STATISTICAL MODELS FOR CAUSAL ANALYSIS

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1 STATISTICAL MODELS FOR CAUSAL ANALYSIS

2 STATISTICAL MODELS FOR CAUSAL ANALYSIS ROBERT D. RETHERFORD MINJA KIM CHOE Program on Population East-West Center Honolulu, Hawaii A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York Chichester Brisbane Toronto Singapore

The content of thli book U Identical to ptevjoeiprfndnfi. This text is printed on acid-free paper. Copyright Q 1993 by John Wiley & Sons, Inc. All rights reserved. Published simultaneously in Canada.

3 A NOTE TO THE READER: TU* book hai bwa electronically rapradneed from digital information Mond at John Wiley * Sea*. Inc. We we pleated tbsi the at* of loi» new teehaolofy wul enable nt to keep wofkt of enduing tcholarly value in prim a* Ion» at then 1* e rehonable demand for them. The content of thli book U Identical to ptevjoeiprfndnfi. This text is printed on acid-free paper. Copyright Q 1993 by John Wiley & Sons, Inc. All rights reserved. Published simultaneously in Canada. Reproduction or translation of any part of this work beyond that permitted by Section 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY Ubmty of ih~ress w i n g in Publicalion Ikta: Retherford, Robert D. Statistical models for causal analysis/robert D. Retherford, Minja Kim Choe. P. cm. Includes bibliographical references and index. ISBN (cloth) 1. Multivariate analysis. I. Choe, Minja Kim. 11. Title. QA278.R '35-dc CIP

PREFACE This book provides a quick overview of statistical models commonly used in causal analyses of nonexperimental data in the social and biomedical sciences.

4 PREFACE This book provides a quick overview of statistical models commonly used in causal analyses of nonexperimental data in the social and biomedical sciences. Topics covered are simple bivariate regression, multiple regression, multiple classification analysis, path analysis, logit regression, multinomial logit regression, and survival models (proportional hazard models and hazard models with time dependence). The goal is to impart an intuitive understanding and working knowledge of these models with a minimum of mathematics. The strategy is to simplify the treatment of statistical inference and to focus primarily on how to specify and interpret models in the context of testing causal theories. General principles and procedures of statistical inference and model testing are covered, but derivations of sampling distributions and standard errors are omitted, thereby circumventing a great deal of complex mathematics. On the other hand, a fair amount of mathematical detail is retained in the discussion of model specification and interpretation. It is assumed that the reader knows elementary algebra and has mastered the equivalent of a graduate-level introductory course in social or biomedical statistics. Calculus is occasionally used, mainly in illustrative derivations involving minimization or maximization (e.g., minimizing a sum of squares or maximizing a likelihood function) and in the development of survival models in Chapters 7-9. The computer programs by means of which illustrative examples were produced are from BMDP, LIMDEP, SAS, and SPSS, because V

vi PREFACE these major statistical computing packages tend to be generally available. For each type of example, at least one and usually two computer programs are provided in Appendix A of this book.

5 vi PREFACE these major statistical computing packages tend to be generally available. For each type of example, at least one and usually two computer programs are provided in Appendix A of this book. Whenever possible, one of these is from LIMDEP, in order to make things easier for the reader who is new to computers and wishes to learn how to use only one package. The programs may be implemented either on a mainframe computer or on a microcomputer. Some models, such as hazard models with time dependence, may take considerable time to execute on a microcomputer and are best done on a mainframe computer. The examples are taken from the 1974 Fiji Fertility Survey, which was part of the World Fertility Survey. The individual-record tape from the Fiji Fertility Survey is in the public domain and provides a convenient source of constructed examples. Although the examples are taken from the field of population studies, the methods are generally applicable to a wide variety of problems in the social and biomedical sciences as well as other fields. Social science and biomedical professionals wishing to upgrade their methodological skills will find this book useful. The book is also suitable for use as either a core or a supplementary textbook for courses in social or biomedical statistics, following an introductory course. Normally these would be graduate-level courses. However, because of the highly simplified treatment, the book may sometimes be appropriate for advanced undergraduate courses as well. We have used the book as the textbook for a one-semester graduate course, with good results. ACKNOWLEDGMENTS We are grateful to John Bauer, S. I. Laskar, Norman Luther, Andrew Mason, and Mahmudul Haq Khan for helpful comments and suggestions, and to Robin Loomis for assistance in preparing graphs and figures. Hondulu, Hawaii ROBERT D. RETHERFORD MINJA KIM CHOE

6 1 Bivariate Linear Regression 1.1. Terminology, Fitting a Least-Squares Line, The Least-Squares Line as a Causal Model, The Bivariate Linear Regression Model as a Statistical Model, Simplifying Assumptions, Statistical Inference: Generalizing from Sample to Underlying Population, Hypothesis Testing, Confidence Intervals, c Values and Z Values, p Value, Importance of a Good Spread of Values of the Predictor Variable, Beware of Outliers!, Beware of Selection on the Response Variable!, Presentation of Results, Goodness of Fit, Standard Error of the Estimate, s, Coefficient of Determination, r2, and Correlation Coefficient, r, Further Reading, 28 1 vii

7 viii 2 Multiple Reqpvssion The Problem of Bias in Bivariate Linear Regression, Multiple Regression with Two Predictor Variables, Multiple Regression with Three or More Predictor Variables, Dummy Variables to Represent Categorical Variables, Categorical Variables with Two Categories, Categorical Variables with More Than Two Categories, Multicollinearity, Interaction, Model Specification, More Complicated Interactions, Correlation without Interaction, Interaction without Correlation, Nonlinearities, Quadratic Specification, Dummy Variable Specification, Goodness of Fit, Standard Error of the Estimate, s, Coefficient of Determination, R2, and Multiple Correlation Coefficient, R, Corrected R2 and Corrected R, Partial Correlation Coefficient, Statistical Inference, Hypothesis Testing, Confidence Intervals, and p Values for a Single Regression Coefficient, Testing the Difference Between 'Ibo Regression Coefficients, pi and B,, Testing Effects When There Is Interaction, Testing Effects When There Is a Nonlinearity, The ANOVA Table, The Omnibus F Test of the Hypothesis = p2 = *. a =Pk=O, Test of the Hypothesis That Some of the pi Are Zero, Stepwise Regression, Illustrative Examples, Example 1, Example 2, Further Reading, 68

8 ix 3 Multiple Classification Analysis The Basic MCA Table, Unadjusted Values, Adjusted Values, Unadjusted and Adjusted, R, A Numerical Example, The MCA Table in Deviation Form, First Approach to Table Set-up, Second Approach to Table Set-up, A Numerical Example, MCA with Interactions, Table Set-up, A Numerical Example, MCA with Additional Quantitative Control Variables, Table Set-up, A Numerical Example, Expressing Results from Ordinary Multiple Regression in an MCA Format (all Variables Quantitative), Table Set-up, A Numerical Example, Why Present Ordinary Regression Results in an MCA Format?, Presenting MCA Results Graphically, Further Reading, 92 4 Path Analysis 4.1. Path Diagrams and Path Coefficients, Path Models with More Than One Exogenous Variable, Path Models with Control Variables, Saturated and Unsaturated Path Models, Path Analysis with Standardized Variables, Standardized Variables and Standardized Path Coefficients, Path Models in Standardized Form, Standardized Versus Unstandardized Coefficients, Path Models with Interactions and Nonlinearities, Further Reading, Logit Regression 5.1. The Linear Probability Model,

9 X 5.2. The Logit Regression Model, The Logistic Function, The Multivariate Logistic Function, The Odds and the Logit of P, Logit Regression Coefficients as Measures of Effect on Logit P, Odds Ratios as Measures of Effect on the Odds, The Effect on the Odds When the Predictor Variable Is Categorical with More Than Two Categories, The Effect of the Predictor Variables on the Risk P Itself, Interactions, Nonlinearities, Statistical Inference, Tests of Coefficients, Testing the Difference Between Two Models, Goodness of Fit, MCA Adapted to Logit Regression, An Illustrative Example, Table Set-up and Numerical Results, Fitting the Logit Regression Model, The Maximum Likelihood Method, Derivation of the Likelihood Function, Some Limitations of the Logit Regression Model, Further Reading, Multinomial Logit Regression From Logit to Multinomial Logit, ' The Basic Form of the Multinomial Logit Model, Interpretation of Coefficients, Presentation of Results in a Multiple Classification Analysis (MCA) Format, Statistical Inference, Goodness of Fit, Changing the Reference Category of the Response Variable, Multinomial Logit Models with Interactions and Nonlinearities, 159

10 xi 6.3. A More General Formulation of the Multinomial Logit Model, Reconceptualizing Contraceptive Method Choice as a Two-step Process, Further Reading, Survival Models, Part 1: Life Tables Actuarial Life Table, Statistical Inference, Product-Limit Life Table, Testing the Difference Between Two Survival Curves, The Life Table in Continuous Form, Further Reading, Survival Models, Part 2: Proportional Hazard Models Basic Form of the Proportional Hazard Model, A Simple Example, Addition of a Second Predictor Variable, Redefinition of the Baseline Hazard, Relative Risks as Measures of Effect on the Hazard, Hazard Regression Coefficients as Measures of Effect on the Log Hazard, The Effect on the Hazard When the Predictor Variable Is Categorical with More Than Two Categories, Interactions, Nonlinearities, Changing the Reference Category of a Categorical Predictor Variable, Calculation of Life Tables from the Proportional Hazard Model, The Hazard Model in Survivorship Form, The Problem of Unobserved Heterogeneity, Test for Proportionality, Statistical Inference and Goodness of Fit, A Numerical Example, Multiple Classification Analysis (MCA) Adapted to Proportional Hazard Regression, Further Reading, 205

11 xii 9 Survival Models, Part 3: Hazard Models with Time Dependence Time-Dependent Predictor Variables, Coefficients and Effects, Choosing a Baseline Hazard Function, MCA Adapted to Hazard Regression with Time-Dependent Predictor Variables, Time-Dependent Coefficients, Coefficients and Effects, Survival Functions, MCA Adapted to Hazard Regression with Time-Dependent Coefficients, Further Reading, 220 Appendix A Sample Computer Programs A.1. Description of the FIJIDATA File, 221 A.2. SAS Mainframe Programs, 223 Program 1. SAS Program for Multiple Regression (Program Used for Table 2.21, 223 Program 2 SAS Program for Partial Correlation Coefficient (Program Used for Table 3.4, Adjusted R Column), 224 Program 3. SAS Program for Logit Regression (Program Used for Table 5.11, 225 Program 4. SAS Program for Multinomial Logit Regression (Program Used for Table 6.4), 226 A.3. Preparing Data for Survival Analysis, 227 Program 5. SAS Program to Create Input Data File for BMDP and LIMDEP Programs for Life Table and Hazard Model Analysis of Progression from Fifth Birth to Sixth Birth, 228 A.4. BMDP Mainframe Programs, 230 Program 6. BMDPlL Program for Actuarial Life Table: Progression from Fifth Birth to Sixth Birth 221

12 Xiii for Fijians and Indians (Program Used for Table 7.1 and the Actuarial Life Tables in Table 7.21, 230 Program 7. BMDPlL Program for Product-Limit Life Table: Progression from Fifth Birth to Sixth Birth for Fijians and Indians (Programs Used for Product-Limit Life Tables in Table 7.21, 232 Program 8. BMDP2L Program for Proportional Hazard Model: Progression from Fifth to Sixth Birth for Indians (Program Used for Table 8.11, 233 Program 9. BMDP2L Program for Hazard Model with Time- Dependent Covariate (Program Used for Table 9.21, 235 Program 10. BMDP2L Program for Hazard Model with Time- Dependent Covariate and Time-Dependent Effect (Program Used for Table 9.31, 237 AS. LIMDEP Programs for IBM-Compatible Personal Computers, 238 Program 11. LIMDEP Program for Multiple Regression (Program used for Table 2.21, 238 Program 12. LIMDEP Program for Partial Correlation Coefficient (Program Used for Table 3.4, Adjusted R column), 239 Program 13. LIMDEP Program for Logit Regression (Program Used for Table 5.11, 240 Program 14. LIMDEP Program for Multinomial Logit Regression (Program Used for Table 6.41, 241

13 xiv Program 15. LIMDEP Program for Life Table (Actuarial and Product-Limit): Progression from Fifth Birth to Sixth Birth, Indians (Program Used for Table 7.21, 242 Problem 16. LIMDEP Program for Proportional Hazard Model: Progression from Fifth Birth to Sixth Birth, Indians (Programs Used for Table 8.11, 243 Problem 17. LIMDEP Program for Hazard Model with Time- Dependent Covariate: Progression from Fifth Birth to Sixth Birth, Indians (Program Used for Table 9.21, 243 Problem 18. LIMDEP Program for Hazard Model with Time-Dependent Covariate and Time-Dependent Effect: Progression from Fifth Birth to Sixth Birth, Indians (Program Used for Table 9.31, 244 Appendix B Statistical Reference Tables 245 References 251 Index 254

Market Risk Analysis Volume I

Market Risk Analysis Volume I Quantitative Methods in Finance Carol Alexander John Wiley & Sons, Ltd List of Figures List of Tables List of Examples Foreword Preface to Volume I xiii xvi xvii xix xxiii