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Latent Variable Models: An Introduction to Factor, Path, and Structural Equation Analysis PDF

330 Pages·2003·16.38 MB·English
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Latent Variable Models An Introduction to Factor, Path, and Structural Equation Analysis Fourth Edition This page intentionally left blank Latent Variable Models An Introduction to Factor, Path, and Structural Equation Analysis Fourth Edition John C. Loehlin LAWRENCE ERLBAUM ASSOCIATES, PUBLISHERS 2004 Mahwah, New Jersey London Camera ready copy for this book was provided by the author. Copyright © 2004 by Lawrence Erlbaum Associates, Inc. All rights reserved. No part of this book may be reproduced in any form, by photostat, microform, retrieval system, or any other means, without prior written permission of the publisher. Lawrence Erlbaum Associates, Inc., Publishers 10 Industrial Avenue Mahwah, New Jersey 07430 Cover design by Sean Trane Sciarrone Library of Congress Cataloging-in-Publlcation Data Loehlin, John C. Latent variable models : an introduction to factor, path, and struc- tural equation analysis / John C. Loehlin.—4th ed. p. cm. Includes bibliographical references and index. ISBN 0-8058-4909-2 (cloth : alk. paper) ISBN 0-8058-4910-6 (pbk. : alk. paper) 1. Latent variables. 2. Latent structure analysis. 3. Factor analysis. 4. Path analysis. I. Title. QA278.6.L64 2004 519.5'35—dc22 2003063116 CIP Books published by Lawrence Erlbaum Associates are printed on acid- free paper, and their bindings are chosen for strength and durability. Printed in the United States of America 10 9 8 7 6 5 4 3 21 Contents Preface ix Chapter One: Path models in factor, path, and structural equation analysis 1 Path diagrams 2 Path analysis 8 Factor models 16 Structural equations 23 Original and standardized variables 24 Differences from some related topics 28 Notes 30 Exercises 32 Chapter Two: Fitting path models 35 Iterative solution of path equations 35 Matrix formulation of path models 40 Full-fledged model-fitting programs 44 Fit functions 52 Hierarchical X2 tests 61 Descriptive criteria of model fits 67 The power to reject an incorrect model 70 Identification 73 Missing data 75 Correlations versus covariances in model fitting 78 Notes 80 Exercises 84 Chapter Three: Fitting path and structural models to data from a single group on a single occasion 87 Structural and measurement models 87 Confirmatory factor analysis 92 Some psychometric applications of path and structural models 95 Structural models-controlling extraneous variables 102 Models with reciprocal influences and correlated errors 106 Nonlinear effects among latent variables 111 Notes 116 Exercises 117 Chapter Four: Fitting models involving repeated measures or multiple groups 120 Models of events over time 121 Models comparing different groups 130 Fitting models to means as well as covariances 139 The versatility of multiple-group designs 147 A concluding comment 148 Notes 149 Exercises 150 Chapter Five: Exploratory factor analysis—basics 152 Factor extraction 154 Estimating communalities 154 Determining the number of factors 157 Rotation 169 An example: Thurstone's box problem 177 Factor analysis using packaged programs-SPSS and SAS 181 Notes 183 Exercises 185 Chapter Six: Exploratory factor analysis-elaborations 187 Rescalings-Alpha and Canonical factors 187 Alternative stopping criteria 190 Alternative rotation methods 193 Estimating factor scores 196 Higher order factors 201 Nonlinear factor analysis 206 Notes 210 Exercises 211 Chapter Seven: Issues in the application of latent variable analysis 213 Exploratory modification of a model 213 Alternative models 217 Can path diagrams be constructed automatically? 222 Modes of latent variable analysis 224 Criticisms of latent variable modeling 230 Notes 234 Exercises 236 VI Appendices 238 A. Simple matrix operations 238 B. Derivation of matrix version of path equations 245 C. LISREL matrices and examples 247 D. Various goodness-of-fit indices 251 E. Phantom variables 258 F. Data matrix for Thurstone's box problem 260 G. Table of Chi Square 262 H. Noncentral Chi Square for estimating power 263 I. Power of a test of poor fit and sample sizes needed for powers of .80 and .90 264 Answers to exercises 265 References 276 Index 309 VII This page intentionally left blank Preface This book is intended as an introduction to an exciting growth area in social science methodology--the use of multiple-latent-variable models. Psychologists and other social scientists have long been familiar with one subvariety of such modeling, factor analysis-more properly, exploratory factor analysis. In recent decades, confirmatory factor analysis, path analysis, and structural equation modeling have come out of specialized niches and are making their bid to become basic tools in the research repertoire of the social scientist, particularly the one who is forced to deal with complex real-life phenomena in the round: the sociologist, the political scientist, the social, educational, clinical, industrial, personality or developmental psychologist, the marketing researcher, and the like. All these methods are at heart one, as I have tried to emphasize in the chapters to follow. I have used earlier versions of this book in teaching graduate students from psychology and related disciplines, and have found the particular approach used-via path diagrams-to be effective in helping not-too- mathematical students grasp underlying relationships, as opposed to merely going through the motions of running computer programs. In some sections of the book a certain amount of elementary matrix algebra is employed; an appendix on the topic is provided for those who may need help here. In the interests of accessibility, I have tried to maintain a relatively informal style, and to keep the main text fairly uncluttered with references. The notes at the end of each chapter are intended to provide the serious student with a path into the technical literature, as well as to draw his or her attention to some issues beyond the scope of the basic treatment. The book is not closely tied to a particular computer program or package, although there is some special attention paid to LISREL, EQS, AMOS, and MX. I assume that most users will have access to a latent-variable model- fitting program on the order of LISREL, EQS, CALIS, AMOS, Mplus, MX, RAMONA, or SEPATH, and an exploratory factor analysis package such as those in SPSS or SAS. In some places, a matrix manipulation facility such as that in MINITAB, SAS, or SPSS would be helpful. I have provided some introductory material but have not tried to tell students all they need to know to run actual programs-such information is often local, ephemeral, or both. The IX

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This book introduces multiple-latent variable models by utilizing path diagrams to explain the underlying relationships in the models. This approach helps less mathematically inclined students grasp the underlying relationships between path analysis, factor analysis, and structural equation modeling
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