Structural Equations with Latent Variables

Transcription

1 Structural Equations with Latent Variables

2 Structural Equations with Latent Variables KENNETH A. BOLLEN Department of Sociology The University of North Carolina at Chapel Hill Chapel Hill, North Carolina n WILEY A Wiley-Interscience Publication JOHN WILEY & SONS New York Chichester. Brisbane. Toronto Singapore

3 To Barbara and to my parents Copyright 1989 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. Librar; of Congress Cataloging in Publication Data: Bollen, Kenneth A. Structural equations with latent variables/kenneth A. Bollen. p. em.- (Wiley series in probability and mathematical statistics. Applied probability and statistics section, ISSN ) "A Wiley-Interscience publication." Bibliography: p. Includes index. 1. Social sciences---statistical methods. 2. Latent variables. I. Ti tie. II. Series. HA29.B '.29-dc ISBN CIP Printed in the United States of America

4 Preface Within the past decade the vocabulary of quantitative research in the social sciences has changed. Terms such as "LISREL," "covariance structures," "latent variables," "multiple indicators," and "path models" are commonplace. The structural equation models that lie behind these terms are a powerful generalization of earlier statistical approaches. They are changing researchers' perspectives on statistical modeling and building bridges between the way social scientists think substantively and the way they analyze data. We can view these models in several ways. They are regression equations with less restrictive assumptions that allow measurement error in the explanatory as well as the dependent variables. They consist of factor analyses that permit direct and indirect effects between factors. They routinely include multiple indicators and latent variables. In brief, these models encompass and extend regression, econometric, and factor analysis procedures. The book provides a comprehensive introduction to the general structural equation system, commonly known as the "LISREL model." One purpose of the book is to demonstrate the generality of this model. Rather than treating path analysis, recursive and nonrecursive models, classical econometrics, and confirmatory factor analysis as distinct and unique, I treat them as special cases of a common model. Another goal of the book is to emphasize the application of these techniques. Empirical examples appear throughout. To gain practice with the procedures, I encourage the reader to reestimate the examples, and then to devise and estimate new models. Several chapters contain some of the LISREL or EQS programs I used to obtain the results for the empirical examples. I have kept the examples as realistic as possible. This means that some of the initial specifications do not fit well. Through my experiences with students, v

5 vi PREFACE colleagues, and in my own work, I frequently have found that the beginning model does not adequately describe the data. Respecification is often necessary. I note the difficulties this creates in proper interpretations of significance tests and the added importance of replication. A final purpose is to emphasize the crucial role played by substantive expertise in most stages of the modeling process. Structural equation models are not very helpful if you have little idea about the subject matter. To begin the fitting process, the analysts must draw upon their knowledge to construct a multiequation system that specifies the relations between all latent variables, disturbances, and indicators. Furthermore they must turn to substantive information when respecifying models and when evaluating the final model Empirical results can reveal that initial ideas are in error or they can suggest ways to modify a model, but they are given meaning only within the context of a substantively informed model Structural equation models can be presented in two ways. One is to start with the general model and then show its specializations to simpler models. The other is to begin with the simpler models and to build toward the general model I have chosen the latter strategy. I start with the regression/econometric and factor analysis models and present them from the perspective of the general model This has the advantage of gradually including new material while having types of models with which the reader is somewhat familiar. It also encourages viewing old techniques in a new light and shows the often unrealistic assumptions implicit in standard regression/ econometric and factor analyses. Specifically, I have organized the book as follows. Chapter 2 introduces several methodological tools. I present the model notation, covariances and covariance algebra, and a more detailed account of path analysis. Appendixes A and B at the end of the book provide reviews of matrix algebra and of asymptotic distribution theory. Chapter 3 addresses the issue of causality. Implicitly, the idea of causality pervades much of the structural equation writings. The meaning of causality is subject to much controversy. I raise some of the issues behind the controversy and present a structural equation perspective on the meaning of causality. The regression/econometric models for observed variables are the subject of Chapter 4. Though many readers have experience with these, the covariance structure viewpoint will be new to many. The consequences of random measurement error in observed variable models is the topic of Chapter 5. The chapter shows why and when we should care about measurement error in observed variables. Once we recognize that variables are measured with error, we need to consider the relation between the error-free variable and the observed variable. Chapter 6 is an examination of this relation. It introduces proce-

6 PREFACE vii dures for developing measures and explores the concepts of reliability and validity. Chapter 7 is on confirmatory factor analysis, which is used for estimating measurement models such as those in Chapter 6. Finally, Chapters 8 and 9 provide the general structural equation model with latent variables. Chapter 8 emphasizes the "basics," whereas Chapter 9 treats more advanced topics such as arbitrary distribution estimators and the trea.tment of categorical observed variables. The main motivation for writing this book arose from my experiences teaching at the Interuniversity Consortium for Political and Social Research (ICPSR) Summer Training Program in Methodology at the University of Michigan ( ). I could not find a text suitable for graduate students and professionals with training in different disciplines. Nor could I find a comprehensive introduction to these procedures. I have written the book for social scientists, market researchers, applied statisticians, and other analysts who plan to use structural equation or LISREL models. I assume that readers have prior exposure and experience with matrix algebra and regression analysis. A background in factor analysis is helpful but not essential. Joreskog and Sorbom's (1986) LISREL and Bentler's (1985) EQS are the two most popular structural equation software packages. I make frequent reference to them, but the ideas of the book extend beyond any specific program. I have many people to thank for help in preparing this book. The Interuniversity Consortium for Political and Social Research (ICPSR) at the University of Michigan (Ann Arbor) has made it possible for me to teach these techniques for the last nine years in their Summer Program in Quantitative Methods. Bob Hoyer started me there. Hank Heitowit and the staff of ICPSR have continued to make it an ideal teaching environment. A number of the hundreds of graduate students, professors, and other professionals who attended the courses provided general and specific comments to improve the book. Gerhard Arminger (University of Wuppertal), Jan de Leeuw (University of Leiden), Raymond Horton (Lehigh University), Frederick Lorenz (Iowa State University), John Fox (York University), Robert Stine (University of Pennsylvania), Boone Turchi (University of North Carolina), and members of the Statistical and Mathematical Sociology Group at the University of North Carolina provided valuable comments on several of the chapters. Barbara Entwisle Bollen read several drafts of most chapters, and her feedback and ideas are reflected throughout the book. Without her encouragement, I do not know when or if I would have completed the book. Brenda Le Blanc of Dartmouth College and Priscilla Preston and Jenny Neville of the University of North Carolina (Chapel Hill) performed expert typing above and beyond the call of duty. Stephen Birdsall, Jack Kasarda,

7 viii PREFACE and Larry Levine smoothed the path several times as I moved the manuscript from Dartmouth College to the University of North Carolina. I thank the Committee for Scholarly Publications, Artistic Exhibitions, and Performances at the University of North Carolina, who provided financial support needed to complete the book. KENNETH A. BOLLEN Chapel Hill. North Carolina

8 Contents 1. Introduction Historical Background, 4 2. Model Notation, Covariances, and Path Analysis Model Notation, 10 Latent Variable Model, 11 Measurement Model, 16 Covariance, 21 Covariance Algebra, 21 Sample Covariances, 23 Path Analysis, 32 Path Diagrams, 32 Decomposition of Covariances and Correlations, 34 Total, Direct, and Indirect Effects, 36 Summary, Causality and Causal Models Nature of Causality, 40 Isolation, 45 Association, 57 Direction of Causation, 61 Limitations of "Causal" Modeling, 67 Model-Data versus Model-Reality Consistency, 67 Experimental and Nonexperimental Research, ix

9 x CONTENTS Criticisms of Structural Equation Models, 78 Summary, Structural Equation Models with Observed Variables Model Specification, 80 Implied Covariance Matrix, 85 Identification, 88 t-rule, 93 Null BRule, 94 Recursive Rule, 95 Rank and Order Conditions, 98 Summary of Identification Rules, 103 Estimation, 104 Maximum Likelihood (ML), 107 Unweighted Least Squares (ULS), III Generalized Least Squares (GLS), 113 Other Estimators, 115 Empirical Examples, 116 Further Topics, 123 Standardized and Unstandardized Coefficients, 123 Alternative Assumptions for x, 126 Interaction Terms, 128 Intercepts, 129 Analysis of Variance (ANOVA) and Related Techniques, 130 Summary, 131 Appendix 4A Derivation of FML (y and x Multinormal), 131 Appendix 4B Derivation of FML (S Wishart Distribution), 134 Appendix 4C Numerical Solutions to Minimize Fitting Functions, 136 Starting Values, 137 Steps in the Sequence, 138 Stopping Criteria, 143 Appendix 4D Illustrations of LISREL and EQS Programs,

10 CONTENTS xi LISREL Program, 144 EQS Program, The Consequences of Measurement Error Univariate Consequences, 151 Bivariate and Simple Regression Consequences, 154 Consequences in Multiple Regression, 159 Correlated Errors of Measurement, 167 Consequences in Multiequation Systems, 168 Union Sentiment Example, 168 Unknown Reliabilities, 170 Objective and Subjective Socioeconomic Status (SES), 172 Summary, 175 Appendix 5A Illustrations of LISREL and EQS Programs, 176 LISREL Program, 176 EQS Program, Measurement Models: The Relation between Latent and Observed Variables Measurement Models, 179 Validity, 184 Content Validity, 185 Criterion Validity, 186 Construct Validity, 188 Convergent and Discriminant Validity, 190 Alternatives to Classical V alidi ty Measures, 193 Summary, 206 Reliability, 206 Empirical Means of Assessing Reliability, 209 Alternatives to Classical Reliability Measures, 218 Summary, 221 Cause Indicators, 222 Summary, 223 Appendix 6A LISREL Program for the Multitrait-Multimethod Example,

11 xii 7. Confirmatory Factor Analysis Exploratory and Confirmatory Factor Analysis, 226 Model Specification, 233 Implied Covariance Matrix, 236 Identification, 238 t-rule, 242 Thfee-Indicator Rules, 244 Two-Indicator Rules, 244 Summary of Rules, 246 An Insufficient Condition of Identification, 246 Empirical Tests of Identification, 246 Recommendations for Checking Identification, 251 Political Democracy Example, 251 Estimation, 254 Nonconvergence, 254 Model Evaluation, 256 Overall Model Fit Measures, 256 Component Fit Measures, 281 Does the Sympathy and Anger Model Fit?, 289 Comparisons of Models, 289 Likelihood Ratio (LR) Test, 292 Lagrangian Multiplier (LM) Test, 293 Wald (W) Test, 293 Summary of LR, LM, and W Tests, 295 Respecification of Model, 296 Theory and Substantive-Based Revisions, 296 Exploratory LR, LM, and W Tests, 298 Other Empirical Procedures, 303 Summary of Respecification Procedures, 304 Extensions, 305 Factor Score Estimation, 305 Latent Variable Means and Equation Intercepts, 306 Cause Indicators, 311 Higher-Order Factor Analyses, 313 Summary, 315 Appendix 7A Examples of Program Listings, 316 CONTENTS 226

12 CONTENTS xiii Example 1: Estimating Means and Intercepts, 316 Example 2: Estimating a Higher-Order Factor Analysis, The General Model, Part I: Latent Variable and Measurement Models Combined 319 Model Specification, 319 Implied Covariance Matrix, 323 Identification, 326 I-Rule, 328 Two-Step Rule, 328 MIMIC Rule, 331 Summary of Identification Rules, 331 Industrialization and Political Democracy Example, 332 Estimation and Model Evaluation, 333 Power of Significance Test, 338 Standardized and Unstandardized Coefficients, 349 Means and Equation Constants, 350 Comparing Groups, 355 Invariance of Intercepts and Means, 365 Missing Values, 369 Alternative Estimators of Covariance Matrix, 370 Explicit Estimation, 373 Systematic Missing Values, 375 Total, Direct, and Indirect Effects, 376 Specific Effects, 383 Summary, 389 Appendix 8A Appendix 8B Asymptotic Variances of Effects, 390 Listing of the LISREL VI Program for Missing Value Example, The General Model, Part II: Extensions 395 Alternative Notations/Representations, 395 Equality and Inequality Constraints, 401 Quadratic and Interaction Terms, 403 Instrumental-Variable (IV) Estimators, 409 Distributional Assumptions, 415

13 xiv CONTENTS Consequences, 416 Detection, 418 Correction, 425 Categorical Observed Variables, 433 Assumptions Violated and Consequences, 433 Robustness Studies, 434 Summary of Robustness to Categorization, 438 Corrective Procedures, 439 Summary, 446 Appendix 9A LISREL Program for Model in Figure 9.1(c), 447 Appendix A. Matrix Algebra Review Scalars, Vectors, and Matrices, 449 Matrix Operations, Appendix B. Asymptotic Distribution Theory Convergence in Probability, 466 Convergence in Distributions, References Index