A general structural equation model with dichotomous
A general structural equation model with dichotomous, ordered categorical, and continuous latent variable indicatorsA GENERAL STRUCTURAL EQUATION MODEL
WITH DICHOTOMOUS, ORDERED CATEGORICAL,
AND CONTINUOUS LATENT VARIABLE INDICATORS
BENGT MUTI~N
GRADUATE SCHOOL OF EDUCATION
UNIVERSITY OF CALIFORNIA
LOS ANGELES, CALIFORNIA
A structural equation model is proposed with a generalized measurement part, allowing for
dichotomous and ordered categorical variables (indicators) in addition to continuous ones. A
computationally feasible three-stage estimator is proposed for any combination of observed variable
types. This approach provides large-sample chi-square tests of fit and standard errors of
estimates for situations not previously covered. Two multiple-indicator modeling examples are
given. One is a simultaneous analysis of two groups with a structural equation model underlying
skewed Likert variables. The second is a longitudinal model with a structural model for multivariate
probit regressions.
Key words: polychoric correlations, probit regressions, generalized least-squares, weight matrix.
1. Introduction
This article considers the specification and estimation of multiple-group (population)
structural equation models with latent variables having multiple indicators, not all of
which are continuous. A linear structure for continuous latent variables will be considered.
However, in the measurement part dichotomous and ordered polytomous observed
variables (indicators) will be allowed in addition to continuous indicators. Such
categorical indicators are frequent in many types of applications, and it seems important
to provide the powerful structural equation modeling tool also for these cases.
The methodology to be presented unifies and generalizes several lines of psychometric,
econometric and biometric work. For an overview, see Muthrn (1983). In particular,
the paper extends the Muthrn-Christoffersson methodology for factor analysis of dichotomous
variables (see e.g. Muthrn, 1978; Muthrn and Christoffersson, 1981) to handle
ordered categorical and continuous indicators and general multiple-group structural
equation models with estimation of latent variable means. Hence, the paper also generalizes
the Jrreskog-Srrbom ("LISREL") methodology for structural equation models (see
e.g., Jrreskog, 1973, 1977; Srrbom, 1982) to handle properly categorical indicators in
addition to continuous ones. New results also include a general estimation approach for
all cases of the model. A three-stage, limited information, generalized least-squares (GLS)
estimator is proposed, which gives large-sample chi-square tests of model fit and largesample
standard errors of estimates. Some examples of analyses that the new techniques
make possible are GLS factor analysis with (mixtures of continuous and) ordered polytomous
indicators, testing hypotheses of both correlation and level ("mean") structures in
multiple-group structural equation models, and multivariate structural regression with
ordered categorical response variables (such as multivariate probit regression). In the
This research was supported by Grant No. 81-1J-CX-0015 from the National Institute of Justice, by Grant
No. DA 01070 from the U.S. Public Health Service, and by Grant No. SES-8312583 from the National Science
Foundation. I thank Julie Honig for drawing the figures. Requests for reprints should be sent to Bengt Muthrn,
Graduate School of Education, University of California, Los Angeles, California 90024.
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