Latent variable models : an introduction to factor, path, and structural equation analysis

330 Pages·2004·16.433 MB·English

by  John C Loehlin

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Latent Variable Models
An Introduction to Factor, Path,
and Structural Equation Analysis
Fourth Edition
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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
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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
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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
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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
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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
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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
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