental data. Hence, social scientists concern themselves with definitions, concepts, associative theories, analytically developed theories or models (like in Economics) contrived experiments, case studies, and questionable causal theories (usually based on improperly drawn and highly suspect causal inferences).
III. C. Experimentation
A part of the reason for the creping and slower progress in causal inference in the non-experimental sciences is that methodologists start, as did I, by considering causal inference in the non-experimental sciences, a highly complex environment for causal inference.
As a pedagogical device, I backed in the approach of beginning the presentation of causal inference in its simplest form or environment, to fabricate understanding, and then move on to more complex environments.
In retrospect, it is now obvious to me that this would have been the correct approach for my and others’ research into causal inference, i.e., understand and master the simplest form of causal inference first and then move on to the more complex environments. Try to walk, before trying to run. Unfortunately, no non-experimental research methodologist seems to have taken this “obvious” approach. “Obvious” in the sense that all ideas are obvious once someone tells you about it.
In the physical and biological sciences, experimentation is almost always possible. In general, if a variable can be repeatedly manipulated and another variable can be repeatedly observed to change, causal inferences will be intuitively drawn.
Say that a researcher wanted to study the effect of a force on the motion of a bowling ball. He could set up an experiment in which, on a series of runs, either a constant, horizontal force of 10 newtons (the metric unit of force) or no force is applied to a 5 kilogram ball. The motion of the ball is measured on each run and all else is held constant (i.e., controlled).
The experimenter would find that, on each run in which the force is applied, the ball accelerates horizontally at a constant rate of 2 meters/sec/sec so long as the 10 newton force is applied. When no force was applied, the acceleration would be zero.
In working with this data, a theoretician would likely develop the formula F=ma (10=5x2) to model the data from the experiment and then propose the formula as a theory to explain the relationship among the variables; force, mass, and acceleration.
Based on this experiment, the relationship inferred, F=ma, would be far from certain. But, if other, more varied experiments were performed, the equation would be found to hold up and confidence in it as a theory or ultimately a natural law would increase.
Yet, in experiments where the relative velocity between the bowling ball and the observer approaches the speed of light, a curious thing happens. The formula remains true, but the 10 Newton force results in an acceleration less than 2m/sec/sec and the mass of the bowling ball increases by a compensating amount, as phenomena explained by Einstein’s Theory of Relativity. This shows that gaining “knowledge” about the operation of the universe, especially at the extremes, is not as simple as it might first appear, but that my dear, is a story for another day.
III.C.1. Non-Statistical Causal Inference from Experimental Data
Looked at in one way, what was observed during the experiment was an associated (i.e., correlation) between force and acceleration. It is intuitive and easy for a researcher to make the small leap to the causal inference that the force caused the acceleration.
Speaking of words and “cause,” as noted previously, the logical foundations and development of Classical, Associative, and Bayesian Statistics do not mention, define, or incorporate “cause,” like “red” is not defined or incorporated in Geometry. Hence, any purported causal inference based on any one of the three statistical paradigms alone, would be invalid on its face. Technically, this is true even for their applications to experimental data for purposes of inferring causal connections. This is a surprising conclusion, given the great progress in the experimental sciences (e.g. Physics, Chemistry, and Biology), which has resulted from causal expectation. How can one explain this apparent contradiction?
Consider an experiment in which books were given to some children and not to others, all randomly selected. Suppose that that experiment discovered that; measured three years later, the treatment of group were superior readers. It would be easy and natural for a human brain to conclude that book ownership causes improvements in reading, but this conclusion would be technically invalid.
Cause is not a component of Associative Statistics, so this brain is making an intuitive jump and using Associative Statistics plus something else to draw the causal conclusion.
To make this causal conclusion logically tight (i.e. deductively complete), the valid experiment would have the define “cause”, (2) as that we live in micro causal universe, (3) as that no outside variables are causing both the experiment to manipulate ownership and the treatment group to read better, and (4) as that the correlation between ownership and reading is not a result of either random or measurement error.
Note that we have chosen these assumptions quite judiciously in such a way that, if they are true, there is one and only one explanation for the observed data. That explanation is that bank ownership causes improved reading in children. Therefore, if the definition and assumptions had been explicitly stated, the causal conclusion that that ownership causes improved reading would be valid and if the Associative Statistics are all true, the conclusion should be correct.
For 100’s of years, experiments have made causal inferences from experimental data without defining “cause” and with no reference to the required assumptions. Technically, this is improper Causal Statistics and is not valid. But, looked at it in another way, these required assumptions generally are acceptable to researchers that they don’t bother to state them, considering the assumptions to be implicit and the causal inferences, therefore, valid. This reasoning generally comports with observed behavior in the experimental sciences and, therefore, is probably the best explanation of what actually happened.
It’s a small leak, with generally acceptable and accepted assumptions. Thats why the experimental sciences have been so successful with their causal conclusions.
The non-experimental sciences are dealing with another kettle of fish. Causal inferences, from non-experimental data using Associative Statistics require a huge and almost always incorrect leap of intuition. This is the reason for the relative lack of success in general in causal theory building in particular in the social sciences, epidemiology, etc.
The essence of this difference in causal inference between experimental and non-experimental studies resides in one factor, the vastly different nature of assumption (3), above, between experimental vs. non-experimental research.
In experimentation, assumption (3) simply asserts that nothing cause both the experimenter to manipulate the independent variable (ownership) and the change in the dependent variable (reading scores), resulting in a non-casual correlation between the two variables; an eminently reasonable and acceptable assumption.
But, in non-experimentation, assumption (3) becomes “assume” that, no outside (i.e. unconsidered) variable(s) caused both independent and dependent (reading scores) variable, resulting in spurious (i.e., non causal) correlation between them.
It is difficult to imagine that, the correctness, or leak thereof, are little, or even large, assumptions could account for the large difference in success between the physical and social sciences. Actually, the mobility to draw valid causal inferences in the social sciences doesn’t account for all of the difference in success, but I would say at least 50%.
Of course, non-experimental researchers and research consumers recognize this and attempt to draw causal inferences with the tools they have. Unfortunately, the tools of Classical, Bayesian, and Associative Statistics are so inadequate to this task in non-experimental research, that most such causal conclusions are incorrect and therefore not only useless, but counter productive.
A select, very few such researchers also recognize the intuitive leak they are required to make to get from non-experimental data and Associative Statistics to causal inferences and attempt to fill in the intuitive gap in one way or another. Such researchers are attempting to go beyond the standard statistical paradigms, to a new statistical realm which I call Causal Statistics. A few such researchers and methodologists have moved some distance into that new statistical paradigm, but not many and not far.
Note that researchers who attempt to draw causal conclusions from experimental data using Classical Statistics are making only a small leap of intuition beyond the tenets of the field. This is why the physical (experimental) sciences have not generally encountered significant problems in drawing causal conclusions. On the other hand, the social and other non-experimental sciences have been largely paralyzed by their difficulties in making valid and correct causal inferences.
As an example of how one might move toward Causal Statistics from Classical Statistics, consider the second error, made by those who claim (either explicitly or implicitly and either knowingly or unknowingly) to have discovered a causal connection; i.e. the assumptions, upon which causal conclusions much necessarily be based, are virtually never stated or even known.
In the RIF example, they could have validly concluded that B causes reading scores, R, if they were willing to state the assumptions (1) that R did not cause B and (2) that no other variables caused both R and B. Under these assumptions, a researcher could make the valid, but likely incorrect causal inference that B causes R. if these assumptions are stated, few would be deceived into accepting the conclusion that B causes R because most people would be unwilling to accept one or both of the required assumptions, if even one of the assumptions is incorrect, then the causal conclusions are highly likely to be incorrect, at least in terms of these magnitude.
This approach would be an attempt on the part of a very few sophisticated researchers in the field, to create a needed part of Causal Statistics by backing into it from Associative Statistics. The attempt is not improper, if executed correctly. But correct and complete execution along this path is very difficult, although not impossible. For this and other reasons, this is not the best approach, because (1) based on what I have observed over the last 45 years, not 1 in 1000 researchers can reach a valid and complete causal inference in this manner; (2) the complete Causal Statistics paradigm would virtually never be developed in this way; and (3) no partial formulations would lead to a complete understanding of the nature, validity, accuracy, sensitivity, and weaknesses of the conclusions reached.
In contrast, I have gone through the front door in developing and deriving Causal Statistics in my dissertation, which is presented on the Causal Statistics website, CausalStatistics.org
I began, as Euclid began with Geometry, by using a Meta language to state primitives, definitions, axioms, etc. From there, the whole of Causal Statistics was derived using both verbal and symbolic logic. Such an approach, allows complete understanding of the paradigm to anyone who is willing to put in the effort to follow the steps of its development, i.e., the derivation of Causal Statistics.
For a less rigorous understanding of Causal Statistics, but adequate for a knowledgeable application to non-experimental research and utilization of the conclusions, the following should suffice.
Causal Statistics begins with a common sense, useful definition of “cause,” a non-trivial philosophical issue, given the numerous, inappropriate, and inadequate definitions forwarded by various philosophers and researchers over the past 3000 years.
Additionally, Causal Statistics is based on three sets of assumptions. The first and most fundamental set postulates that we live in a causal universe, at least at the macro level, where cause and effect governs the behavior of all variables.
The second assumption set is for the purpose of isolating the variables considered by the specific research study. As an example of the types of assumptions in Set #2, note the two assumptions presented above, in conjunction with the RIF program. These assumptions were necessary to make the causal inference i.e. , that book ownership causes improved reading; valid. Nevertheless, these assumptions in no way make the causal conclusion correct, if one or more of these assumptions are incorrect. We can be confident that the causal conclusion is correct, only if the assumptions are correct.
The second assumption set need not totally rule out all influences from outside variables on variables inside the system of study. These assumptions need only prohibit outside influences that would exert just certain types of influences. To understand which set two assumptions are required and which are not, one must understand the components of association.
The association between two variables in a study have the major sources; association due to (1) causal connection (e.g., B causes R and/or R causes B), (2) spurious correlation (e.g., S, socioeconomic status, causes both R and B or S causes X, any outside variable, and B and X causes R), and (3) definitional overlay of internal variables (e.g., book ownership and school book ownership), and (4) study error (e.g. measurement bias) statistical error. An observed association will result from one or a combination of all of these sources.
Associations due to causal connection, i.e., source (1), are what researchers are looking for. Spurious correlation, source (2), are what researchers want to avoid and is what assumption set two needs to rule out.
To be a little more precise, some spurious correlations are acceptable and others are not. It is not acceptable for any external variable(s) to cause a spurious correlation within an internal variable pair, if the variable pair is the subject of a causal hypothesis test in the study. Such a spurious correlation must be eliminated by assumption(s) or by bringing such external variable(s) into the study to make the causal conclusion valid. Yet, if the assumption is erroneous, the causal conclusion will be similarly erroneous and the problem external variable(s) must be brought into the study. External sources of spurious correlation within any non-hypothesis variable pair are not a problem to the study and need not be eliminated.
Definitional overlap, source (3), is simply an error in research design and should be avoided. Study error, source (4), is no more likely in causal studies than any other experimental research and should be handled with the usual techniques.
If the set two assumptions are properly designed and all else in the research is handled correctly, the causal inferences drawn should be “valid.” But if these assumptions are not satisfied (i.e., false), the causal conclusion may be incorrect or at least the accuracy of the causal inferences will be diminished. The extent of diminution is usually related to the degree of error in the isolating assumptions.
The first two assumption sets are required only by Causal Statistics and not by other types of Statistics, but there is a third potential assumption set required in all paradigms. This third set assumes no measurement error, no sampling bias, etc. Since the third assumption set impacts all forms of Statistics equally and our primary purpose here is to highlight the differences, we will simply consider the third set to be satisfied and not deal with these assumptions further.
In the standard statistical paradigms, with the third set of assumptions satisfied, a positive result supports a non-causal hypothesis, e.g., an association, a mean value, etc., about the population, subject only to the error inherent in random sampling (i.e., statistical error). The probability of statistical error can be calculated and specified precisely. Such standard statistical results are very close to simply reporting on what was observed, i.e., only slightly more inferential than reporting on what was observed.
In Causal Statistics, one or more causal hypothesizes are proposed; a study is designed; data are collected; and the causal hypothesizes are either confirmed or not confirmed; subject (1) to all three assumption sets (although we have, for purposes of this discussion, stipulated that the third assumption set is satisfied.), (2) to a given definition of “cause,” (3) to calculated probabilities of statistical error, and (4) to the requirement that all assumptions be explicitly stated.
Note that causal conclusions are not directly observable and therefore Causal Statistics requires more input to obtain its results (i.e., the causal inferences). The standard statistical paradigm require only data and no additional input, except assumption set number three (which was already considered satisfied), to obtain its conclusions, i.e., inferences to population associations. The additional [burden inherent] inputs required by Causal Statistics are a useful definition of “cause” and two additional sets of assumptions.
III.C.1.a. Validity and Correctness
See [IV.B]
“Intuitive and easy,” yes, but valid? No and yes, in that order. What about “correct?” For such an experiment, very likely.
Technically, the causal inference that force causes accelerations draws from the experiment, is not valid, because one cannot logically/deductively arrive at the causal inference using only the data from the experiment, as the researcher seems to have done. The additional inputs needed are an appropriate definition of “cause“ (see …) and a set of assumptions about the causal nature of the universe and the conditions extant during the experiment.
Specifically, the required assumptions for the experiment are as follows:
(1) We live in a causal universe (This means that the behavior of our universe is not simply an innumerable series of random events that only appear to be causal and that no greater power is intervening and making the behavior appear causal.)
(2) The operation (i.e., causal and any other “laws”) of the universe does not change over time,
(3) No variable caused both the experimenter to exert the force and the ball to, independently, accelerate,
(4) None of the correlation between force and acceleration is the result of random fluctuation in one or both of these variables. {Make this assumption or accept 99.5 confidence in the causal connection or causal inference.}
(5) the observed accelerations are not due to measurement error.
What a result! We did not use intuition to leap across a chasm of unknowns, to the conclusion that force causes acceleration. We used stated assumptions to build a plank bridge across the gorge and walked across to deductive causal inference in the way that Euclid started with definitions and assumptions and deductively concluded (i.e., prove, derived) that all Triangles contain 180 degrees. If the assumptions are true, the conclusion is true. [It must be noted that these assumptions (planks) are always suspect and, if even one assumption is not true, there is a real possibility of falling to the bottom of the gorge.]
Note that, in the non-experimental sciences the situation with regard to drawing of valid causal inferences is similar in form, but vastly different in confidence. The required assumptions are similar, but the analogue to as. (3) is for more suspect [assumptions (4) and (5) are less true and control is less].
For 100’s of years, experimenters have made causal inferences from experimental data with no reference to the required assumptions. Technically, these are improper and are not valid causal inferences. But, looked at it in another way, these required assumptions so are generally acceptable to researchers that they don’t bother to state them, considering the assumptions implicit and the causal inferences, therefore, valid. This reasoning generally comports with observed behavior in the experimental sciences and, therefore, is probably the best explanation of what actually happened.
Now, what about the question as to weather or not the inferred causal connection is “correct?” If one is 99% confident (Bayesian probability) that the assumptions are correct, then he can be at least 99% confident that the causal inference is correct. I say “at least” because there is a possibility that an assumption may be false and yet the conclusion (causal connection) be true or correct. For the person in the about example, the causal inference is valid, but he is not certain that the causal assertion is correct, but almost, i.e., greater than or equal to 99%.
III.C.1.b. Causal Modeling/Theory Building
A causal theoretician, looking at these experimental results, would likely conclude that force causes acceleration in an amount equal to F/m or according to the equation, a=F/m.
III.C.2. Statistical Causal inference
In the above experiment, there is no need for statistical analysis because there is no significant random component or random error in the experiment.
The only relevant variables which vary, from run to run, are the force and the acceleration, which are the variables of interest in the study. All other relevant variables (e.g., mass of the bowling ball, wind speed, direction of the force) are controlled to be unchanged from run to run. Also, there was no significant measurement error.
In experiments with random error, the greater the stochastic component relative to the size of the effect, the greater the need for statistical analysis. Consider the above experiment, but designed by a researcher who failed to control some of the relevant variables, like mass.
Say that, due to lack of knowledge of the subject, the researcher used a different bowling ball, randomly selected, on each run and hence did not control for the different masses. The force would lead to various measured accelerations, rather than the identical accelerations observed in the first experimental design.
If on a given run the mass was 4 kg, the measure acceleration would be 2.5 m/sec (a= F/m = 10/4= 2.5 m/sec). In the original experiment, the mass was 5 kg and the measured acceleration was 2 m/sec. Hence, the new, measured acceleration contains and error term equal to .5 m/sec (a=2.5= 2+e, e=.5 m/sec.)
Say that the mass, for the next run, is 6 kg. The measured acceleration would be 1.67 m/sec (a=F/m =10/6 = 1.67), which would contain an error term equal to -.33 m/sec (a= 1.67= 2+e, e= -.33).
Note that the distribution of masses, from which the balls are randomly selected, could be used to calculate the distribution of the error term.
As an aside, there is pedagogical advantage to this procedure. Advanced Statistics classes are taught starting from the error terms, without looking to the origin of the error terms. Starting form the source of the error and then determining what the error should look like, is a good way to teach someone how to go the opposite the direction, i.e., error distribution to source.
Note that, in an experiment where the relative magnitude of the stochastic processes are large, there is some likelihood that even though the sample correlation is positive, the true correlation might be zero. This would require that a hypothesis test be done to determine if the experimental data yields a sample correlation between force and acceleration significantly different from zero.
If a Bayesian hypothesis test results in a probability of 95% that the true correlation is greater than zero, the researcher could validly conclude that force is a positive cause of acceleration with 95% confidence, based on the previously stated assumptions.
“Valid” does not mean “correct” or “certain.” It means the process (e.g., for making causal inferences) has been followed properly. A Classical Statistical inference, from a random sample of people, that the mean population age in the U.S. is greater than 35, may or may not be correct, but, if all statistical procedures were followed, the inference is “valid.”
If the research consumer believed with 90% confidence that the assumptions were true, his confidence that this causal connection is correct would be .95 x .90 = .855 or 85.5%
The apparently random error would increase, if the number of uncontrolled items increased, e.g., the experimenter did not control the direction of the force. In the first effort, where the force was horizontal, gravity did not affect the measure acceleration. But, in an experiment in which direction is uncontrolled and also not controlled for analytically, more stochastic error is introduced. [If the choice of directive on each run is randomly selected the error in measured acceleration will also be random. But, if the direction is selected systematically, the error term will also be systematic. For example, if the direction of force is always more or less down, ranging form 3:00 to 9:00, the effect of gravity will always be greater than or equal to zero. Therefore, the error in measured acceleration due to gravity will always be greater or equal to zero. This will lead to a positive bias in the measured acceleration, with the error distributed around a positive number, rather then zero.
? III. C. 2. a. Validity and Correctness
III. C. 2. b. Modeling/Theory Building
If the researchers or theoretician wished to calculate the strength of the causal connection, i.e., the amount of change in acceleration caused by a unit change in force, he/she could use Classical Statistics in the form of regressive analysis: a= c_{1} + c_{2 }(F/m) + e, where c_{1} and c_{2} are regression parameters, estimated form the experimental data.
If all of the error assumptions of regression analysis are satisfied, c_{1}, should be close to zero and c_{2 }should be close to one, yielding the following equation.:
a @ F/m +e
which is consistent with the equation developed from the zero error experiment, namely a = F/m.
III.C.3. Conclusions
Causal inference from experiments, even highly stochastic experiments, are much more difficult and much more complicated. These difficulties have lead to philosophical, definitional, and validity concerns that have permeated work with and the understanding of causality in non-experimental research.
The essence of this difference in causal inference between experimental and non-experimental studies, is (1) the difference in assumption (3), above, and (2) the usually diminished ability of the researcher to control (e.g., hold constant) other variables.
In experimentation, assumption (3) simply asserts that nothing causes both the experimenter to manipulate the independent variable (force) and the change in the dependant variable (acceleration), resulting in a non-causal correlation between the two variables; an eminently reasonable and acceptable assumption.
But, in non-experimentation, assumption (3) becomes, “assume that, during the study, no other variable or variables caused both independent and dependent variables, resulting in a spurious (i.e., non-causal) correlation between them;” a vastly more suspect assumption.
In the first experiment, above, the research was perfectly controlled. The only two relevant variables that changed were force and acceleration. In non-experimental research….
In fact, causal inferences are generally so easy to draw from experiments that, most of the time, no or only minor thought is required and, in stochastic experimental, only rudimentary statistical analysis is required. In such cases, Classical Statistics has generally been use to assist in making these causal inferences from experimental data, but Classical Statistics alone would not be sufficient for drawings valid causal inferences from even experimental data because “causality” is not defined in or considered by Classical Statistics.
Therefore, something additional is necessary. It turns out that that that extra something is human’s natural, innate intuition for making causal inferences. It is so natural for us that we don’t even realize that we are making a leap of intuition to causality, something, as Hume said, that we cannot sense directly, but can only infer.
But, such intuition cannot be a logical basis for valid causal inference. Hence, as was shown earlier, valid causal inferences from experimental data must additionally be based on a generally acceptable set of assumptions.
III.D. Non-Experimentation
In non-experimental research, everything changes. Causal inference is extremely difficult, leading to philosophical, definitional, and validity concerns that permeate work with and the understanding of causality in non-experimental research.
In the Social Sciences and in Epidemiology, experimentation is seldom feasible. In these and other non-experimental sciences, the situation with regarding to drawing causal inferences is vastly different. It is much more difficult and much more complicated. These difficulties lead to philosophical, definitional, and validity concerns that permeate work with and the understanding of causality in non-experimental research.
The essence of this difference in causal inference between experimental and non-experimental data is (1) the difference in assumption (3), above, and (2) the usually diminished ability of the researcher to control (e.g., hold constant) other variables.
In experimentation, assumption (3) simply asserts that nothing causes both the experimenter to manipulate the independent variable (force) and the change in the dependant variable (acceleration), resulting in a non-causal correlation between the two variables; and eminently reasonable and acceptable assumptions.
But, in non-experimentation, assumption (3) becomes, “assume that, during the study, no other variable or variables caused both independent and dependent variables, resulting in a spurious (i.e., non-causal) correlation between them;” a vastly more suspect assumption.
In the first experiment, above, the research was perfectly controlled. The only two relevant variables that changed were force and acceleration. In non-experimental research….
No valid causal conclusion can be drawn from this study by the use of Classical Statistics because the only way to make valid causal inferences from this non-experimental study is to use Causal Statistics, either explicitly or implicitly.
A management researcher, unsatisfied with the problems inherit in non-experimental studies, could avail himself of the option to perform an experimental study, likely in a lab and using students. Such a study would be experimental and allow for valid causal inferences, but the artificial, over simplified environment would greatly limit the generalizability(sp?) of any causal findings. This is often the dilemma in the non-experimental sciences, (1) to experiment in an artificial environment and obtain causal inferences of questionable generalizability or (2) to carry out a non-experimental study in out carry a rich environment and relinquish the possibility of reaching valid causal conclusions.
Enter Causal Statistics. The proper use of Causal Statistics can distinctly diminish these problems for non-experimental researchers. I developed Causal Statistics 40 years ago, spend 10 years, when I wasn't teaching, trying to push the idea and to get funding, without notable success, and eventually went on to other things.
Now, I’m back, with a fresh look and new insights and approaches. Sad to say, 40 years ago, I thought people, even highly educated people, were a whole lot smarter than they turned out to be. Also, I now know that 99% of academics, funders, etc. look at a new, eclectic, and revolutionary idea to find what might be wrong with it and not how important it would be, if successful, and not how any apparent problems could be overcome to make it successful.
I now know, even highly educated, smart people have to be lead by the hand through the whole continent of a huge, revolutionary idea and through solutions to all the potential problems.
Consider the 50 year fight to “prove” that tobacco use causes lung cancer. If a valid causal inquiring system had been used, rather than Associative Statistics, it shouldn’t have taken more than 10 years, even with the disinformation campaigns mounted by tobacco companies. How many lives would have been saved? The ignorance of and/or the act of ignoring Causal Statistics are not a meaningless condition, equivalent to the argument of how many angles can dance on the of a pen, but a scientific/methodological gaf of high gravity, with great human and financial costs.
A paragon of the problems flowing from the application of Associative Statistics, and the non-use of Causal Statistics to analyze non-experimental data, is a program called Reading is Fundamental (RIF) which has been in existence for 41 years and is “the nations largest children’s literacy organization.” The original research found a correlation between children’s reading performance (R) and the number of he/she owned (B).
RIF was formed to increase book ownership so Johnny would read better, based on the invalid inference, not explicitly stated, that B causes R.
How did the founder of RIF know that correlation between B and R wasn’t due to the fact that R caused B? Looking a little further a field, how did they know that socioeconomic level of the child (S) didn’t cause both B and R, leading to a spurious (i.e., non-causal) correlation between B and R?
40 years, 300 million books distributed, and millions of people duped by bad research, bad scientists, bad research consumers, and bad statistical inference.
RIF exemplifies two errors in causal inference that are common in non-experimental research. The first error is that causal inferences are often drawn, both explicitly and implicitly, by scientists who almost as often deny, cover up, and/or are unaware that, in their results, causal connections between variables are either stated or implied.
The RIF error was accomplished by (1) not specifically stating that B causes R, but saying that kids who own books are better readers, misleading the majority of human minds to jump to their own invalid causal inferences, and (2) then proceeding to develop a program and establish an organization, as if it had been proved that B causes R.
Other researchers hide the fact that they are making causal inferences (typically in an invalid manor) by not using the word “cause,” but by using synonyms like “yields,” “results in,” “produces,” “brings forth,” “brings out,” “creates,” “effectuates,” “elicits,” “is due to,” “generates,” “induces,” “leads to,” “makes,” and more.
The second error made, by those who claim (either explicitly or implicitly and either knowingly or unknowingly) to have discovered a causal connection, is that the assumptions, upon which the causal conclusions are based, are virtually never stated or even known. In the RIF example, they could validly conclude that B causes R if they were willing to state the assumptions (1) that R did not cause B and (2) that no other variables caused both R and B. Under these assumptions, a researcher could make the valid causal inference that B causes P, but few would be deceived into accepting the conclusion because most people would be unwilling to accept one or both of the required assumptions and, of course, if even one of the assumptions is invalid, then the conclusions are invalid.
IV. A Brief History of Non-Experimental Causal Inference
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Dissertation, Part I
WP7
V. The Need for a New Causal Inference Tool
Give Researchers algorithm for applying Causal Statistics
WP6 Paragraph 1-4
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VI. Advancing by Returning to the Beginning
The dissertation. Started the process.
Here we shall extract and surpass.
Induction vs. deduction.
[ III. Foundations of Causal Statistics]
At its core, Causal Statistics is a philosophically, logically, and statistically based causal inquiring system. Knowledge, techniques, and original developments from these three disparate fields are blended to form the foundation upon which Causal Statistics is constructed.
Causal Statistics is an axiomatic/deductive construct, in the way that Euclidian Geometry is an axiomatic/deductive construct. Like Geometry, Causal Statistics begins with definitions and basic assumptions. From these “first principles” and through the application of logic, the construct of Causal Statistics was derived deductively in my 1972 dissertation, entitled Foundations of Mathematical Epistemology: a Derivation of Causal Statistics, presented above.
But why would research methodologists, empirical researchers, theoreticians, and research and theory consumers want to have Causal Statistics derived? especially since the derivation resulted in mathematical forms very similar to those of regression analysis and econometrics.
Similarly, one might ask why is it desirable for Euclid to derive Geometry (Certainly the theorems came before the axioms and derivations.), for Einstein to derive Relativity, for Whitehead and Russel to derive Arithmetic and Algebra in Principia Mathematica, and others to derive Classical and Bayesian Statistics?
Consider Einstein’s contribution with his derivation of Relativity. Much of the mathematical formulations of Special Relativity, the Lorentz Equations, were already known from experimentation. But nobody understood why these equations were the way they were; i.e., why, for fast moving objects, their length shortened; their mass increased; and time slowed down. So the equations couldn’t be applied with understanding. Einstein, rather judiciously, formulated a set of assumptions (e.g., the measured velocity of light is constant, no matter what the velocity of the observer) about the nature of the universe and derived the Lorentz Equations and then went beyond the empirically known, to derive E=mc².
Causal inference was in a somewhat analogical situation, in that those few people who dared consider causation thought they knew some of the mathematical formulations, but they couldn’t be sure and couldn’t apply these formulations with understanding. My derivation of Causal Statistics, in combination with my present task, i.e. the extraction from the dissertation of a generally usable formulation of a complete causal inquiring system, should enable researchers to make causal inferences with understanding and confidence. The derivation within the dissertation is logically sufficient for this, but the extraction of the essence of Causal Statistics from the dissertation is far too complicated for most non-experimental research methodologists and researchers. So, I will do the extraction herein and present Causal Statistics with an intellectually accessible structure, which can be readily applied.
The application of this Causal Statistics paradigm to non-experimental data will enable researchers to draw valid causal inferences with complete understanding of the basic definitions and assumptions utilized, the use of prior information, and the proper interpretation of the resulting causal coefficients.
[IV. Foundation, Codification, and Exemplification of Causal Statistic]
At its core, Causal Statistics is a philosophically, logically, and statistically based causal inquiring system. Knowledge, techniques, and original insights in these three disparate fields are blended to form a foundation from which Causal Statistics can be derived.
My dissertation formally derived the mathematical structure of Causal Statistics from a particular, useful definition of cause and a set of basic assumptions about the nature of the universe, e.g., the basic laws upon which the universe operates are micro causal laws acting between adjacent fundamental particles.
It is important to note that the structure of Causal Statistics is not dependent on the exact assumption set used in the dissertation. I could have assumed that the most fundamental operation of the universe was (1) quantum mechanical in nature, with the micro causal laws being stochastic in nature or (2) based on micro causal interactions between string theory “particles.” Such varied micro causal assumptions would lead deductively, to the same mathematical macro structure for Causal Statistics, because random fluctuations at the micro level would average to virtually zero at the macro level, in accordance with the Law of Large Numbers.
That is a mathematical/statistical way of looking at the jump from the micro world to the macro world. Another approach to the same conclusion would be: if the three different theories about the fundamental operation of the universe (exact, Quantum Mechanical, and String Theory micro causes) lead to different macro mathematical formulations of Causal Statistics, they would also lead to differences in the operation of the macro universe. Therefore, by studying the macro universe, one could infer back to which of the three theories of the fundamental operation of the universe was correct.
But we can’t do that, so the three theories must be observationally equivalent at the macro level and the macro mathematical structure of Causal Statistics would also be the same for all three. Hence, the derivation of Causal Statistics is quite robust, with regard to differences in the fundamental micro assumptions about the nature of the universe.
VII. Attempting to Move Beyond the Beginning
Dissertation in Library of Congress and University of California Berkeley Graduate Library
Best year of my life. I learned so much out of my own head, I could almost become a rationalist.
WP6
During the 10 years I was working on and with Causal Statistics, beginning 40 years ago, I expected, with increasing disillusionment, that funding agencies, like the NSF, would shower me with research money and that non-experimental research methodologists, researchers, and research consumers would immediately grasp the importance and benefits of the subject and extract from the dissertation an application oriented formulation of Causal Statistics because of the great need for and the exceptional importance of this inquiring system. I expected that they would analyze all the philosophical details, expand and further develop the formulation of Causal Statistics, and apply it in almost all non-experimental research studies.
Boy! Was I mistaken. Most scientists showed little or no interest or understanding. The most common reactions were avoidance or ridiculous, naive criticisms. Research funders were polite, but felt that I should try some one else, anyone else, but them. Causal Statistics was much too eclectic for them.
I began wondering of they were all insane. Oh, that didn’t sound to good. Insane people usually think everyone else is crazy. Maybe it was me.
After ten years, I give up and went on to other things. Now, I’m back. When I looked at what’s been done in the intervening years, I was surprised. There is much more talk now, but not that much improvement in insight or over all progress.
As I read over the dissertation, I was motivated anew at the need for and importance of Causal Statistics. Some times you read something you wrote 40 years ago and it seems hopelessly off the beam and naive. There was a little of that, but, overall, I was struck by the insights and clarity with which the whole thing was pulled off and the feeling that the axiomatic/deductive approach to the development and overall understanding of Causal Statistics was the best one and that this approach is far superior to anything in the literature.
Maybe I was crazy then, but if so, I still am. I believe 100 years from now, assuming I don’t die before I can get the word out, that Causal Statistics will be considered the most important contributor in the development of the non-experimental sciences.
I’m back, but back with some differences. I now don’t see the reticent scientists and funders as insane. I now capabilities, their understanding of the needs, their ability to grasp such a large and complicated system as Causal Statistics, and their ability to rap their minds around such an eclectic, revolutionary, all-encompassing approach to the subject.
(Should have written papers, published empirical c. s. res., etc. That didn’t help S. Wright and others.)
VIII. Extractions and Additions\
Give Researchers algorithm for applying Causal Statistics
VIII. A. Philosophical and Epistemological Foundations (Background)
[IV. A. The Philosophy and Nature of Causality
IV. A. 1 Background]
Causality is a philosophical subject of long standing, with definitive of cause, typologies of causation, methods for establishing cause, etc. all over the ballpark.
2300 years ago, Aristotle posited four types of causes. His typology was largely based on the Greek language usage of the word and has little relevance to the conception of causality useful for developing scientific theories.
VIII. A. 1. [ IV. A. 1. a.] The Empiricists
Extreme Empiricism is a theory of knowledge which holds that all knowledge comes from experience, i.e., though sense impressions of the natural world.
In the 17the Century, empiricism was centered in Britain, with devotes like Lock, Hume, and Berkeley. Lock recorded the first explicit formulation of the doctrine of empiricism. Lock felt that at birth the mind was a clean slate (tabula rasa) upon which experience writes. Such beliefs would deny innate ideas, like the concept of causality. So, if the concept of causality is not innate, it must somehow spring from he natural world, through the senses, and into the mind as a fully formed concept, since, in the extreme empiricist's view, rational thinking or analysis can add nothing to an idea or concept that was not already there in the sense impressions.
VIII. A. 2. [IV. A. 1. b.] The Rationalists
In its purest (i.e., most extreme) form, rationalism looks to reason alone as the source of all knowledge. For rationalist philosophers, Euclidian Geometry was the paragon of their discipline, beginning with definitions and axioms and using deductive reasoning to derive Geometry.
In this process, they saw no use of information obtained through the senses from the physical universe. Hence, rationalism was their science a method, based on pure reason. Causality is not observable.
Extreme rationalist philosophers believed (1) the concept of causality to be a prior knowable, without reference to experience and (2) that knowledge of specific cause and effect relationships was determined a prior]??
Experiment: push ball 1 into ball 2 once. Assume nothing caused both balls to move. It could be that a que from below, pushing thru the table. Therefore, we cant deduce causal connection between ball 1 and 2.
VIII. A. 3. [IV. A. 1. c.] The Eclectics
[IV. B.???? Certainty of any Causal Connection and the Concept of Causality]
Hume presented two different arguments, both or which yield the same conclusion, namely that it is impossible to be certain that any two objects are related causally. This conclusion is considered by an amazingly large number of scholars to wield a death blow to the concept of causality.
Even though correct, Hume's conclusion is not necessarily a fatal blow to the usefulness of the concept. What percentage of the "knowledge" employed daily is know with certainty and/or exactitude? Epistemology recognizes relatively few items of certain knowledge and, in fact, there may be none. Can you be sure that the sun will rise tomorrow?
If Bayesian Statistics were employed to infer from a sample correlation to its population correlation, the hypothesis that the population correlation is zero, may be rejected at the .05 level. Have we established, with certainty, that the population correlation is not zero. No! But, based on the assumption of no measurement error and other assumptions, we can conclude, with a confidence of 95% that the population correlation is greater than zero. This is not certainty, but useful information.
Hume believed that a billiard ball could be perceived through the senses, without further mental or logical manipulation. If a rolling billiard ball hits another and the second ball moves off toward a pocket, the observer will likely attribute cause and effort to the event. As before, the balls are perceived through the senses, but the attribution of cause and effect is a result of metal processing and not simple sensation or observation. Hume would say that causality cannot be observed and is, therefore, a mental, i.e., rational, construct or concept. That being said, the inference of a specific cause and effect relationship is the result of the alchemy of (1) the rational concept of causality (and its definition); (2) various assumptions about the nature of our universe, about the relationships among variables (both considered and unconsidered), and about the research design (e.g., no measurement error); (3) the observable co variations among the study variables (i.e., data) (empiricism).
[ 4. Causal Usefulness]
Hume denied our ability to obtain certain knowledge of a causal connection. Here, I would agree with Hume and go much further to assert that we cannot obtain certain knowledge of almost any thing, including associations. Associations possess statistical error, measurement uncertainties, sampling error, etc.
But, from a pragmatic point of view, Hume admits the usefulness and universality of the concept of causality:
"...it may still, perhaps, be rash to conclude positively that the subject, therefore, pass all human comprehension....It is certain that the most ignorant and stupid peasants--nay, infants; nay, even brave beasts--improve by experience, and learn the qualities of natural objects, by observing the effects which result from them. When a child has felt the sensation of pain from touching a flame of a candle, he will be careful not to put his hand near any candle; but will expect a similar effect from a cause which is similar in its sensible qualities and appearance.”*
------------------------------------------------------------------------------------------------------------*Hume, David: ENQUIRIES, Second Edition, Oxford at the Clarendon Press, MDCCCCII, p. 38-39. ------------------------------------------------------------------------------------------------------------
This is the point at which many scholars misinterpret Hume. They see his conclusion that there can be no certainty of causal connections, but do not comprehend the distinction he draws between certainty and usefulness.
In statistical terms, we cannot prove causal connections, but we can be 99+% confident of the usefulness of a causal influence, based on an appropriate set of assumption.
Hume explains the seeming conflict between the philosophic and the pragmatic points of view, asserting that, based upon the experience of a constant conjunction between flame and heat, "the mind is carried by custom to expect heat"** from a flame. ------------------------------------------------------------------------------------------------------------**Ibid., p.46 ------------------------------------------------------------------------------------------------------------ "All inferences from experience, therefore, are effect of custom, not of reasoning."*** ------------------------------------------------------------------------------------------------------------***I bid., p.43
------------------------------------------------------------------------------------------------------------
Hume asserts that the effect of custom upon the mind, in overcoming reason is "an operation of the soul."**** ------------------------------------------------------------------------------------------------------------****Ibid., p.46 ------------------------------------------------------------------------------------------------------------
With today’s understands, we would simply state that human and even animal minds evolved to infer causal connections, even though uncertain, because such inferences were useful for their survival.
[5. Observability
6. Observationally Equivalent Concepts/Theories]
Causality is one concept to explain the observed activity in the universe. An alternative concept is God. One could assert that God controls everything and there is no cause and effect relationship operating between the 2 billiard balls. God only makes it look that way.
These are two observationally equivalent theories of the operation of the universe. Either concept could be correct and the observed universe would look the same.
This puts the lie to the empiricists’ assertion that causality and causal connections are observable. Obviously, causality is a mental construct, generated to explain (account for) the observed behavior of objects within the natural world.
Hence, the empiricists and rationalists were both partly correct and partly incorrect. The concept of causality is a non-observable, rational construct, but the concept was created to explain observed activity in the universe. Therefore, the concept of causality ultimately arises from both rational and empirical inputs.
As with the concept of causality, specific causal connections cannot be directly observed either, but nor can valid specific causal correlations be inferred without empirical input.
Sample correlations are empirical results. One could correctly argue that the calculations of correlation coefficient are a mental process. But associates can be perceived without the calculation of correlation coefficients. On the other hand, rationalists might claim that remembering multiple observations and putting the multiple observations together to perceive an association are mental acts. I would agree with this, but a causal inference is a different level of mental processing. Association or correlation requires memory and calculation. Causal inferences require these plus an appropriate definition of cause plus assumptions about the causal nature of the universe plus, in non-experimental research, assumptions about unconsidered variables and the relationship of all variables to each other.
What the construct of Causal Statistics does is to routinize (or codify or make into an algorithm) the handling of data (mostly correlations), the definition of cause, the assumptions, and the calculations required to arrive at valid causal inferences.
One could say that valid causal inferences could be drawn without Causal Statistics, simply using Classical Statistics to calculate correlations and then by doing all of the other above activities correctly without reference to the construct of Causal Statistics. This is true and that is what researchers have been trying to do, using Classical Statistics in the absence of the knowing that Causal Statistics exists, without much success. First, in the best cases, they don’t know all of the inputs required. More likely, they aren’t even aware that these additional inputs are required.
Attempting non-experimental causal inferences without Causal Statistics is like not using algebra to determine four unknowns, given sufficient relationships among the variables. But if algebra exists, no one would attempt the task without using algebra. Some with Causal Statistics.
Other of observationally equivalent theories are possible. Consider the possibility that there is no causality and no God and that the behavior of the universe is simply a result of extremely highly improbable random occurrence.
One such random occurrence might be that a billiard ball roles up and stops next to (i.e., touching) a second billiard ball and, on a random fluke, the second billiard ball immediately begins to rolled away toward a pocket. No causality, no God; just random occurrence. Note again that causal connections cannot be observed, only inferred.
Bayesian Statistics would consider it highly improbable that the operation of our universe could be based on the occurrence of so many highly improbable fluke random events, like the probability of getting heads one Trillion Trillion… times in a row. Highly unlikely, but the possibility cannot be rules out with certainty.
VIII. A. 4.
[7.
]
Definition of Caus
VIII. B. [B.] Logic
Secondly, Causal Statistics is a logical construct, in the sense that Euclidean Geometry is a logical construct. It is founded on appropriately chosen definitions and assumptions and its final formulation is derived through deductive logic.
The necessary assumptions can be grouped into three sets. The three assumptions sets could be broadly, but somewhat imperfectly (i.e., over simplified), labeled as follows: fundamental assumptions, isolating assumptions, and statistical assumptions.
VIII. B. 1. The First Assumptions Set
The first and most fundamental assumption set postulates that we live in a causal universe, where cause and effect governs the behavior of variables, at least at the macro level. These assumptions are presented in chapters 7, of the dissertation, below.
VIII. B. 2. Second Assumptions Set
The second assumption set is for the purpose of isolating and/or for limiting the interactive freedom of variables considered in a non-experimental study. These assumptions must be judiciously chosen in ways that will allow valid causal inferences.
VIII. B. 3. The Third Assumption Set
The third assumption set deals with the typical concerns of any statistical study: measurement error, sampling bias, etc. With regard to this set, there is no real difference between Causal Statistics and Classical Statistics.
IX. [1)]
A Two Variable
Eexample
of Causal Inference
Identification pb
Recursive equations
Endogenous
Exogenous, etc Dissertation, Chapter 13
Estimations
Parameter interpretations
Error handling
Structure and structural Delta
Start with general form of Causal Statistics
Recommend econometric sources and note their avoidance of “cause”
Simultaneous equation parameters partial out the considered correlation
Give researchers algorithm for applying Causal Statistics
Consider an empirical study in which variables B (i.e., number of book owned by the family) and R (i.e., reading capability of the child) are found to be correlated. If we assume (1) that no outside variables causally affect both B and R in such a way as to change their correlation, (2) that R does not cause B, and (3) that causal relationship are linear; one can validly conclude, subject to the usual statistical error, that B causes R with standardized strength equal to the correlation coefficient.
A “valid” causal inference is a causal inference which is a logically necessary result of the definitions stated, the assumptions made, and the data collected. In other words, a “valid” causal connection results from a causal inference arrived at by the proper application of Causal Statistics.
Note that a valid causal inference does not necessarily result in a correct causal connection. If one or more of the assumptions is incorrect, the causal connection will be incorrect. The greater the degree to which the assumptions are in error, generally, the greater the error in the causal connection drawn. This second assumption set can be very restrictive and questionable, yet, that is not the fault of Causal Statistics. It is, if you will, the fault of logic and the universe we exist in. Gravity can be inconvenient, if you want to fly, but the field of physics is not at fault. In fact physics can assist in overcoming the problem; same with Causal Statistics.
X. [2)] A Three Variable Example of Causal Inference
Give researchers algorithm for applying Causal Statistics
Explain Hypothesis, testing vs. correlation matrices and the use of matrices to choose hypothesis.
Exemplify the 2 various research and associations, leading to the 3 variable research, leading to 10 variable research, etc. Using prior data. Someone else can figure out how or I’ll do it when I have time
Simultaneous experimental parameters partial out the considered correlation
XI. A Ten Variable Example of Causal Inferences (A Ten Variable Follow-on Example of Causal Inference)
In the above example, one way to decrease the stringency of the required assumptions, would be to bring one or more of the relevant outside variables, inside. For example, collect data on B, R, and S (i.e., socioeconomic status of the family).
In this case, it would be perfectly reasonable to assume that B does not cause S and that R does not cause S. Further, we could assume that no outside variables causally affect B, R, or S in such a way as to change the correlations among any of the three considered variables and that any causal connection that do exist among the inside variables are linear.
The resulting possible causal connections can be represented by two simultaneous equations:
B = a + br + cs + d
R = g + hB + is + e
where a, b, c, g, h, and i are causal coefficients which can be determined by applying econometric estimation techniques to the studies data, and d and e are error terms. The error terms might be assumed to be normally distributed around a zero mean.
To use econometric estimation techniques, the system of equations must be “identified” (Google “econometric identification.”), analogous to having the same number of equations and unknowns in algebra. The above equation set is under identified. Therefore, econometric estimation cannot really occur until further assumptions are made and/or prior information (i.e., knowledge) is inserted, resulting in exact identification.
?? 3) Causal Statistics is not for the faint of Heart, nor the weak of Mind
Oh, so you think this sounds difficult, complicated, and renders even valid causal inferences highly suspect? I’m sorry but making valid causal inferences from non-experimental data is not for the faint of heart, nor the weak of mind.
Again, it is not Causal Statistics’ fault. Causal Statistics is an optimized system, doing the best it can with the logical and physical universe it was given. Maybe some parallel universe would make Causal Statistics as simple to apply as Classical Statistics, but getting to that universe would likely be even more difficult than properly applying Causal Statistics.
I dare say, you have probably never before seen non-experimental causal inferences drawn in this way. What do you think this means? I would suggest that you have probably never before seen valid causal inferences drawn.
This is not to say that no correct, non-experimental causal connections have been established in the past 100 years. But, I would say that the number that are as accurate as would have been possible with the proper application of Causal Statistics, is far less than 1% and the number of non-experimental causal inferences that are correct to any level of accuracy is probably less than 10%.
Further, our efficiency in determining causal connections has been dismal. Consider the 50 year debate over the causal connection between smoking and lung cancer.
Now, it should be clear why I have complained of the vast financial and intellectual waste in the non-experimental sciences over the past 100 years.
As you may or may not remember, we were discussing the second assumption set. Now, on to the third.
?? C. Statistics
Now, back to the three disparate fields upon which Causal Statistics is based. Thirdly, the final formulation of Causal Statistics is statistical, in that the results are subject
to calculatedly [ ] sampling error, just like the results from Classical and Bayesian Statistical studies.}
(Add parts of WP3)
XII. [V.] How does C S Fit into the field of Statistics, into the Social Sciences, and into Epistemology
XII. [V.] A. The Relationship among the Three Statistical Paradigms
Classical Statistics, Bayesian Statistics, and Causal Statistics; like virtually every major mathematical discipline; are a/d constructs. These three statistical paradigms are similar to, but different from, each other; akin to the way that Euclidian and the various non-Euclidian Geometry’s are similar to, but distinct from, each other.
Classical Statistics is concerned with making inferences from sample statistics to population statistics; like means, correlations, and standard deviations. Causality is not considered by Classical Statistics; nor is “cause” or any of its synonyms within the vocabulary of Classical Statistics. Therefore, any consideration of causal inference is outside the domain (i.e., the area of applicability) of Classical Statistics. Researchers, who attempt to use Classical Statistics alone to draw valid causal inferences from non-experimental data, as many have done, are bound to fail.
Causality and causal inference is at the heart of the domain of Causal Statistics and it is the only complete inquiring system designed specifically for making causal inferences.
At this point the question often arises, “Which one of the three is correct?” The answer is that each is correct or useful with in its own domain, like an individual’s depression is an issue for psychology, but not for sociology or economics. Causal Statistics is absorbed with making valid causal inferences and Classical Statistics is totally unconcerned with causality in any way.
Classical Statistics and Causal Statistics are like Physics and Chemistry or like Sociology and Psychology. The pairs are not in conflict with each other; they deal mostly with different issues; and each is applicable and useful within its own domain. Classical Statistics and Causal Statistics are not different pieces of the same thing. They are different things; different definitions, different assumptions, different concepts, different interpretations, different domains; with some similarities in mathematical form.
So, it’s not a question of which one is right or which one is wrong. Both Classical Statistics and Causal Statistics are right and correct and usefully and usable in their own domains. It’s when scientists use one in a domain it’s not suited for, that the validity of their research suffers, like using Classical Statistics on non-experimental data, in an attempt to draw causal inferences.
On the other hand, it would not be improper to use both statistical paradigms together, if each addresses itself only to elements within its own domain.
For example, a researcher might use Causal Statistics to extract a causal coefficient from non-experimental data and utilize Classical Statistics to infer from the sample causal coefficient to the population causal coefficient: Alternatively, one could use Bayesian Statistics rather than Classical Statistics and calculate a Bayesian confidence interval for the causal coefficient.
Aside: Ultimately, these are simply two different paradigms, based on different definitions of probability. So who is right? The answer is that neither of these logical constructs is absolutely right or wrong. The appropriate question is, “Which one is most useful and in what situations, when exposed to empirical applications (i.e., reality)?” Both Euclidian and some Non-Euclidian Geometries work well in our everyday experience and with Newtonian Mechanics, but Euclidian Geometry breaks down when applied to Einstein’s Relativistic universe. There, it turns out, that one of the Non-Euclidian Geometries works best for both. It turned out that Euclidian Geometry is just a very good approximation in our everyday experience, but ultimately wrong.
Carrying things a little further, even Relativity breaks down when applied to particles at the atomic scale and the quantum mechanics paradigm takes over. Yet, neither can explain the behavior of matter at the center of a black hole, a place somewhat removed from everyday experience.
?? Aside: Are the assumptions true?
Are the def’s true? True is not relevant, useful is better.
?? Aside: Plane Euclidean Geometry. 3D by solid Geometry. Extension because no conflict. Thought to explain the world Non-Euclidean Geometry.
?? Aside: Causal Statistics could use Classical Statistics or Bayesian Statistics for handling the statistical element of Causal Statistic, like hypothesis Test, Type I and II error, confidence intervals, etc. I will use Classical Statistics for that because it is much more familiar, although a little less meaningful.
WP3
XII. A. [VII.] How does Causal Statistics fit into the Pantheon (Mosaic) of other Statistical Paradigms?
To understand the importance of the fact that Causal Statistics and Classical Statistics are different axiomatic/deductive constructs, let us step back for a moment and look more broadly at the fields within the realm of statistics. For a discussion of how Causal Statistics and Associative Statistics are related and how the aforementioned causal inquiring system is founded on these three, apparently dispirit, disciplines, see Working Papers #3 and #4, below.
XII. B. [A.] Understanding the Nature of Mathematical and scientific “Knowledge.”
?? B. Where Does the Dissertation Fit in?
In science and math, the big, encompassing theories and mostly logical, deduct paradigms; like Causal Statistics in my dissertation, like Euclid derived Geometry, like Einstein derived the Theory of Relativity, like Whitehead and Russel derived Arithmetic and Algebra in Principia Mathematica, and others to derive Classical and Bayesian Statistics.
The interested reader might ask, “So what?” and the uninterested reader has probably already moved on to taking out the trash or to a porn site. Anyway, for you that me left, back to, “So what?”
Consider Einstein’s contribution with his derivation of Relativity. Much of the mathematical formulation of Special Relativity, the Lorentz Equations, were already known from experimentation. But nobody understood why these equations were the way they were; i.e., why, for fast moving objects, their length shortened; their mass increased; and time slowed down. So the equations couldn’t be applied with understanding. Einstein, rather judiciously, formulated a set of assumptions (e.g., the measured velocity of light is constant, no matter what the velocity of the observer) about the nature of the universe and derived the Lorentz Equations and then went beyond the empirically known, to derive E=mc².
I did a similar thing in deriving a causal inquiring paradigm, which I call Causal Statistics. The application of the Causal Statistics paradigm to non-experimental data can now enable researcher to draw valid causal inferences with complete understanding of the basic definitions utilized, the assumptions made, the use of prior information, and the proper interpretation of causal coefficients. See Working Papers #___ for a simple, but complete, application oriented formulation of Causal Statistics.
XII. A. [C.] The Logical Constructs within Statistics and their Relations to each other
Some will still argue that Causal Statistics conflicts with Classical Statistics and, hence, Causal Statistics must be wrong or that Causal Statistics is an invalid use of Statistics. My response is that Causal Statistics deals with causality and Classical Statistics does not. So they are not in conflict for the same reason that chemistry is not in conflict with nuclear physics and sociology is not in conflict with psychology. They are different paradigms dealing with different aspects of the universe. Classical Statistics and Causal Statistics are not different pieces of the same thing. They are different things (different definitions, different assumptions, different interpretations, different domains) with similarity in mathematical form.
Concerning the certainty that people will call Causal Statistics an invalid statistical technique: it is a logical construct, like Relativity or Classical Statistics are logical constructs. Therefore, it is not per se valid, because it is internally consistent. Yet, the real proof of the pudding for any logical construct, like Geometry is in the eating. When applied to the real world, are the results useful? I believe that Causal Statistics will eventually be proved so useful that it will revolutionize non-experimental research.]??
?? Aside: Objective: to give all people all understandings necessary to understand context, nature, etc of Causal Statistics and to apply Causal Statistics.
??[IV.]
?? VIII. How does Dissertation, fit into the field of Statistics
Causal Statistics is axiomatic/deductive constructs like virtually all other math. Dissertation derives Causal Statistics like contribution of Einstein.] ??
(No??) [IX. The Importance of Causal Statistics]
XIII. [X.] Why is it Taking so Long? And how much longer will it take?
Dissertation in Library of Congress and University of California Berkeley Graduate Library
Best year of my life. I learned so much out of my own head, I could almost become a rationalist.
XIV. [XI.] Where do we go from Here?
Dissertation, Chapters 14, 1, or 2
A. Get researchers using Causal Statistics
1. Develop a simple presentation of the Causal Statistics paradigm.
2. Answer all objections.
3. Publicize.
4. Free application consulting.
5. Book? With basics and examples
6. Funding: only after it’s no longer innovated and is acceptable. When it’s too late, there will then be adequate money.
Causal Inference via Causal Statistics: Causal Inference with Complete Understanding [no loose ends, with deductive certainty]
Table of Contents
I[c29]. Background, Setting, and Context
the purpose of this book is to design an algorithm for properly applying causal statistics in nonexperimental research so that, after reading this book, any nonexperimental researcher will be able to apply causal statistics to their work correctly and with understanding. In furtherance of that purpose, the book should answer any questions and handle any problems that the researcher is likely to encounter in either application or understanding.
I.1
[c30]II. The Need for Non-Experimental Causal Inference
Dissertation in Library of Congress and University of California Berkeley Graduate Library
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WP6 Paragraph 1-4
WP7
e
III. [c31]Causality and the Empirical Research Environment
e
Aside:
It is my tendency to present a subject by starting at the beginning and
proceeding in chronological/logical order to the end/conclusion, Q.E.D.
Although logical, this approach is pedagogically problematic. The problem
is that, at best, the reader usually knows only generally where he is
going, but not precisely. Hence, subtleties in the step-by-step
presentation are not appreciated when they are read, and, as a result, not
committed to memory and/or not place in the proper juxtaposition to other
elements for reaching a well-reasoned, logically tight conclusion.
Hence, you might consider reading the concluding chapter
and
then return to chapter II with a comprehension of exactly where you are
going.
III. A. Prediction, without Intervention
III. B. Prediction, with Intervention
III. C. Experimentation
III.C.1. Non-Statistical Causal Inference from Experimental Data
III.C.1.a. Validity and Correctness
See [IV.B]
e
III.C.1.b. Causal Modeling/Theory Building
A causal theoretician, looking at these experimental results, would likely conclude that force causes acceleration in an amount equal to F/m or according to the equation, a=F/m.
III.C.2. Statistical Causal inference
? III. C. 2. a. Validity and Correctness
III. C. 2. b. Modeling/Theory Building
III.C.3. Conclusions
III.D. Non-Experimentation
[c32] IV. A Brief History of Non-Experimental Causal Inference
WP5
Dissertation, Part I
WP7
[c33] V. The Need for a New Causal Inference Tool
Give Researchers algorithm for applying Causal Statistics
WP6 Paragraph 1-4
WP7
[c34] VI. Advancing by Returning to the Beginning
The dissertation. Started the process.
Here we shall extract and surpass.
Induction vs. deduction
[ III. Foundations of Causal Statistics]
[IV. Foundation, Codification, and Exemplification of Causal Statistics]
[c35] VII. Attempting to Move Beyond the Beginning
Dissertation in Library of Congress and University of California Berkeley Graduate Library
Best year of my life. I learned so much out of my own head, I could almost become a rationalist.
WP6
e
[c36] VIII. Extractions and Additions
Give Researchers algorithm for applying Causal Statistics
VIII. A. Philosophical and Epistemological Foundations (Background)
[IV. A. The Philosophy and Nature of Causality
IV. A. 1 Background]
VII. A. 1. [ IV. A. 1. a.] The Empiricists
VIII. A. 2. [IV. A. 1. b.] The Rationalists
e
experiment: push ball 1 into ball 2 once. Assume nothing caused both balls to move. It could be that a que from below, pushing thru the table. Therefore, we cant deduce causal connection between ball 1 and 2.
VIII. A. 3. [IV. A. 1. c.] The Eclectics
[IV. B.???? Certainty of any Causal Connection and the Concept of Causality]
[ 4. Causal Usefulness]
Hume denied
[5. Observability
6. Observationally Equivalent Concepts/Theories]
VIII. A. 4. [7. ]Definition of Cause
VIII. B. [B.] Logic
Secondly, Causal Statistics is a logical construct, in the sense that Euclidean Geometry is a logical construct. It is founded on appropriately chosen definitions and assumptions and its final formulation is derived through deductive logic.
Assumptions
The necessary assumptions can be grouped into three sets. The three assumptions sets could be broadly, but somewhat imperfectly (i.e., over simplified), labeled as follows: fundamental assumptions, isolating assumptions, and statistical assumptions.
VIII. B. 1. The First Assumptions Set
The first and most fundamental assumption set postulates that we live in a causal universe, where cause and effect governs the behavior of variables, at least at the macro level. These assumptions are presented in chapters 7, of the dissertation, below.
VIII. B. 2. Second Assumptions Set
The second assumption set is
for the purpose of isolating and/or for limiting the interactive freedom
of
variables considered in a non-experimental study. These
assumptions must be judiciously chosen in ways that will allow valid causal
inferences.
VIII. B. 3. The Third Assumption Set
The third assumption set deals with the typical concerns of any statistical study: measurement error, sampling bias, etc. With regard to this set, there is no real difference between Causal Statistics and Classical Statistics.
[c37] IX. [1)] A Two Variable Example of Causal Inference
Identification pb
Recursive equations
Endogenous
Exogenous, etc Dissertation, Chapter 13
Estimations
Parameter interpretations
Error handling
Structure and structural Delta
Start with general form of Causal Statistics
Recommend econometric sources and note their avoidance of
“cause”
Simultaneous equation parameters partial out the considered correlation
Give researchers algorithm for applying Causal Statistics
e
[c38]X. [2)] A Three Variable Example of Causal Inference
Give researchers algorithm for applying Causal Statistics
Explain Hypothesis testing vs. correlation matrices and the use of matrices to choose hypothesis.
Exemplify the 2 various research and associations, leading to the 3 variable research, leading to 10 variable research, etc. Using prior data. Someone else can figure out how or I’ll do it when I have time
Simultaneous experimental parameters partial out the considered correlation
e
[c39]XI. A Ten Variable Example of Causal Inferences(A Ten Variable Follow-on Example of Causal Inference)
?? 3) Causal Statistics is not for the faint of Heart, nor the weak of Mind
?? c. The Third Assumption Set
The third assumption set deals with the typical concerns of any statistical study: measurement error, sampling bias, etc. With regard to this set, there is no real difference between Causal Statistics and Classical Statistics.
?? C. Statistics
Now, back to the three disparate fields upon which Causal Statistics is based. Thirdly, the final formulation of Causal Statistics is statistical, in that the results are subject
to calculatedly [ ] sampling error, just like the results from Classical and Bayesian Statistical studies.}
(Add parts of WP3)
[c40] XII. [V.] How does Causal Statistics Fit into the field of Statistics, into the Social Sciences, and into Epistemology
XII. [V.] A. The Relationship among the Three Statistical Paradigms
e
?? Aside: Ultimately, these are simply two different paradigms, based on different definitions of probability. So who is right? The answer is that neither of these logical constructs is absolutely right or wrong. The appropriate question is, “Which one is most useful and in what situations, when exposed to empirical applications (i.e., reality)?” Both Euclidian and some Non-Euclidian Geometries work well in our everyday experience and with Newtonian Mechanics, but Euclidian Geometry breaks down when applied to Einstein’s Relativistic universe. There, it turns out, that one of the Non-Euclidian Geometries works best for both. It turned out that Euclidian Geometry is just a very good approximation in our everyday experience, but ultimately wrong.
Carrying things a little further, even Relativity breaks down when applied to particles at the atomic scale and the quantum mechanics paradigm takes over. Yet, neither can explain the behavior of matter at the center of a black hole, a place somewhat removed from everyday experience.
?? Aside: Are the assumptions true?
Are the def’s true? True is not relevant, useful is better.
?? Aside: Plane Euclidean Geometry. 3D by solid Geometry. Extension because no conflict. Thought to explain the world Non-Euclidean Geometry.
?? Aside: Causal Statistics could use Classical Statistics or Bayesian Statistics for handling the statistical element of Causal Statistic, like hypothesis Test, Type I and II error, confidence intervals, etc. I will use Classical Statistics for that because it is much more familiar, although a little less meaningful.}
WP3
XII. A. [VII.] How does Causal Statistics fit into the Pantheon (Mosaic) of other Statistical Paradigms?
To understand the importance of the fact that Causal Statistics and Classical Statistics are different axiomatic/deductive constructs, let us step back for a moment and look more broadly at the fields within the realm of statistics. For a discussion of how Causal Statistics and Associative Statistics are related and how the aforementioned causal inquiring system is founded on these three, apparently dispirit, disciplines, see Working Papers #3 and #4, below.
XII. B. [A.] Understanding the Nature of Mathematical and scientific “Knowledge.”
?? B. Where Does the Dissertation Fit in?
XII. A. [C.] The Logical Constructs within Statistics and their Relations to each other
?? Aside: Ultimately, these are simply two different paradigms, based on different definitions of probability. So who is right? The answer is that neither of these logical constructs is absolutely right or wrong. The appropriate question is, “Which one is most useful and in what situations, when exposed to empirical applications (i.e., reality)?” Both Euclidian and some Non-Euclidian Geometries work well in our everyday experience and with Newtonian Mechanics, but Euclidian Geometry breaks down when applied to Einstein’s Relativistic universe. There, it turns out, that one of the Non-Euclidian Geometries works best for both. It turned out that Euclidian Geometry is just a very good approximation in our everyday experience, but ultimately wrong.
Carrying things a little further, even Relativity breaks down when applied to particles at the atomic scale and the quantum mechanics paradigm takes over. Yet, neither can explain the behavior of matter at the center of a black hole, a place somewhat removed from everyday experience.
?? Aside: Are the assumptions true?
Are the def’s true? True is not relevant, useful is better.
?? Aside: Plane Euclidean Geometry. 3D by solid Geometry. Extension because no conflict. Thought to explain the world Non-Euclidean Geometry.
?? Aside: Causal Statistics could use Classical Statistics or Bayesian Statistics for handling the statistical element of Causal Statistic, like hypothesis Test, Type I and II error, confidence intervals, etc. I will use Classical Statistics for that because it is much more familiar, although a little less meaningful.}
?? Aside: Objective: to give all people all understandings necessary to understand context, nature, etc of Causal Statistics and to apply Causal Statistics
?? [IV.]
[How does the dissertation fit?
?? { VIII. How does Dissertation, fit into the field of Statistics
Causal Statistics is axiomatic/deductive constructs like virtually all other math. Dissertation derives Causal Statistics like contribution of Einstein.] ??
(No??) [ IX. The Importance of Causal Statistics ]
[c41] XIII. [X.] Why is it Taking so Long? And how much longer will it take?
Dissertation in Library of Congress and University of California Berkeley Graduate Library
Best year of my life. I learned so much out of my own head, I could almost become a rationalist.
[c42] XIV. [XI.] Where do we go from Here?
Dissertation, Chapters 14, 1, or 2
A. Get researchers using Causal Statistics
1. Develop a simple presentation of the Causal Statistics paradigm.
2. Answer all objections.
3. Publicize.
4. Free application consulting.
5. Book? With basics and examples
6. Funding: only after it’s no longer innovated and is acceptable. When it’s too late, there will then be adequate money.
OLDer Table of Contents
0.1 In a Nutshell, What is Causal Statistics and How Important Is It?
I. Origin, Background, Setting, and Context of Causal Statistics
I.1 Causal Statistics: A Two Variable Example
I.2 Causal Statistics: A Three Variable Example
II. The Need for Non-Experimental Causal Inference (Put in I and or III)
III. Causality and the Empirical Research Environment
III. A. Prediction, without Intervention
III. B. Prediction, with Intervention
III. C. Experimentation
III.C.1. Non-Statistical Causal Inference from Experimental Data
III.C.1.a. Validity and Correctness
III.C.1.b. Causal Modeling/Theory Building
III.C.2. Statistical Causal inference
? III. C. 2. a. Validity and Correctness
III. C. 2. b. Modeling/Theory Building
III.C.3. Conclusions
III.D. Non-Experimentation
IV. A Brief History of Non-Experimental Causal Inference
V. The Need for a New Causal Inference Tool
VI. Advancing by Returning to the Beginning
VII. Attempting to Move Beyond the Beginning
VIII. Extractions and Additions
VIII. A. Philosophical and Epistemological Foundations (Background)
VIII. A. 1. The Empiricists
VIII. A. 2. The Rationalists
VIII. A. 3. The Eclectics
VIII. A. 4. Definition of Cause
VIII. B. Logic
VIII. B. 1. The First Assumptions Set
VIII. B. 2. Second Assumptions Set
VIII. B. 3. The Third Assumption Set
[c9]IX. A Two Variable Example of Causal Inference
[c10]X. A Three Variable Example of Causal Inference
[c11]XI. Ten Variable Example of Causal Inferences (A Ten Variable Follow-on Example of Causal Inference)
?? 3) Causal Statistics is not for the faint of Heart, nor the weak of Mind
?? c. The Third Assumption Set
?? C. Statistics
[c12]XII. How does Causal Statistics Fit into the field of Statistics, into the Social Sciences, and into Epistemology
XII. A. The Relationship among the Three Statistical Paradigms
XII. A. How does Causal Statistics fit into the Pantheon (Mosaic) of other Statistical Paradigms?
XII. B. [A.] Understanding the Nature of Mathematical and scientific “Knowledge.”
?? B. Where Does the Dissertation Fit in?
XII. A. [C.] The Logical Constructs within Statistics and their Relations to each other
[c13]XIII. Why is it Taking so Long? And how much longer will it take?
[c14]XIV. Where do we go from Here?
XIV. A. Get researchers using Causal Statistics
1. Develop a simple presentation of the Causal Statistics paradigm.
2. Answer all objections.
3. Publicize.
4. Free application consulting.
5. Book? With basics and examples
6. Funding: only after it’s no longer innovated and is acceptable. When it’s too late, there will then be adequate money.
Old Preface
This book is intended for a broad range of readers: academics who specialize in causal inference, research methodologists, non-experimental researchers, non-experimental research consumers, statisticians, people interested in epistemology and/or the philosophy of causality, those interested in inductive and deductive systems, and others.
There are many ways to organize the book, in an attempt to effectively and efficiently communicate with readers. Unfortunately, the sequence of chapters that is best for one type of reader is not best for another.
Hence, I would suggest that you proceed in the following manner. First, you should read Chapter 1, i.e., Background, Setting, and Context, to get an idea of what lead up to the book. Then go to the Table of Contents and read the chapter titles and deeper if necessary, to get a feeling for the overall flow and content of the book. With that background, you can decide which chapters to read in what order, given your knowledge and background.
For example, a causal inference expert might have only scanned Chapter 2, The Need for Non-experimental Causal Inference. Many people might want to read Chapters 9, Two Variable Example of Causal Inference, and Chapter 10, A Three Variable Example of Causal Inference, early on to get a clear understanding of where they are going.
The prerequisite for reading and understanding this book is one course in statistics. Therefore this book can be read and understood by almost any college student. But it is intended for a broad range of readers: from any college student who has had one course in statistics to causal inference specialists.
This book is not basically a mathematical statistics, as one might initially think. The mathematics is basically the same as classical statistics or Bayesian statistics. Classical statistics measures the degree of Association or correlation. Correlation is simply a measure of what is observed.
A causal connection between two variables is not simply a matter of observation. Causality is many times more complex and complicated than Association. First, philosophers have posited many definitions of cause over the past 3000 years. I found none to be inadequate.
Second, causal inference requires some assumptions about the way the universe works, namely that it is causal. From the above, one could make causal inferences from experimental studies, properly controlled. But, to draw causal inferences from non-experimental data, a series of additional assumptions to make sure that the correlation between the two variables of interest is not a result of some outside, that is to say, unconsidered variable or variables causing both of the variables of interest and resulting in the observed correlation.
Causal Statistics is a deductive system like Euclidian geometry is a deductive system. The causal inference drawn from the non-experimental study is true if all the assumptions are satisfied, just like one can conclude that a triangle contains 180° if all the postulates, that is to say, assumptions, of Euclidian geometry are true.
In geometry, performing logical manipulations on Euclid's definitions and postulates results in the logical certainty that a triangle contains 180°, if the assumptions are correct. Similarly, if the assumptions of causal statistics and the assumptions made about the variables in the non-experimental study are correct, the definition of cause and the observed data, logically manipulated, results in a determinable degree of causal connection between the two variables of study.
The conclusion that A causes B, with a regression coefficient of .7 is a certain causal connection, almost. The causal connection is a certainty, logically flowing from the combination of the deductive system of causal statistics and observed data from the non-experimental study. But there are two flies in this ointment which removed a certainty. First, the assumptions required to perform the derivation may be incorrect. Second, he calculated causal regression coefficient of .7 will certainly contain statistical error.
The first deviation from certainty is the same as if one of Euclid's assumptions were determined to be incorrect. In the causal statistics case an incorrect assumption concerning omitted variables can be corrected by changing and/or adding other assumptions by performing a new and larger empirical study considering more variables, and in a number of other ways.
The second movement toward uncertainty is in the nature of all statistical studies and can be handled with the usual classical or Bayesian statistical techniques.
Aside: It is my tendency to present a subject by starting at the beginning and proceeding in chronological/logical order to the end/conclusion, Q.E.D. Although logical, this approach is pedagogically problematic. The problem is that, at best, the reader usually knows only generally where he is going, but not precisely. Hence, subtleties in the step-by-step presentation are not appreciated when they are read, and, as a result, not committed to memory and/or not place in the proper juxtaposition to other elements for reaching a well-reasoned, logically tight conclusion. Hence, you might consider reading the concluding chapter and then return to chapter II with a comprehension of exactly where you are going.
When I began to work on causal inference, the mathematics was largely standard and no real problem. It was the philosophical issues, definitions, assumptions, interpretations of the calculated parameters, etc. which were the sources of my initial confusion. These questions, as they applied to causal inference, were not answered anywhere in the literature.
I spent literally years thinking about these and many other issues related to causal inference. I would pose a question about one of the issues and then write a stream of consciousness monologue as to my thinking on the issue. Usually that monologue resulted in several other questions which were duly noted in the writing for return consideration at a later time or, if necessary, consideration prior to consideration of the original issue.
For example, what is the interpretation of a regression parameter in a causal study? After much thinking and writing, I concluded that a regression parameter is simply a measure of what is observed, that is to say, Association. It is the job of causal statistics to determine what percentage of that association is due to A causing B, the percentage you to B causing A (note that, if a causes be positively and be causes a negatively, the percentage of the Association attributable to these two causal connections will be less than the sum of percentage attributable to each causal connection), what percentage is due to spurious correlation, what percentage is due to statistical error, etc.
This process led to many new ideas and questions. One thing I found is that, if I didn't write down some new idea immediately, I often lost it, to be recaptured days or weeks later or in some cases not at all.
I had a measure of how much progress I was making my noting how many new questions I generated for every question I attacked. At first it was about five to one. I remember the day I decided that the ratio was less than one; I knew I was on the downhill slope.
In a logical/deductive sense, Causal statistics is more like probabilities theory than like classical statistics. The interpretation of probabilistic results is crucially tied to the definition of probability. Frequentist probability theory and classical statistics derived from one definition of probability and Bayesian probability theory and Bayesian statistics derived from another definition of probability.
Both classical and Bayesian statistics are fairly straightforward mathematical deductions from classical and Bayesian probability theories, respectively. Logically/deductively, causal statistics is more like probability theory, but causal statistics is far more logically/deductively complicated than probability theory. Additionally, causal statistics possesses a complicated philosophical component not present in probability theory.
Noting this broad range of complications and the eclectic nature of causal statistics, it is now much more understandable why the problem of completely understandable, non-experimental causal inference has proved to be so intractable to even exceptional statisticians and research methodologists. Few people have the breath of training and/or the breath of mind to expand outside their field and into all of the disciplines required for proper handling of non-experimental causal inference.
Euclid and Einstein combined logical/deductive derivation with their fields, geometry and physics, respectively. To solve the causal inference problem, one must utilize logical/deductive derivation, statistics, non-experimental research methodology, and philosophy.
Einstein's great advantage over other physicists was that he chose to use the logical/deductive derivation approach. This was considered to be part of the field of mathematics and possibly philosophy, but not part of the field of physics. Therefore, logical/deductive derivation was not a part of the usual bag of tricks for a physicist. But it was undoubtedly the proper solution for the problem.
This connection between my approach and those of Euclid and Einstein boldly occurred to me in retrospect, while trying to explain myself and to explain causal statistics. While wondering in the deep forest, which was causal inference in the late 60s, I likewise didn't see the solution as baking a cake from a recipe: a cup of logic, two tablespoons of philosophy, half a cup of non-experimental research methodology, etc.
At the beginning, [at the time, these analogies did not occur to me either. Only in retrospect I see how well the analogy fits my state of mind] I was entering this forest with no idea what was there nor how deep it was nor its extent. As I walked through the forest, I explored its entirety. As I explored the forest I solve many problems but in doing so, generated or discovered many many more.
At this point I could see the project as a whole and wrap my mind around it as a megaproject, but I saw problems that still existed everywhere. But at the same time I saw the deductive approach as the only route to real understanding. At this point the mega-solution began to take shape.
Now, it was just a matter of solving the remaining problems along the path of the solution and hoping that the solved problems could be cobbled together in a continuous chain of logical reasoning and derivation from the beginning assumptions and definitions to the final form of causal statistics.
There were several points along the chain of logical derivation where I was concerned that I might not be able to get from one step to the next: the repeatability of causal impulses through a micro causal chain, the aggregation of micro variables into macro variables, and the jump of micro causal chains to macro causal chains. These and all other nodes along the logical/deductive chain were eventually resolved and all other necessary problems were solved. Then it was just a matter of writing it up: presenting the derivation and explaining all the why's and wherefores in my dissertation.
OLD I. Origin, Background, Setting, and Context of Causal Statistics
In 1967, I was at the University of California (Berkeley), finishing an M.S. in Nuclear Engineering and beginning a Ph.D. in Business. The dissertation research plan was to carry out a non-experimental management study, in an attempt to determine what managerial behaviors caused increased productivity of scientists and engineers engaged in physical sciences research and development.
As work on the dissertation progressed, I discovered that causal inferences were difficult to impossible to draw from non-experimental studies. I attempted to use the causal inquiring systems available at the time, but could not apply them with complete understanding, insight, or confidence -- e.g., many of the assumptions implicit in the various systems were unknown, the nature of statistical causal connections was not understood, etc.
Recognizing that causal results were, by far, the most desired and that most research in the social sciences, epidemiology, management, etc. was non-experimental; I realized that the development of a non-experimental causal inquiring system, which could be applied with complete understanding and confidence, was of transcendent importance. Hence, I discontinued the original empirical study and turned my efforts to the development of a definitive causal inquiring system.
This statement rolls off the tongue very easily, but what kind of ego maniac would so cavalierly set about solving one of the most important problems in philosophy and THE most important and difficult problem in the social sciences, epidemiology, and statistics: first the problem of causality--a concept challenged with only a modicum of complete success by philosophers for over 3000 years--and second, the problem of non-experimental causal inference, a task never adequately handled by social scientists, epidemiologists, nor statisticians? Answer: a young graduate student who didn’t know better than to believe that he could ultimately solve any problem that had a rational solution.
Aside: I would note that any researcher, who doesn't believe he can solve his research problem, is probably right. He probably couldn't solve it. Einstein, Enrico Fermi, and R. A. Fisher certainly attacked their respective, important problems with the belief that they could solve them.
After trying many approaches to the development of a complete causal inquiring system, I eventually did as Euclid did for Geometry and Einstein did for Relativity and no one had or has done for causality. I went back to basics and--beginning with axioms, definitions, primitives, etc., about the nature of the universe and the nature of empirical research--performed a logical derivation of the general formulation of a causal inquiring system, I called Causal Statistics. This research was reported in my Ph.D. dissertation, entitled Foundation of Mathematical Epistemology: A Derivation of Causal Statistics, published in 1972.
During the time I was working on the dissertation and after its completion -- when I was looking for funding to carry the research to the next stage--I was amazed at the negative reactions of many ostensibly intelligent people (professors, funders, etc.) to my research. Over 40 years, I have heard it all. It couldn't be done; it needn't be done; Hume has already considered it and proved that it couldn't and needn't; it should be done, but someone else should fund its further development (e.g., it's too interdisciplinary, too different, too eclectic, too innovative, or conflicting with accepted paradigms to be funded here or evidently anywhere); it's too risky (i.e., it might fail) for results oriented departments and especially for the untenured; etc.; etc.; etc.
There were about as many different negative criticisms as there are people reacting and, in the final analysis, almost none of the negative opinions held any water. If there had been some convergence of opinion about what was wrong with my research, the critiques would have been more believable.
Nevertheless, I analyzed every criticism and modified my results for those very few with merit. Even so, such corrections almost never satisfied the doubter whose critique was accepted and corrected. These critics would then come up with new, off-the-wall criticism. I called these critiques the “yes-buters.” "Yes, but what about this new criticism?"
Over the years, I've had many different thoughts about why people with the right educational background, have produced so many negative and incorrect reactions. I have come to several different conclusions: (1) Many just can't think out of the box, even when led by the hand. (2) Many fear, or just naturally resist, the new and different, i.e., anything more than 1 millimeter deviant from what they were taught at their professor’s knee. (3) Most don't have the breadth of mind to comprehend the whole of a large and complicated project (what I call a mega project) all at one time. They can only see distinct pieces. (4) A surprisingly large number just don’t have the mental where-with-all to even make the aforementioned three errors, e.g., "We like the way we've been doing statistics previously."
As a consequence, we are 40 years down the road and Causal Statistics is still on a Ph. D. dissertation and generally unavailable to non-experimental researchers: 40 years of lost time, billions of wasted dollars in the non-experimental sciences, untold waste in competent human resources--i.e., social science and epidemiological researchers getting 10% of what they could get from their research efforts--and many lives lost (e.g., from lung and other environmental cancers) and destroyed (e.g., due to the dearth of good causal research and theories to correct criminal, social, and health problems).
If I had been able to spend a substantial portion of the last 35 years carrying out the beyond-dissertation steps in the Causal Statistics project, I believe that today Causal Statistics would be in broad usage. But, for the lack of less funding than that required for one government secretary, almost no lasting progress has been made in the last 35 years. Such collective stupidity on the part of funders and funding agencies is beyond belief.
At least one other explanation is possible, maybe I'm insane. The mental case is always the last to know. Well, then my dissertation chairman, Professor C. West Churchman and the dissertation committee must also have been insane. Further, if I was insane then, I still am. The research makes just as much sense to me now as it did in 1972.
As I said before, the dissertation derived the general form of Causal Statistics. Originally, I had planned to get research funding to extract an application oriented formulation of Causal Statistics from the dissertation. When the funding didn't materialize, I went on to other things, with the belief that some extremely analytic and dedicated researcher would either do the extraction or develop his/her own application oriented formulation of Causal Statistics.
About two years ago, as I started moving toward retirement, I looked at the developments in the field over the past 35 years and found that a great deal had been written about causal inference and causal theory construction. Virtually all attempts were laudable and most were correct, as far as they went. These writers were obviously people who were (1) sharp enough to see the need and (2) insightful enough to say something meaningful and correct about the subject. Nevertheless, they were about where I was in 1968, a year after I began researching the subject and before realizing the need to return to fundamentals and take the deductive approach.
A few methodologists--like Pearl, Rubin, and others--have gone further, but no one has taken the deductive approach, which I believe to be the optimal. My research is still the only effort to do what Euclid and Einstein did; i.e., go backwards, down to basics, start from definitions, axioms, primitives, etc. and derive the whole field. This approach gives a logical foundation to causal inference and allows a complete understanding of the inferential process, just as Euclid's and Einstein's deductive approaches to Geometry and Relativity, respectively, were the only routes to real understanding in their fields. Their derivations effectively ended controversies concerning origins in their fields and their colleagues move on to research in other aspects of their fields.
Now you should have a ballpark understanding of the content of the 1972 dissertation and where the field of causal inquiring systems is today. [GOAL:] The initial impetus for this book is to extract from the dissertation the elements necessary for a minimal, yet sufficient and usable, formulation of Causal Statistics and present it herein, along with examples of its application in non-experimental research.
But, as long as I'm at it, I might as well take the opportunity to handle all other relevant issues that I can think of. Hence, the presentation will be interspersed with background information (necessary for understanding), answers to potential objections, explanations of how the various components being presented relate to each other, and delineations of how these components fit into their philosophical, logical, and statistical environments and into the overall mosaic of human knowledge.
[the purpose of this book is to design an algorithm for properly applying causal statistics in non-experimental research so that, after reading this book, any non-experimental researcher will be able to apply causal statistics to their work correctly and with understanding. In furtherance of that purpose, the book should answer any questions and handle any problems that the researcher is likely to encounter in either application or understanding.
In a]
Some areas of the website are still under construction.
©2008 Center for Applied Social Science
B. Research Methodology
C. The Derivation of Causal Statistics??
D. The Assumptions and the General Model of Causal Statistics
1. The First Assumption Set
2. The Second Assumption Set
3. The Third Assumption Set
E. Mathematical statistics (The General Form of Causal Statistics)
F. A Causal Calculus/Algebra
G. Inductive Logic in Causal Statistics
VIII. Causal Statistics: A Two Variable Example
Classical and Bayesian Statistical Analyses of the Example Data (The Mathematics and Interpretation of Classical and Bayesian Association)
History of Correlation
IX. Causal Statistics: a Three Variable Example
Classical and Bayesian Statistical Analyses of the Example Data (The Mathematics and Interpretation of Classical and Bayesian Association)
X. Causal Inference in Experimental Research
A. Non-Stochastic Causal Inference
1. Validity and Correctness
2. Causal Modeling and Theory Building
B. Statistical Causal Inference
1. Validity and Correctness
2. Causal Modeling and Theory Building
XI. Causal Inference in Non-experimental Research
A. The Relationship among the Three Statistical Paradigms
A. How does Causal Statistics fit into the Pantheon (Mosaic) of other Statistical Paradigms?
B. [A.] Understanding the Nature of Mathematical and scientific “Knowledge.”
?? B. Where Does the Dissertation Fit in?
A. [C.] The Logical Constructs within Statistics and their Relations to each other
XIV. The Design of Non-experimental Causal Studies and Causal Study Sequences
XV. History of Causal Philosophy and Causal Inference, as Seen from the High Ground
A. Conditionals, Counterfactuals, etc.
XVI. My Post Dissertation Attempts to do Causal Statistics and Obtain Funding
XVII. Why is it Taking so Long? And how much longer will it take?
XVIII. Where do we go from Here?
A. Get researchers using Causal Statistics
1. Develop a simple presentation of the Causal Statistics paradigm.
2. Answer all objections.
3. Publicize.
4. Free application consulting.
5. Book? With basics and examples
6. Funding: only after it’s no longer innovated and is acceptable. When it’s too late, there will then be adequate money.
Part I Introduction
I. In a Nutshell, What is Causal Statistics and How Important Is It?
II. Origin, Background, Setting, and Context of Causal Statistics
III. The Needs for Non-Causal and Causal Inference
IV. Empirical Research, Associative and Causal (An Example)
A. Non-experimental Research, Associative and Causal (An Example)
B. Experimental Research, Associative and Causal (An Example)
-- OR --
A. Experimental Research, Associative and Causal (An Example)
B. Non-experimental Research, Associative and Causal (An Example)
V. Example of Large Scale Causal Statistics Study
A. Applicability of Causal Statistics
B. Application of Causal Statistics
C. Assumptions Required, 3 Assumption Sets
D. Further Complications
Non Linearity
Time Lags ( e.g., psychological problems in kids occur later or was it an operationalization error or measurement error or a study design error that we didn't see psychological problems earlier)
Show various techniques for increasing identification ( e.g., results of previous studies, assumptions, etc.) and the effects of ~ 0 causal parameters.
E. Reporting
F. Follow on Studies
Part II Foundations
VI. Foundations of Causal Statistics (Philosophy)
A. Epistemology/Philosophy of Science
1. The Empiricists
2. The Rationalists
3. The Eclectics
4. Observability
5. Un-observability (Metaphysics)
B. Logic
1. Deductive
2. Inductive
3. Deductive/Inductive (Combination) Systems
C. Philosophy of Causality & Definition of “Cause”
Hume
Mill
Definitions of "Cause"
D. Attempts at Causal Inference
1. God
2. Hobbs
3. Bacon
4. Mill
5. Conditionals, Counterfactuals, etc.
A. Nuclear Physics
B. Fundamental Particles
C. Quantum Mechanics
D. Statistical Mechanics
E. Etc.
F. Final Definitions of “Cause”
VIII. Causal Inference in Experimental Research
A. Non-Stochastic Causal Inference
1. Validity and Correctness
2. Causal Modeling and Theory Building
B. Statistical Causal Inference
1. Validity and Correctness
2. Causal Modeling and Theory Building
IX. Causal Inference in Non-experimental Research
X. Derivation of Causal Statistics
A. Deductive Logic
B. Research Methodology
C. The Derivation of Causal Statistics??
D. The Assumptions and the General Model of Causal Statistics
1. The First Assumption Set
2. The Second Assumption Set
3. The Third Assumption Set
E. Mathematical statistics (The General Form of Causal Statistics)
F. A Causal Calculus/Algebra
G. Inductive Logic in Causal Statistics
XI. History of Causal Philosophy and Causal Inference, as Seen from the High Ground
A. Conditionals, Counterfactuals, etc.
Part III Simple (or Linear) Single Equation Models
XII. Causal Statistics: A Two Variable Example
Classical and Bayesian Statistical Analyses of the Example Data (The Mathematics and Interpretation of Classical and Bayesian Association)
A. Curve Fitting
Scatter Diagrams
Possible Criteria for Curve Fitting
The Least Squares Criteria
B. Statistical Inference for a Fixed Independent Variable
The Fundamental Assumptions of Regression Analysis
The Fundamental Assumptions of Causal Statistics
Causal Parameter Estimation
The Means, Variance, and Distributions of Causal Parameter Estimators
Confidence Intervals for Causal Parameters
Hypothesis Tests for Causal Parameters
Confidence Intervals for Causal Predictions
C. Statistical Inference for Random Independent Variables
D. Application to a Bivariate Normal Population
E. Interpretation of Results
XIII. Correlation
A. Preliminary Remarks
B. Simple Correlation
The Population Correlation Coefficient
The Sample Correlation Coefficient
Interpretation of the Correlation Coefficient
C. Relationship between the Correlation Coefficient and the Causal Parameter
D. Partial Correlation
E. Relationship between the Partial Correlation Coefficient and the Partial Causal Parameter
F. Multiple Correlation
G. Canonical Correlation
H. History of Correlation
XIV. Causal Statistics: a Three Variable Example
Classical and Bayesian Statistical Analyses of the Example Data (The Mathematics and Interpretation of Classical and Bayesian Association)
XV. Multivariable Models
A. Preliminary Remarks
B. Graphical Representation of the Three Dimensional Case
C. Assumptions
D. Causal Parameter Estimation
E. Confidence Intervals and Hypothesis Testing
F. Multicollinearity
G. Stepwise Parameter Estimation
H. Computer Analysis
I. Interpretation of Results
PART IV Complexities
XVI. Nonlinear Models
A. Preliminary Remarks
B. Nonlinearity in the Variables
C. Nonlinearity in the Causal Parameters
D. Differential Models
E. Intractable Nonlinear Models
XVII. Relaxation of Various Assumptions
A. Preliminary Remarks
B. Heteroschedasticity
C. Specification Error
D. Data Snooping
E. Omitted Relevant Variables
F. Included Irrelevant Variables
G. Superfluous Variables
H. Serial correlation in the Error Term
I. Lagged Variables
J. Correlation between the Error Term and an Independent Variable
K. Measurement Error
L. Transcription Error
M. Causal Relationships between Independent Variables
N. Reciprocal Causation
XVIII. Special Topics and Techniques in Causal Inference
A. Preliminary Remarks
B. Operational Variables
C. Proxy Variables
D. Ordinal Variables
E. Dichotomous Variables
F. Qualitative Variables
G. Moderator Variables
H. Scanning
I. Dummy Variables Used as a Jackknife
J. Time as an Independent Variable
K. Inclusion of Associative Variable
L. Dummy Variables for Seasonal Variations
M. Clustered Residuals
XIX. Performing a Causal Analysis Study
A. Preliminary Remarks
B. State the Problems
C. Construct a Hypothesized Model
D. State the Assumption Set
E. Identify the Equations
F. Operationalize Variables
G. Collect Data
H. Estimate Causal Parameters
I. Interpret Results
J. Write-up Study
K. Criticize and Redo Study
Part V Simultaneous Equation Models
XX. Limited Information Estimation Techniques
A. Preliminary Remarks
B. Ordinary Least Squares
C. Indirect Least Squares
D. Instrumental Variables
E. Two Stage Least Squares
F. Least-Variance Ratio
XXI. Identification
A. Preliminary Remarks
B. Under-identification
C. Identification by Addition of Exogenous Variables to the Model
D. Identification Using A Priori Information
E. Necessary Conditions for Identification
F. Necessary and Sufficient Conditions for Identification
G. Over-identification
XXII. Full Information Estimation Techniques
A. Preliminary Remarks
B. Three Stage Least Squares
C. Iterated Three Stage Least Squares
D. Full Information Maximum Likelihood
XXIII. Comparison of Estimation Techniques
A. Preliminary Remarks
B. Small Sample Distributions
C. Summary of Overall Comparisons
XXIV. The Design of Non-experimental Causal Studies and Causal Study Sequences
Part VI Conclusions
A. The Relationship among the Three Statistical Paradigms
A. How does Causal Statistics fit into the Pantheon (Mosaic) of other Statistical Paradigms?
B. [A.] Understanding the Nature of Mathematical and scientific “Knowledge.”
?? B. Where Does the Dissertation Fit in?
A. [C.] The Logical Constructs within Statistics and their Relations to each other
XXVI. My Post Dissertation Attempts to do Causal Statistics and Obtain Funding
XXVII. Why is it Taking so Long? And how much longer will it take?
XXVIII. Where do we go from Here?
A. Get researchers using Causal Statistics
1. Develop a simple presentation of the Causal Statistics paradigm.
2. Answer all objections.
3. Publicize.
4. Free application consulting.
5. Book? With basics and examples
6. Funding: only after it’s no longer innovated and is acceptable. When it’s too late, there will then be adequate money.
Foundations and Understandings of Causal Statistics
Part I Introduction
I. In a Nutshell, What is Causal Statistics and How Important Is It?
II. Origin, Background, Setting, and Context of Causal Statistics
III. The Needs for Non-Causal and Causal Inference
IV. Empirical Research, Associative and Causal (An Example)
A. Non-experimental Research, Associative and Causal (An Example)
B. Experimental Research, Associative and Causal (An Example)
-- OR --
A. Experimental Research, Associative and Causal (An Example)
B. Non-experimental Research, Associative and Causal (An Example)
V. Example of Large Scale Causal Statistics Study
A. Applicability of Causal Statistics
B. Application of Causal Statistics
C. Assumptions Required, 3 Assumption Sets
D. Further Complications
Non Linearity
Time Lags ( e.g., psychological problems in kids occur later or was it an operationalization error or measurement error or a study design error that we didn't see psychological problems earlier)
Show various techniques for increasing identification ( e.g., results of previous studies, assumptions, etc.) and the effects of ~ 0 causal parameters.
E. Reporting
F. Follow on Studies
Part II Foundations
VI. Foundations of Causal Statistics (Philosophy)
A. Epistemology/Philosophy of Science
1. The Empiricists
2. The Rationalists
3. The Eclectics
4. Observability
5. Un-observability (Metaphysics)
B. Logic
1. Deductive
2. Inductive
3. Deductive/Inductive (Combination) Systems
C. Philosophy of Causality & Definition of “Cause”
Hume
Mill
Definitions of "Cause"
D. Attempts at Causal Inference
1. God
2. Hobbs
3. Bacon
4. Mill
5. Conditionals, Counterfactuals, etc.
A. Nuclear Physics
B. Fundamental Particles
C. Quantum Mechanics
D. Statistical Mechanics
E. Etc.
F. Final Definitions of “Cause”
VIII. Causal Inference in Experimental Research
A. Non-Stochastic Causal Inference
1. Validity and Correctness
2. Causal Modeling and Theory Building
B. Statistical Causal Inference
1. Validity and Correctness
2. Causal Modeling and Theory Building
IX. Causal Inference in Non-experimental Research
X. Derivation of Causal Statistics
A. Deductive Logic
B. Research Methodology
C. The Derivation of Causal Statistics??
D. The Assumptions and the General Model of Causal Statistics
1. The First Assumption Set
2. The Second Assumption Set
3. The Third Assumption Set
E. Mathematical statistics (The General Form of Causal Statistics)
F. A Causal Calculus/Algebra
G. Inductive Logic in Causal Statistics
XI. History of Causal Philosophy and Causal Inference, as Seen from the High Ground
A. Conditionals, Counterfactuals, etc.
Part III(VI) Conclusions
A. The Relationship among the Three Statistical Paradigms
A. How does Causal Statistics fit into the Pantheon (Mosaic) of other Statistical Paradigms?
B. [A.] Understanding the Nature of Mathematical and scientific “Knowledge.”
?? B. Where Does the Dissertation Fit in?
A. [C.] The Logical Constructs within Statistics and their Relations to each other
XIII(XXVI). My Post Dissertation Attempts to do Causal Statistics and Obtain Funding
XIV(XXVII). Why is it Taking so Long? And how much longer will it take?
XV(XXVIII). Where do we go from Here?
A. Get researchers using Causal Statistics
1. Develop a simple presentation of the Causal Statistics paradigm.
2. Answer all objections.
3. Publicize.
4. Free application consulting.
5. Book? With basics and examples
6. Funding: only after it’s no longer innovated and is acceptable. When it’s too late, there will then be adequate money.
Causal Statistics for Applications
Part I(III) Simple (or Linear) Single Equation Models
I(XII). Causal Statistics: A Two Variable Example
Classical and Bayesian Statistical Analyses of the Example Data (The Mathematics and Interpretation of Classical and Bayesian Association)
A. Curve Fitting
Scatter Diagrams
Possible Criteria for Curve Fitting
The Least Squares Criteria
B. Statistical Inference for a Fixed Independent Variable
The Fundamental Assumptions of Regression Analysis
The Fundamental Assumptions of Causal Statistics
Causal Parameter Estimation
The Means, Variance, and Distributions of Causal Parameter Estimators
Confidence Intervals for Causal Parameters
Hypothesis Tests for Causal Parameters
Confidence Intervals for Causal Predictions
C. Statistical Inference for Random Independent Variables
D. Application to a Bivariate Normal Population
E. Interpretation of Results
II(XIII). Correlation
A. Preliminary Remarks
B. Simple Correlation
The Population Correlation Coefficient
The Sample Correlation Coefficient
Interpretation of the Correlation Coefficient
C. Relationship between the Correlation Coefficient and the Causal Parameter
D. Partial Correlation
E. Relationship between the Partial Correlation Coefficient and the Partial Causal Parameter
F. Multiple Correlation
G. Canonical Correlation
H. History of Correlation
III(XIV). Causal Statistics: a Three Variable Example
Classical and Bayesian Statistical Analyses of the Example Data (The Mathematics and Interpretation of Classical and Bayesian Association)
IV(XV). Multivariable Models
A. Preliminary Remarks
B. Graphical Representation of the Three Dimensional Case
C. Assumptions
D. Causal Parameter Estimation
E. Confidence Intervals and Hypothesis Testing
F. Multicollinearity
G. Stepwise Parameter Estimation
H. Computer Analysis
I. Interpretation of Results
PART II(IV) Complexities
V(XVI). Nonlinear Models
A. Preliminary Remarks
B. Nonlinearity in the Variables
C. Nonlinearity in the Causal Parameters
D. Differential Models
E. Intractable Nonlinear Models
VI(XVII). Relaxation of Various Assumptions
A. Preliminary Remarks
B. Heteroschedasticity
C. Specification Error
D. Data Snooping
E. Omitted Relevant Variables
F. Included Irrelevant Variables
G. Superfluous Variables
H. Serial correlation in the Error Term
I. Lagged Variables
J. Correlation between the Error Term and an Independent Variable
K. Measurement Error
L. Transcription Error
M. Causal Relationships between Independent Variables
N. Reciprocal Causation
VII(XVIII). Special Topics and Techniques in Causal Inference
A. Preliminary Remarks
B. Operational Variables
C. Proxy Variables
D. Ordinal Variables
E. Dichotomous Variables
F. Qualitative Variables
G. Moderator Variables
H. Scanning
I. Dummy Variables Used as a Jackknife
J. Time as an Independent Variable
K. Inclusion of Associative Variable
L. Dummy Variables for Seasonal Variations
M. Clustered Residuals
VIII(XIX). Performing a Causal Analysis Study
A. Preliminary Remarks
B. State the Problems
C. Construct a Hypothesized Model
D. State the Assumption Set
E. Identify the Equations
F. Operationalize Variables
G. Collect Data
H. Estimate Causal Parameters
I. Interpret Results
J. Write-up Study
K. Criticize and Redo Study
Part III(V) Simultaneous Equation Models
IX(XX). Limited Information Estimation Techniques
A. Preliminary Remarks
B. Ordinary Least Squares
C. Indirect Least Squares
D. Instrumental Variables
E. Two Stage Least Squares
F. Least-Variance Ratio
X(XXI). Identification
A. Preliminary Remarks
B. Under-identification
C. Identification by Addition of Exogenous Variables to the Model
D. Identification Using A Priori Information
E. Necessary Conditions for Identification
F. Necessary and Sufficient Conditions for Identification
G. Over-identification
XI(XXII). Full Information Estimation Techniques
A. Preliminary Remarks
B. Three Stage Least Squares
C. Iterated Three Stage Least Squares
D. Full Information Maximum Likelihood
XII(XXIII). Comparison of Estimation Techniques
A. Preliminary Remarks
B. Small Sample Distributions
C. Summary of Overall Comparisons
XIII(XXIV). The Design of Non-experimental Causal Studies and Causal Study Sequences
[An Exposition toward the Ultimate Goal of this Web site, i.e., the Extraction, Formulation, Explanation, and Exemplification of Causal Statistics
Rough Draft
A critical introduction to the methods used to collect data in social science: surveys, archival research, experiments, and participant observation. Evaluates "facts and findings" by understanding the strengths and weaknesses of the methods that produce them. Case based.
Make statistical def of cause
Simi exp’al, quasi exp’al, etc
Who should be able to understand this book
CAUSAL INFERENCE VIA CAUSAL STATISTICS
The remainder of this web page is dedicated to accomplishing the goal of making a sea change in the way non-experimental scientists conduct their research. Specifically, the goal is that social scientists, epidemiologists, and other non-experimental researchers--when appropriate--utilize Causal Statistics in the design, conduct, analysis, and reporting of their empirical research. The more precise purpose of this Exposé is to tackle Objective 4, above, i.e., to present Causal Statistics in an easily accessible (intellectually) way, even multiple ways.
This article draws extractions from the dissertation and presents a minimal, yet sufficient, formulation of Causal Statistics, along with examples of its application in non-experimental research. Additionally, the presentation is interspersed with background information (necessary for understanding), answers to potential objections, explanations of how the various components being presented relate to each other, and delineations of how these components fit into their philosophical, logical, and statistical environments and into the overall mosaic of human knowledge.
This presentation will proceed via the following sections:
{Click to go back to: Table of Contents, I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII, XIII, XIV, XV, XVI, XVII, XVIII}
I. In a Nutshell, What Is Causal Statistics and How Important Is It?
“Causal Statistics is a mathematical inquiring system which enables empirical researchers to draw causal inferences from non-experimental data, based upon the minimum required assumptions, explicitly stated.” 1968
"Causal Statistics is the only causal inquiring system which is a deductive mathematical construct, in the sense that Euclidian geometry is an axiomatic, deductive, logical construct. It's derivation is founded in causal philosophy, physics, epistemology, symbolic logic, statistics, and non-experimental research methodology." 1976
“The development and utilization of Causal Statistics will eventually be as important to the non-experimental sciences as the codification and utilization of the scientific method was to the physical (i.e., experimental) sciences." 1969
“100 years from now, research results and theories in the non-experimental sciences will consist mostly of large arrays of variables, connected by multi-equation causal models, inferred from a single large or a compounded succession of smaller applications of Causal Statistics in empirical research studies.” 2007
{Click to go back to: Table of Contents, I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII, XIII, XIV, XV, XVI, XVII, XVIII}
II. Origin, Background, Setting, and Context of Causal Statistics
In 1967, I was at the University of California (Berkeley), finishing an M.S. in Nuclear Engineering and beginning a Ph.D. in Business Administration. The dissertation research plan was to carry out a non-experimental management study, in an attempt to determine what managerial behaviors caused increased productivity of scientists and engineers engaged in physical sciences research and development.
. [causal inferences were difficult to impossible to draw from non-experimental studies.] [ I had expected a completely understandable, logically tight causal inquiring system to be one of the ]
As work on the dissertation progressed, I was amazed to discover that there was no well-defined technique which social scientists used to make causal inferences. My sojourn in the physical sciences led me to think that such an important and needed component of research methodology would be a standard arrow in the quiver of all social science researchers and, if it wasn't, the whole field of social science would be moving heaven and earth to discover and/or develop such an arrow. [As a recent convert from the physical sciences,] [as a recent refugee from the physical sciences,] [What seemed even more amazing to me was that no one seemed to be working on the development of an understandable, logically tight causal inquiring system.]
Well, there was no such standard arrow. As I would discover later, there were ongoing efforts on the parts of some nonexperimental research methodologists to develop causal inquiring systems, but the frenzied, almost frantic, intensity which I expected was not observable. Most researchers and research consumers seemed to accept the limitations on drawing causal inferences as a given, as part of the nature of the field. Many even felt that any discussion of causality was somehow inappropriate and even unscientific. Even the social scientists and others who wished to have a complete, understandable, and well-founded causal inquiring system generally felt that such a thing was impossible. [There any significant pressure or effort to develop one. So I set about finding out what work had been done in causal inference. ]
After some time I found that the available causal inquiring systems broke down into three, more or less distinct, mathematical forms, all forms of classical statistics. All three were attempts to stretch classical statistics in a way that causal inferences could be produced. One of the problems with these attempts was that they were basically intuitive and not based on logical derivation or deduction, resulting in an understandable, logically tight causal inquiring system.
"Cause" is not a term defined in classical statistics; therefore, causal inferences cannot be established from classical statistics. Each of the three forms of causal inquiring system utilized a slightly different mathematical form from classical statistics. Each then gave an intuitive, handwaving argument to arrive at its causal inquiring system, utilizing the initial mathematical form. No wonder the systems, assumptions, definitions, etc. were not completely understandable; they were never actually stated nor was any continuous logical argument to derived them from classical statistics ever made.
So what you had was the domain of classical statistics and then three, somewhat overlapping causal inquiring systems some distance away, in the logical space, from classical statistics and no real, logical, deductive connection between classical statistics and the causal inquiring systems.
This is not to impugn the work done by these authors. Even to see the problem put them far ahead of 90% of the other professionals in the field. Then, to develop a causal inquiring system, no matter how intuitive, was godlike compared to everyone else. The point that I am making is only that these causal inquiring systems were not derived analytically, but intuitively, and therefore could not be applied with complete understanding or confidence.
What I did was (1) in a sense, to fill that opening (Actually, I started from scratch and fill the space from zero to the general form of causal statistics with definitions assumptions the deck of logic etc.) with deductive logic; (2) derive a general, rather than specific, form for causal inquiry; (3) generate all the required definitions; (4) produce all the required assumptions; and (5) explain all of the above in a manner hopefully understandable to a careful reader.
Now, in this book, I am engaged in simplifying and extracting all of the above from the dissertation; adding some explanations, global and epistemological information; and presenting a complete, understandable, and well-founded algorithm (which I call causal statistics) for making causal inferences from nonexperimental studies.
{2,5 Extant Causal Inquiring Systems
2.5.1 Summary
At present, there are three, more-or-less distinct,
causal inquiring systems. They are path analysis,
econometrics, and the Simon-Blalock approach. In actual
fact they are virtually identical to one another.
2.5.2 Path Analysis
Path analysis was introduced in a phenomenally
innovative paper by Sewall Wright* in 1921. Since that
*Wrlght, Sewall: "Correlation and Causation,"
J. of Agricultural Research, Vol. 20, 1921, pp.
557-85.
time additional innovations and, also, acceptance have
been amazingly slow. Path analysis considered only oneway
causation until 1954 when John Tukey** introduced
**Tnkey, John Wilder: "Causation, Regression, and
Path Analysis," in Oscar Keinpthorne, T. A.
Bancroft, J. W. Gowen, and J. L. Lush, eds.,
Statistics and Mathematics in Biology, Ames: Iowa
State College Press, 1954, pp. 35-66
two-way path analysis. This is the only innovation in
path analysis of major importance since 1921.
Basically, path analysis is a linear regression or
simultaneous linear regression technique in which the
coefficients are causal, assuming that the basic
assumptions of the model are valid. These coefficients
cj’s are called path regression coefficients. See
WrIght* 1960 for a summary of path analysis.
*Wright, Sewall: "Path Coefficients and Path
Regressions: Alternative or Complementary
Concepts?" Biometrics, Vol. 16, 1960, pp. 189-202.
2.5.3 Econometrics
Econometrics employs regression and simultaneous
equation models. It is far more advanced mathematically
than path analysis, but there are comparatively few
papers in the field which consider the causal implica
tions of these mathematical techniques.
Econometricians try to avoid the word "cause"
because of’ their misinterpretation of Humian philosophy
on the subject. Due to their avoidance of this word,
econometricians have failed to consider sufficiently a
many of the causal implications and proertIes of
econometrics and b many of the problems and benefits
connected with causal prediction.
Two good econometric references are Johnston** and
Goldberger***.
**Johnston, J.: Econometric Methods. New York,
McGraw-Hill, 1963.
***Goldberger, Arthur S. z Econometric Theory. New
York, John Wiley & Sons, 1964.
2.5.4 The Simon-.Bialock Approach
The Simon-Blalock approach began with a 1954 paper
by Herbert Simon*. This paper served as the foundation
*Simon, Herbert A.: "Spurious Correlation: A
Causal Interpretation," J. of the American Statis
tical Association, Vol. 9, 19514, pp. 467-47.
for a great deal of later work by Hubert Blalock.
Basically, this approach gives causal Interpreta
tion to some of the more elementary f’ormalizations of
econometrics. An exception is Blalock** 1969 in which
**Blalock, Hubert N., Jr.: Theory Construction:
From Verbal to Mathematical Formulati, Englewood
Cliffs, Prentice-Hall, 1969.
he gives preliminary consideration to some simple
differential equation models.}
mathematical treatise on the subject was published by Sewall Wright, with his innovative, 1921 article, “Correlation and Causation”, in J. of Agricultural. Research, Vol. 20, 1921, pp. 557-85. Dr. Wright presented a regression analysis approach to causal inference. That should have gotten the ball rolling, but amazingly it didn't.
33 years later, John Tukey published “Causation, Regression, and Path Analysis,” in Oscar Kempthorne, T. A. Bancroft, J. W. Gowen, and J.L. Lush, eds., Statistics and Mathematics in Biology Ames: Iowa State College Press, 1954, pp. 35-66 and Herbert Simon published “Spurious Correlation: A Causal Interpretation,” in J. of the American Statistical Association, Vol. 49, 1954, pp. 467-479.
These and other causal inquiring systems were incomplete and their foundations in philosophy and axiomatic logic not established. Nevertheless, these insightful initial efforts should have triggered a tidal wave of research into causal inquiring systems and their foundations.
Yet, there was no more than a diminishing ripple on the ponds of non-experimental research, statistics, and research methodology.
Econometrics, with its multi-equation and sometimes recursive forms, was mathematically superior to, i.e., more general than, the other forms of the day for making causal inferences. Econometrics employs regression and simultaneous equation models. It is more advanced mathematically than path analysis, but the number of papers in that field which consider the causal implications of these mathematical techniques is small.
Econometricians try to avoid, the word “cause” because of their misguided belief that Hume put a stake in the heart of causality. Due to their avoidance of this word, econometricians have failed to consider sufficiently (a) many of the causal implications and properties of econometrics and (b) many of the benefits that could be gleaned from facing cause inference from econometric analysis honestly and straight forwardly.
In 1968, I attempted to utilize the aforementioned systems to draw causal inferences in my nonexperimental management study, but I found that I could not apply these systems for causal inference with complete understanding, insight, or confidence. For example, many of the assumptions implicit in the various systems were unknown, the nature of statistical causal connections was not understood, etc.
13 years after Tukey’s and Simon’s first articles, I stumbled into the field of causal inquiring systems because of a desire to do good empirical research, rather then me-too research, with inferior and inappropriate statistical tools.
Recognizing that causal results were, by far, the most desired and that most research in the social sciences, epidemiology, management, etc. was non-experimental; I realized that the development of a non-experimental causal inquiring system, which could be applied with complete understanding and confidence, was of transcendent importance. Hence, I discontinued the original empirical study and turned my efforts to the development of a definitive causal inquiring system.
This statement rolls off the tongue very easily, but what kind of ego maniac would so cavalierly set about solving one of the most important problems in philosophy and THE most important and difficult problem in the social sciences, epidemiology, and statistics: first the problem of causality--a concept challenged with only a modicum of success by philosophers for over 3000 years--and second, the problem of non-experimental causal inference, a task never adequately handled by social scientists, epidemiologists, nor statisticians? Answer: a young graduate student who didn’t know better than to believe that he could ultimately solve any problem that had a rational solution.
{ In 1967-8 I suspected that the problem of causal inference must be pretty difficult, in that it was the most important problem in the field and it hadn't been solved. Even so I didn't realize how extremely hard it was; I spent at least two man years posing questions to myself, thinking, doing library research, and writing down the answers.
In another sense, this was an easy project for me because of the sheer joy of attacking such difficult and such interesting unsolved problems and, over time, seeing them yield to the relentless assault of pure reason. . I had no idea how much I could learn out of my own head by just thinking.
The concept of a theoretical dissertation was appealing to me from the first time I heard that such a thing was possible, sitting on the football/softball intramural field at Berkeley and talking to a Ph.D. candidate in nuclear engineering who was doing a theoretical dissertation. It turned out to be all I imagined and more.
I attacked the problem with all the confidence of one who didn't know better.}
Aside: I would note that any researcher, who doesn't believe he can solve his research problem, is probably right. He probably couldn't solve it. Einstein, Enrico Fermi, and R. A. Fisher certainly attacked their respective, important problems with the belief that they could solve them.
{I studied causal philosophy from Plato to Hume.}
After trying many approaches to the development of a complete causal inquiring system, I eventually did as Euclid did for Geometry and Einstein did for Relativity and no one had or has done for causality. I went back to basics and--beginning with axioms, definitions, primitives, etc., about the nature of the universe and the nature of empirical research--performed a logical derivation of the general formulation of a causal inquiring system, I called Causal Statistics. This research was reported in my Ph.D. dissertation, entitled Foundation of Mathematical Epistemology: A Derivation of Causal Statistics, published in 1972.
During the time I was working on the dissertation and after its completion -- when I was looking for funding to carry the research to the next stage--I was amazed at the negative reactions of many ostensibly intelligent people (professors, funders, etc.) to my research. Over 40 years, I have heard it all. It couldn't be done; it needn't be done; Hume has already considered it and proved that it couldn't and needn't; it should be done, but someone else should fund its further development (e.g., it's too interdisciplinary, too different, too eclectic, too innovative, or conflicting with accepted paradigms to be funded here or evidently anywhere); it's too risky (i.e., it might fail) for results oriented departments and especially for the untenured; etc.; etc.; etc.
There were about as many different, negative criticisms as there are people reacting and, in the final analysis, almost none of the negative opinions held any water. If there had been some convergence of opinion about what was wrong with my research, the critiques would have been more believable.
Nevertheless, I analyzed every criticism and modified my results for those very few with merit. Even so, such corrections almost never satisfied the doubter whose critique was accepted and corrected. These critics would then come up with some new, off-the-wall criticism. I called these critiques the “yes-buters.” "Yes, but what about this new criticism?"
Over the years, I've had many different thoughts about why people with the right educational background, have produced so many negative and incorrect reactions. I have come to several different conclusions: (1) Many just can't think out of the box, even when led by the hand. (2) Many fear, or just naturally resist, the new and different, i.e., anything more than 1 millimeter deviant from what they were taught at their professor’s knee. (3) Most don't have the breadth of mind to comprehend the whole of a large and complicated project (what I call a mega project) all at one time. They can only see distinct pieces. (4) A surprisingly large number just don’t have the mental where-with-all to even make the aforementioned three errors, e.g., "We like the way we've been doing statistics previously."
{In all fairness, might some of the blame for the retarded acceptance of causal statistics properly be placed at my door?
Could I have explain things better? With regard to the dissertation itself, I think things were explained quite well. But concerning the relationship of causal statistics to other statistical paradigms, my early explanations were lack.
Could I have published in journals to get the word out there? Maybe, but there were no journals in any way related to or interested in causal inference. Getting something in a journal which the publisher and referees feel are off-topic, is difficult in the best of circumstances. Doing it with a paper on causal inference, a topic of anathema to most academics, was multipally harder, apropos my experience with funding applications.
Would it have been better to apply causal statistics to nonexperimental research data and use the resulting report as an example of the benefits of causal statistics? I did exactly that with a 10 variable analysis of DDT, DDE, hypertension, pesticides, etc. on the data of a nationwide EPA study. They had collected data for about 10 years and, in all that time, had been incapable of sorting out the causal connections. I did exactly that in about two months and wrote a research report. The local scientists were happy, but those in Washington couldn't be less interested. Again, I was amazed. Is all the world mad or just most of it?}
As a consequence, we are 40 years down the road and Causal Statistics is still a Ph. D. dissertation and generally unavailable to non-experimental researchers: 40 years of lost time, billions of wasted dollars in the non-experimental sciences, untold waste in competent human resources--i.e., social science and epidemiological researchers getting 10% of what they could get from their research efforts--and many lives lost (e.g., from lung and other environmental cancers) and destroyed (e.g., due to the dearth of good causal research and theories to correct criminal, social, and health problems).
If I had been able to spend a substantial portion of the last 35 years carrying out the beyond-dissertation steps in the Causal Statistics project, I believe that today Causal Statistics would be in broad usage. But, for the lack of less funding than that required for one government secretary, surprisingly little progress toward codifying an understandable, complete, and algorithmic arrow for causal inference has been made in the last 35 years. Such collective stupidity on the part of funders and funding agencies is beyond belief.
At least one other explanation is possible, maybe I'm insane. The mental case is always the last to know. Well, then my dissertation chairman, Professor C. West Churchman and the dissertation committee must also have been insane. Further, if I was insane then, I still am. The research makes just as much sense to me now as it did in 1972.
As I said before, the dissertation derived the general form of Causal Statistics. Originally, I had planned to get research funding to extract an application oriented formulation of Causal Statistics from the dissertation. When the funding didn't materialize, I went on to other things, with the belief that some extremely analytic and dedicated researcher would either do the extraction or develop his/her own application oriented formulation of Causal Statistics.
About two years ago, as I started moving toward retirement, I looked at the developments in the field over the past 35 years and found that a great deal had been written about causal inference and causal theory construction. Virtually all attempts were laudable and most were correct, as far as they went. These writers were obviously people who were (1) sharp enough to see the need and (2) insightful enough to say something meaningful and correct about the subject. Nevertheless, they were about where I was in 1968, a year after I began researching the subject and before realizing the need to return to fundamentals and take the deductive approach.
A few methodologists--like Pearl, Rubin, and others--have gone further, but no one has even come close to the total package, i.e., an understandable, complete, algorithmic causal inquiring system for nonexperimental research. Neither has anyone taken the deductive approach, which I believe to be the optimal. My research is still the only effort to do what Euclid and Einstein did; i.e., go backwards, down to basics, start from definitions, axioms, primitives, etc. and derive the whole field. This approach gives a logical foundation to causal inference and allows a complete understanding of the inferential process, just as Euclid's and Einstein's deductive approaches to Geometry and Relativity, respectively, were the only routes to real understanding in their fields. Their derivations effectively ended controversies concerning origins in their fields and their colleagues move on to research in other aspects of their fields.
Not all deductive constructs are as earth shattering as Euclid's and Einstein's. Alfred North Whitehead and Bertrand Russell used symbolic logic to derive arithmetic and algebra in their book, Principia Mathematica. Few people who use arithmetic or algebra refer to or even know about their work.
In one sense, the derivation of causal statistics is of greater utility to its field than the work of Whitehead and Russell was to their field. Euclid's deductive construct was earth shattering in terms of its importance to deductive and symbolic logic, but less so to the field of plane geometry. Interestingly, Euclid's derivation of plane geometry was more important for its contribution to the development of non-Euclidian geometries then for Euclidian geometry.
As everyone knows, Einstein's derivation of relativity was and is unequaled in its effect on the field of physics and in popular culture (the public consciousness). At the turn of the 20th century, physics was where nonexperimental causal inference was in 1968. Most of the mathematical formulations have been developed, but they could not be applied with confidence or understanding because no one understood why the formulas were as they were. Einstein's logical construct, relativity, derived the Lorentz equations and specified the assumptions and concepts on which they were based. After Einstein published his work on relativity, physicists progressively moved toward the accept that's of relativity and its assumptions as the intellectual foundations for the Lorentz equations and moved on to other problems in physics.
In the above sense, the derivation of causal statistics is not comparable in importance to the derivations of Euclid and Einstein. Yet, in the effect on human lives in the long run, the derivation of causal statistics will arguably be more important than any of the others. If causal statistics were used in all appropriate nonexperimental research, the increase in medical, social, etc. knowledge would be so great and the application of this knowledge so earth shattering in its effects that, in that sense, the derivation of causal statistics would surpass the others.
Now you should have a ballpark understanding of the content of the 1972 dissertation and where the field of causal inquiring systems is today. About two years ago I started a website called causalstatistics.org. The initial purpose of the web site was to make my dissertation easily accessible. A downloadable, selectable, and searchable copy of the dissertation is presented on that website.
The dissertation presents Causal Statistics at a level that extremely analytic and dedicated researchers could apply the paradigm in non-experimental research and obtain valid causal inferences. Nevertheless, greater simplification is necessary for the vast majority of social science researchers to utilize Causal Statistics with complete understanding and confidence.
{ The remainder of this web page is dedicated to accomplishing the goal of making a sea change in the way non-experimental scientists conduct their research. Specifically, the goal is that social scientists, epidemiologists, and other non-experimental researchers--when appropriate--utilize Causal Statistics in the design, conduct, analysis, and reporting of their empirical research. The more precise purpose of this Exposé is to tackle Objective 4, above, i.e., to present Causal Statistics in an easily accessible (intellectually) way, even multiple ways.
This article draws extractions from the dissertation and presents a minimal, yet sufficient, formulation of Causal Statistics, along with examples of its application in non-experimental research. Additionally, the presentation is interspersed with background information (necessary for understanding), answers to potential objections, explanations of how the various components being presented relate to each other, and delineations of how these components fit into their philosophical, logical, and statistical environments and into the overall mosaic of human knowledge.}
But, as long as I'm at it, I might as well take the opportunity to handle all other relevant issues that I can think of. Hence, the presentation will be interspersed with background information (necessary for understanding), answers to potential objections, explanations of how the various components being presented relate to each other, and delineations of how these components fit into their philosophical, logical, and statistical environments and into the overall mosaic of human knowledge. (Moved this paragraph to preface?)
[the purpose of this book is to design an algorithm for properly applying causal statistics in non-experimental research so that, after reading this book, any non-experimental researcher will be able to apply causal statistics to their work correctly and with understanding. In furtherance of that purpose, the book should answer any questions and handle any problems that the researcher is
Hence, as my work has progressed, my ultimate goal has become more far-reaching. The goal has evolved toward making a sea change in the way non-experimental scientists conduct their research. Specifically, it is desired that social scientists, epidemiologists, and other non-experimental researchers, when appropriate, utilize Causal Statistics in the design, conduct, analysis, and reporting of their empirical research.
In an effort to accomplish this goal, I have established five objectives (“Objectives” are steps on the path toward accomplishing the overall goal.):
1. To make the dissertation readily available to all (accomplished via the presentation of the dissertation at causalstatistics.org),
2. To extract from the dissertation portions that are, in sum, necessary and sufficient for formulating a physics/logically/epistemology/ statistically/research methodology/philosophically based causal inquiring system,
3. To utilize these extractions to formulate Causal Statistics in a complete, coherent, and interrelated (i.e. with consideration of how Causal Statistics related to its epistemological environment) form,
4. To present this formulation of Causal Statistics in an easily accessible (intellectually) way (even multiple ways) to present and future research methodologists, to the researchers themselves, and to research consumers. (The initial presentation will be accomplished through the development of this book),
5. To challenge non-experimental scientists and research methodologists to do the hard work to study, understand, analyze, critique, extend, and apply Causal Statistics
[GOAL:] The initial impetus for this book is to extract from the dissertation the elements necessary for a minimal, yet sufficient and usable, formulation of Causal Statistics and present it herein, along with examples of its application in non-experimental research.
{But, as long as I'm at it, I might as well take the opportunity to handle all other relevant issues that I can think of. Hence, the presentation will be interspersed with background information (necessary for understanding), answers to potential objections, explanations of how the various components being presented relate to each other, and delineations of how these components fit into their philosophical, logical, and statistical environments and into the overall mosaic of human knowledge.
[the purpose of this book is to design an algorithm for properly applying causal statistics in non-experimental research so that, after reading this book, any non-experimental researcher will be able to apply causal statistics to their work correctly and with understanding. In furtherance of that purpose, the book should answer any questions and handle any problems that the researcher is likely to encounter in either application or understanding.
In a]}
Take your recent example. A researcher found a correlation between lack of sleep and poor grades, on the part of children. The newscaster said, “You should make sure your children get lots of sleep so they will get better grades."
Is it possible that parents who don't care enough to make sure that their children get sufficient sleep also don't care to push or to help them make better grades? Or could it be a spurious correlation due to a genetic factor?
What about correlations between book ownership and reading scores of children? Or between....
What about the causes and the effects of mental illness? What about the environmental causes cancer? What about the effects, of lax law enforcement in low income areas and on young people, on the future criminal behavior of these young people. Etc., etc., etc.
I have virtually never seen a non-experimental study reported in the media correctly. The original research report almost never uses the word cause, but the media people almost invariably give an inappropriate causal interpretation, even if the word cause is not used by them either.
[ IV. A Brief History of Non-Experimental Causal Inference
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Dissertation, Part I
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[V. The Need for a New Causal Inference Tool
Give Researchers algorithm for applying Causal Statistics
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[I.1 Philosophical and Logical Foundations of Causal Statistics: A Two Variable Example
Identification pb
Recursive equations
Endogenous
Exogenous, etc Dissertation, Chapter 13
Estimations
Parameter interpretations
Error handling
Structure and structural Change
Start with general form of Causal Statistics
Recommend econometric sources and note their avoidance of “cause”
Simultaneous equation parameters partial out the considered correlation
Give researchers algorithm for applying Causal Statistics
Consider an empirical study in which variables B (i.e., number of book owned by the family) and R (i.e., reading capability of the child) are found to be correlated. If we assume (1) that no outside variables causally affect both B and R in such a way as to change their correlation, (2) that R does not cause B, and (3) that causal relationship are linear; one can validly conclude, subject to the usual statistical error, that B causes R with standardized strength equal to the correlation coefficient.
A “valid” causal inference is a causal inference which is a logically necessary result of the definitions stated, the assumptions made, and the data collected. In other words, a “valid” causal connection results from a causal inference arrived at by the proper application of Causal Statistics.
Note that a valid causal inference does not necessarily result in a correct causal connection. If one or more of the assumptions is incorrect, the causal connection will be incorrect. The greater the degree to which the assumptions are in error, generally, the greater the error in the causal connection drawn. This second assumption set can be very restrictive and questionable, yet, that is not the fault of Causal Statistics. It is, if you will, the fault of logic and the universe we exist in. Gravity can be inconvenient, if you want to fly, but the field of physics is not at fault. In fact physics can assist in overcoming the problem; same with Causal Statistics.
Questions which I had about causal inquiring systems that existed in 1968.
When assumptions were these systems based on?
To the systems you'll causal parameters? Only causal parameters? Partially causal parameters? What determines these things?
What is the definition of cause?
Why are causal inferences drawn so effortlessly from experimental studies? and seem almost impossible to draw from nonexperimental studies?
Many people assert that Hume sounded the death knell for causality and proved that it was a word and concept which should not be used. Hume showed that causal connections cannot be proved.
Our causal connections immutable?
How can you have causal connections in the social sciences which are separated in time and space?
What is statistical causality? Sometimes it causes and sometimes it doesn't?
Are all causes scientific laws built into this fabric of the universe?
Our mathematical connections causal laws? Like, people who are taller in inches are also taller in feet.
Our definitional connections causal laws?
How about variables or objects, they are almost always composed of smaller variables or objects? Are the universes causal laws working between macro variables? What if the structure of the macro variables are different from one situation to the next, like thrown rocks or depression?
Many people argued that causality was incompatible with theory building in the social sciences. Are they correct or not?
In causal inference, you are not discovering fundamental laws of the universe, but regularities observable in the macro world, flowing from the basic micro causal laws of the universe. These macro causal inferences are like the lumpy nature of objects in the universe, like asteroids, planets, stars, galaxies, galaxy clusters, etc.
The concept of regularity or lumpiness is not limited to causal inference. Consider a correlation. The Association discovered between two variables is not likely a connection built into the fundamental micro laws of the universe. It is simply a regularity, observable in the macro universe.
What's the relationship between macro causal laws and statistical causal laws?
Another reason I couldn't use the existing techniques for drawing causal inferences for my dissertation, was that I didn't know how to determine if regression parameters were causal or not. You get a regression parameter of 3. Is it causal or associative? Or is it somewhat causal and the rest associative, and if so, how much?]
{Click to go back to: Table of Contents, I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII, XIII, XIV, XV, XVI, XVII, XVIII}
III. The Needs for Non-Causal and Causal Inference
Dissertation in Library of Congress and University of California Berkeley Graduate Library
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A government planner, as a consumer of social science research, might want to predict recidivism rates as a function of the hours per week an inmate spends in self-improvement and educational programs, in order to plan for future prison space requirements. For a non-experimental study to produce the information that the government planner needs, the researcher need only obtain the association (i.e., correlation) between prison program participation and recidivism (a dichotomous variable) or the amount of time prior to return (a continuous variable). In either event, the data analysis would require only a straight forward application of Associative Statistics, a subset of Classical Statistics, and no information about causation is needed.
From the correlation or, more desirably, the regression results, government planners could predict recidivism rates and, from that and other information, prison space requirements.
But more importantly, government planners, counselors, administrators, etc. would like to know how to intervene to reduce the recidivism rate. Such manipulation would require knowledge of the causes, both positive and negative, of recidivism rate for one or more of the variables which government can control.
The demand by research consumers for prediction, without intervention, is miniscule compared to the demand (generally, unsatisfied in the Social Sciences) for research results enabling prediction, with intervention or manipulation, i.e., causal results. Accurate causal theories enable us to control the future rather than just forecast it.
Unfortunately, no complete or totally understandable research tool is available or drawing valid causal inferences from non-experimental data. The dominant, almost exclusive, inquiring tool is Classical Statistics, which yields correlations and/or regression coefficients. And, as basic statistics texts will tell you, if they deal with the subject at all, correlation does not imply causation. “Cause” is not even a term in the vocabulary of Classical Statistics; therefore, Classical Statistics alone could not draw valid causal inferences. Yet, the need for causal results and theories is so great that many researchers and research consumers have incorrectly purported to do just that, utilize Classical Statistics to draw valid causal inferences from non-experimental data.
Now, stop a moment to let this lunacy sink in. Causal inferences are the most desired and needed conclusions in the non-experimental sciences. But, there is no complete or logically consistent inquiring tool available to draw the needed valid causal inferences from non-experimental data. [Yet, little funding or effort is being expended to remedy this gross and debilitating short coming. Is such mass stupidity possible? Regretfully Dorothy, it is.]
Presented with these facts, any intelligent person would imagine that a tremendous amount of research money and effort would have been devoted to the development of a complete, understandable, and valid causal inquiring system. But this intelligent person would be, not only wrong, but grossly wrong. No significant money and only slightly more effort has been expended on the methodological problem of causal inference.
Aside: One might argue to the contrary, in that a significant portion of my dissertation was supported by the National Aeronautics and Space Administration through the Space Sciences Laboratory at the University of California (Berkeley). While true, and even though I had worked at the Manned Spacecraft Center (now the Johnson Space Center) for NASA in the late 60's, in the Theoretical Physics Branch analyzing the Bremsstrahlung and other radiation hazards to Apollo astronauts, it was not NASA's intent to fund me to develop Causal Statistics. The original funding was for an empirical study to determine the causes (both positive and negative) of productivity in scientific research and development.
Since the study was non-experimental, causation was difficult to establish. I attempted to use the causal inquiring systems available at that time, but could not apply them with understand, insight, or confidence--e.g., the assumptions implicit in the various systems were unknown, there were no proofs of the appropriateness of such systems, and it was unclear how to disentangle causal components from associative components in coefficients (e.g., regression coefficients) inferred.
I then turned my efforts to the development of a causal inquiring system which could overcome the aforementioned problems. As this work proceeded, I saw a far-reaching importance of this line of statistical research. Its significance dwarfed that of the original R & D study. For this reason the R & D study was discontinued, with the causal statistics project taking its place.
At this point, NASA would likely have cut my funding. But Professor C. West Churchman--the head of Berkeley's Space Sciences Laboratory, my dissertation chairman, and all around intellect and great human being, renewed my funding, probably without saying a word to NASA.