Table Of ContentSpringer Texts in Statistics
Advisors:
George Casella Stephen Fienberg Ingram Olkin
Springer
New York
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Springer Texts in Statistics
Alfred: Elements of Statistics for the Life and Social Sciences
Berger: An Introduction to Probability and Stochastic Processes
Blom: Probability and Statistics: Theory and Applications
Brockwell and Davis: An Introduction to Times Series and Forecasting
Chow and Teicher: Probability Theory: Independence, Interchangeability,
Martingales, Third Edition
Christensen: Plane Answers to Complex Questions: The Theory of Linear
Models, Second Edition
Christensen: Linear Models for Multivariate, Time Series, and Spatial Data
Christensen: Log-Linear Models and Logistic Regression, Second Edition
Creighton: A First Course in Probability Models and Statistical Inference
Dean and Voss: Design and Analysis of Experiments
du Toit, Steyn, and Stumpf" Graphical Exploratory Data Analysis
Edwards: Introduction to Graphical Modelling
Finkelstein and Levin: Statistics for Lawyers
Flury: A First Course in Multivariate Statistics
Jobson: Applied Multivariate Data Analysis, Volume I: Regression and
Experimental Design
Jobson: Applied Multivariate Data Analysis, Volume II: Categorical and
Multivariate Methods
Kalbfleisch: Probability and Statistical Inference, Volume I: Probability,
Second Edition
Kalbfleisch: Probability and Statistical Inference, Volume II: Statistical
Inference, Second Edition
Karr: Probability
KeyJitz: Applied Mathematical Demography, Second Edition
Kiefer: Introduction to Statistical Inference
Kokoska and Nevison: Statistical Tables and Formulae
Lehmann: Elements of Large-Sample Theory
Lehmann: Testing Statistical Hypotheses, Second Edition
Lehmann and Casella: Theory of Point Estimation, Second Edition
Lindman: Analysis of Variance in Experimental Design
Lindsey: Applying Generalized Linear Models
Madansky: Prescriptions for Working Statisticians
McPherson: Statistics in Scientific Investigation: Its Basis, Application, and
Interpretation
Mueller: Basic Principles of Structural Equation Modeling
Nguyen and Rogers: Fundamentals of Mathematical Statistics: Volume I:
Probability for Statistics
Nguyen and Rogers: Fundamentals of Mathematical Statistics: Volume II:
Statistical Inference
(Continued after index)
Ralph O. Mueller
Basic Principles of
Structural Equation
Modeling
An Introduction to LISREL and EQS
With 25 Illustrations
Springer
Ralph O. Mueller, PhD
Department of Educational Leadership
Graduate School of Education and
Human Development
The George Washington University
Washington, DC 20052
USA
rmueller@gwis2.circ.gwu edu
Editorial Board
George Casella Stephen Fienberg Ingram Olkin
Biometrics Unit Department of Statistics Department of Statistics
Cornell University Carnegie Mellon University Stanford University
Ithaca. NY 14853-7801 Pittsburgh, PA 15213-3890 Stanford, CA 94305
USA USA USA
On the cover: The artwork is based on an illustration of the basic matrices for structural equation
models.
Library of Congress Cataloging-in-Publication Data
Mueller, Ralph O.
Basic principles of structural equation modeling: an
introduction to LISREL and EQS/Ralph O. Mueller.
p. cm.-(Springer texts in statistics)
Includes bibliographical references (pp. 216-221) and index.
ISBN-13: 978-1-4612-8455-0 e-ISBN: 978-1-4612-3974-1
DOl: 10.1007/978-1-4612-3974-1
1. LISREL. 2. EQS (Computer file) 3. Path analysis-Data
processing. 4. Social sciences-Statistical methods. I. Title.
II. Series.
QA278.3.M84 1996
519.5'35-dc20 95-15043
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© 1996 Springer-Verlag New York, Inc.
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To Mama and Papa;
to Dan, Paula;
and to Rob:
My family of choice
Preface
During the last two decades, structural equation modeling (SEM) has emerged
as a powerful multivariate data analysis tool in social science research
settings, especially in the fields of sociology, psychology, and education.
Although its roots can be traced back to the first half of this century, when
Spearman (1904) developed factor analysis and Wright (1934) introduced
path analysis, it was not until the 1970s that the works by Karl Joreskog and
his associates (e.g., Joreskog, 1977; Joreskog and Van Thillo, 1973) began to
make general SEM techniques accessible to the social and behavioral science
research communities. Today, with the development and increasing avail
ability of SEM computer programs, SEM has become a well-established
and respected data analysis method, incorporating many of the traditional
analysis techniques as special cases. State-of-the-art SEM software packages
such as LISREL (Joreskog and Sorbom, 1993a,b) and EQS (Bentler, 1993;
Bentler and Wu, 1993) handle a variety of ordinary least squares regression
designs as well as complex structural equation models involving variables
with arbitrary distributions.
Unfortunately, many students and researchers hesitate to use SEM
methods, perhaps due to the somewhat complex underlying statistical repre
sentation and theory. In my opinion, social science students and researchers
can benefit greatly from acquiring knowledge and skills in SEM since the
methods-applied appropriately-can provide a bridge between the theo
retical and empirical aspects of behavioral research. That is, interpretations
of SEM analyses can assist in understanding aspects of social and behavioral
phenomena if (a) a "good" initial model is conceptualized based on a sound
underlying substantive theory; (b) appropriate data are collected to estimate
the unknown population parameters; (c) the fit of those data to the a priori
hypothesized model is assessed; and (d) if theoretically justified, the initial
model is modified appropriately should evidence of lack-of-fit and model
misspecification arise. Structural equation modeling thus should be under
stood as a research process, not as a mere statistical technique.
VII
V1I1 Preface
Many social science graduate programs now offer introductions to SEM
within their quantitative methods course sequences. Several currently avail
able textbooks (e.g., Blalock, 1964; Bollen, 1989; Duncan, 1975; Hayduk,
1987; Kenny, 1979; Loehlin, 1992) reflect the rapidly increasing interest and
advances in SEM methods. However, most treatments of SEM are either
outdated and do not provide information on recent statistical advances and
modern computer software, or they are of an advanced nature, sometimes
making the initial study of SEM techniques unattractive and cumbersome to
potential users. The present book was written to bridge this gap by address
ing former deficiencies in a readily accessible introductory text.
The foci of the book are basic concepts and applications of SEM within
the social and behavioral sciences. As such, students in any of the social and
behavioral sciences with a background equivalent to a standard two-quarter
or two-semester sequence in quantitative research methods (general ANOV A
and linear regression models and basic concepts in applied measurement)
should encounter no difficulties in mastering the content of this book. The
minimal knowledge of matrix algebra that is required to understand the basic
statistical foundations of SEM may be reviewed in this text (Appendix C) or
acquired from any elementary linear algebra textbook. The exercises and
selected references at the end of each chapter serve to test and strengthen the
understanding of presented topics and provide the reader with a starting
point into the current literature on SEM and related topics.
Central to the book is the development of SEM techniques by sequen
tially presenting linear regression and recursive path analysis, confirmatory
factor analysis, and more general structural equation models. To illustrate
statistical concepts and techniques, I have chosen to use the computer pro
grams LISREL (Joreskog and Sorbom, 1993a,b) and EQS (Bentler, 1993) to
analyze data from selected research in the fields of sociology and counseling
psychology. The book can be used as an introduction to the LISREL and
EQS programs themselves, although certainly it is not intended to be a
substitute for the program manuals (LISREL: Joreskog & Sorbom, 1993a,b;
EQS: Bentler, 1993; Bentler and Wu, 1993). While other computer programs
could have been utilized [e.g., Steiger's (1989) EzPATH, MuthCn's (1988)
LISCOMP, or SAS's PROC CALIS (SAS Institute Inc., 1990)], the choice of
introducing LISREL and EQS was based on my perception that they are
the most widely used, known, and accepted structural equation programs
to date.
The notation used to present statistical concepts is consistent with that in
other treatments of SEM (e.g., Bollen, 1989; Hayduk, 1987) and follows the
Joreskog-Keesling-Wiley approach to represent systems of structural equa
tions (e.g., Joreskog, 1973, 1977; Joreskog and Sorbom, 1993a; Keesling, 1972;
Wiley, 1973). Since all versions of LISREL are based on this approach, the
program's matrix-based syntax is easily explained and understood once some
underlying statistical theory has been introduced. However, the LISREL
package also features an alternative command language, SIMPLIS (Joreskog
Preface IX
and Sorbom, 1993b), which allows the researcher to analyze structural equa
tion models without fully understanding and using matrix representations.
While LISREL examples throughout the chapters use the matrix-based syn
tax to illustrate the statistical foundation and representation of SEM, their
equivalents using SIMPLIS are explained in Appendix A.
The EQS language, on the other hand, is built around the Bentler-Weeks
model (e.g., Bentler, 1993; Bentler and Weeks, 1979, 1980); this alternative
way to represent structural equation systems is not used in this book. Like
SIMPLIS, the EQS syntax does not involve matrix specifications and, hence,
knowledge of the matrix form of the Bentler-Weeks model is not required to
understand and use the EQS program. The package's MS Windows version
(Bentler and Wu, 1993) includes the BuilLEQS option that allows the
analyst to create an EQS input file without actually having to write the input
program but, instead, by "clicking" on certain options within various dia
logue boxes. I feel that knowledge of the program's syntax for model specifi
cation is of overall educational value and, thus, discuss actual input files
rather than emphasizing the Build_EQS option.
Initially, any SEM programming language that does not depend on an
understanding of the matrix representation of structural equation models
might seem easier than a traditional matrix-based syntax. To acquire an
"aesthetic" appreciation of SEM, however, it is necessary to grasp some of
the underlying statistical principles and formulations-be it via the approach
used here or the one proposed in the Bentler-Weeks model. Getting
acquainted with the syntax of the matrix- and nonmatrix-based command
languages of LISREL and EQS will aid in this endeavor.
In summary, this book was written with four goals in mind: (1) to help
users of contemporary social science research methods gain an understand
ing and appreciation of the main goals and advantages of "doing" SEM; (2)
to explicate some of the fundamental statistical theory that underlies SEM;
(3) to illustrate the use of the software packages LISREL and EQS in the
analysis of basic recursive structural equation models; and, most impor
tantly, (4) to stimulate the reader's curiosity to consult more advanced
treatments of SEM.
Organization and Historical Perspective
The book contains three chapters, largely reflecting the historical develop
ment of what is known now as the general area of SEM: from a discussion of
classical path analysis (Chapter 1) and an introduction to confirmatory factor
analysis (Chapter 2) to an exploration of basic analyses of more general
structural equation models (Chapter 3). In addition, the book contains five
appendices: Appendix A presents all LISREL examples from the chapters in
the SIMPLIS command language and serves as an introduction to this
x Preface
alternative LISREL syntax. Appendix B briefly reviews the statistical con
cepts of distributional location, dispersion, and variable association mainly
to explain some of the notation used in this book; some elementary concepts
of matrix algebra are presented in Appendix C. Finally, Appendices D and E
contain coding information and summary statistics for all variables used in
the LISREL and EQS example analyses.
Chapter 1 introduces the language and notation of SEM through a discus
sion of classical recursive path analysis. Most modern treatments trace the
beginnings of SEM to the development of path analysis by Sewall Wright
during the 1920s and 1930s (Wright, 1921, 1934). However, it took almost
half a century before the method was introduced to the social sciences with
articles by Duncan (1966) in sociology, Werts and Linn (1970) in psychology,
and Anderson and Evans (1974) in education, among others. Early path
analyses were simply ordinary least squares (OLS) regression applications to
sets of variables that were structurally related. In many ways, SEM tech
niques still can be seen as nothing more than the analysis of simultaneous
regression equations. In fact, if the statistical assumptions underlying linear
regression are met, standard OLS estimation~as available in computer
programs such as SPSS, SAS, or BMDP~can be used to estimate the
parameters in some structural equation models.
Throughout the review of univariate simple and multiple linear regression
in Chapter 1, standard regression notation is replaced by that commonly
used in SEM (the Joreskog-Keesling-Wiley notation system), providing an
easy transition into the basic concepts of SEM. The introduction of path
diagrams and structural equations leads to the First Law of Path Analysis,
which allows for the decomposition of covariances between any two vari
ables in a recursive path model. In turn, such decompositions lead to the
definitions of the direct, indirect, and total effects among structurally ordered
variables within a specific model. Finally, a discussion and demonstration of
the identification problem in recursive path models is included to sensitize
the reader to a complex issue that demands attention before SEM analyses
can be conducted. Throughout the chapter, theoretical concepts are illus
trated by annotated LISREL and EQS analyses of a data set taken from the
sociological literature.
In Chapter 2, the inclusion of latent, unobserved variables in structural
equation models is introduced from a confirmatory factor analysis (CFA )
perspective. Based on Spearman's (1904) discovery of exploratory factor
analysis (EFA), several individuals (e.g., Joreskog, 1967, 1969; Lawley, 1940,
1967; Thurstone, 1947) advanced the statistical theory to represent relation
ships among observed variables in terms of a smaller number of underlying
theoretical constructs, that is, latent variables. It was not until the mid-1950s
that a formal distinction was made between EF A and CFA : While the former
method is used to search data for an underlying structure, the latter tech
nique is utilized to confirm (strictly speaking, disconfirm) an a priori hypoth
esized, theory-derived structure with collected data. In this text, only CFA
Preface Xl
models are discussed; books by Gorsuch (1983), Mulaik (1972), or McDonald
(1985) can be consulted for detailed introductions to EF A.
Based on the assumption that most measured variables are imperfect
indicators of certain underlying, latent constructs or factors, CFA allows the
researcher to cluster observed variables in certain prespecified, theory-driven
ways, that is, to specify a priori the latent construct(s) that the observed
variables are intended to measure. After estimating the factor loadings (struc
tural coefficients in the regressions of observed on the latent variables) from
a variance/covariance matrix of the observed variables, the investigator can
and should assess whether or not the collected data "fit" the hypothesized
factor structure. Since several issues surrounding the question of how best to
assess data-model fit still are unresolved and current answers remain contro
versial, a detailed discussion of some of the currently utilized data-model fit
indices is presented in Chapter 2. The different approaches to data-model fit,
as well as other concepts introduced in Chapter 2 (e.g., model modification),
are illustrated by (a) an extended analysis of the data set introduced in
Chapter 1, and (b) a LISREL and EQS CF A to assess the validity and
reliability of a behavior assessment instrument taken from the counseling
psychology literature.
Finally, Chapter 3 treats the most general recursive structural equation
model presented in this book. The conceptual integration of the classical
path analysis model (Chapter 1) and the CFA model (Chapter 2) is used to
develop the general structure. During the 1970s, several researchers (e.g.,
Keesling, 1972; Joreskog, 1977; Wiley, 1973) succeeded in developing the
statistical theory and computational procedures for combining path analysis
and CF A into a single statistical framework. The various versions of LISREL,
developed by Joreskog and associates dating back to 1970, continuously
incorporated up-to-date statistical advances in SEM. Similarly, EQS (Bentler,
1993) is regarded as one of the most sophisticated and respected software
packages for the analysis of a wide variety of structural equation models,
including path analytical, CFA , and more general models. In addition, its MS
Windows version (Bentler and Wu, 1993) includes many exploratory data
analytical tools, such as outlier analysis and other descriptive statistics with
various plot options, in addition to inferential techniques, such as dependent
and independent t-tests, regression, and ANOYA.
After discussing the specification and identification of general structural
equation models that involve latent variables in Chapter 3, the estimation of
direct, indirect, and total effect components introduced in Chapter 1 is sim
plified by introducing a matrix algorithm to estimate the various effect com
ponents. Also, two of the most common iterative methods for estimating
parameters in general structural equation models are discussed: maximum
likelihood (Joreskog, 1969, 1970; among others) and generalized least squares
(Goldberger, 1971; Joreskog and Goldberger, 1972). While these techniques
have important advantages over more traditional methods such as OLS,
both methods still depend on the multivariate normality assumption; for
