Computer Output for Structural Equation Modeling

Structural Equation Modeling
Overview Structural equation modeling (SEM) grows out of and serves purposes similar
to multiple regression, but in a more powerful way which takes into account
the modeling of interactions, nonlinearities, correlated independents,
measurement error, correlated error terms, multiple latent independents
each measured by multiple indicators, and one or more latent dependents
also each with multiple indicators. SEM may be used as a more powerful
alternative to multiple regression, path analysis, factor analysis, time
series analysis, and analysis of covariance. That is, these procedures may
be seen as special cases of SEM, or, to put it another way, SEM is an
extension of the general linear model (GLM) of which multiple regression is
a part.
Advantages of SEM compared to multiple regression include more flexible
assumptions (particularly allowing interpretation even in the face of
multicollinearity), use of confirmatory factor analysis to reduce
measurement error by having multiple indicators per latent variable, the
attraction of SEM's graphical modeling interface, the desirability of
testing models overall rather than coefficients individually, the ability
to test models with multiple dependents, the ability to model mediating
variables rather than be restricted to an additive model (in OLS regression
the dependent is a function of the Var1 effect plus the Var2 effect plus
the Var3 effect, etc.), the ability to model error terms, the ability to
test coefficients across multiple between-subjects groups, and ability to
handle difficult data (time series with autocorrelated error, non-normal
data, incomplete data). Moreover, where regression is highly susceptible to
error of interpretation by misspecification, the SEM strategy of comparing
alternative models to assess relative model fit makes it more robust.
SEM is usually viewed as a confirmatory rather than exploratory procedure,
using one of three approaches:
1. Strictly confirmatory approach: A model is tested using SEM goodness-
of-fit tests to determine if the pattern of variances and covariances
in the data is consistent with a structural (path) model specified by
the researcher. However as other unexamined models may fit the data as
well or better, an accepted model is only a not-disconfirmed model.
2. Alternative models approach: One may test two or more causal models to
determine which has the best fit. There are many goodness-of-fit
measures, reflecting different considerations, and usually three or
four are reported by the researcher. Although desirable in principle,
this AM approach runs into the real-world problem that in most
specific research topic areas, the researcher does not find in the
literature two well-developed alternative models to test.
3. Model development approach: In practice, much SEM research combines
confirmatory and exploratory purposes: a model is tested using SEM
procedures, found to be deficient, and an alternative model is then
tested based on changes suggested by SEM modification indexes. This is
the most common approach found in the literature. The problem with the
model development approach is that models confirmed in this manner are
post-hoc ones which may not be stable (may not fit new data, having
been created based on the uniqueness of an initial dataset).
Researchers may attempt to overcome this problem by using a cross-
validation strategy under which the model is developed using a
calibration data sample and then confirmed using an independent
validation sample.
Regardless of approach, SEM cannot itself draw causal arrows in models or
resolve causal ambiguities. Theoretical insight and judgment by the
researcher is still of utmost importance.
SEM is a family of statistical techniques which incorporates and integrates
path analysis and factor analysis. In fact, use of SEM software for a model
in which each variable has only one indicator is a type of path analysis.
Use of SEM software for a model in which each variable has multiple
indicators but there are no direct effects (arrows) connecting the
variables is a type of factor analysis. Usually, however, SEM refers to a
hybrid model with both multiple indicators for each variable (called latent
variables or factors), and paths specified connecting the latent variables.
Synonyms for SEM are covariance structure analysis, covariance structure
modeling, and analysis of covariance structures. Although these synonyms
rightly indicate that analysis of covariance is the focus of SEM, be aware
that SEM can also analyze the mean structure of a model.
See also partial least squares regression, which is an alternative method
of modeling the relationship among latent variables, also generating path
coefficients for a SEM-type model, but without SEM's data distribution
assumptions. PLS path modeling is sometimes called "soft modeling" because
it makes soft or relaxed assumptions about data... Key Concepts and Terms . The structural equation modeling process centers around two steps:
validating the measurement model and fitting the structural model. The
former is accomplished primarily through confirmatory factor analysis,
while the latter is accomplished primarily through path analysis with
latent variables. One starts by specifying a model on the basis of
theory. Each variable in the model is conceptualized as a latent one,
measured by multiple indicators. Several indicators are developed for
each model, with a view to winding up with at least three per latent
variable after confirmatory factor analysis. Based on a large (n>100)
representative sample, factor analysis (common factor analysis or
principal axis factoring, not principle components analysis) is used
to establish that indicators seem to measure the corresponding latent
variables, represented by the factors. The researcher proceeds only
when the measurement model has been validated. Two or more alternative
models (one of which may be the null model) are then compared in terms
of "model fit," which measures the extent to which the covariances
predicted by the model correspond to the observed covariances in the
data. "Modification indexes" and other coefficients may be used by the
researcher to alter one or more models to improve fit.
. LISREL, AMOS, and EQS are three popular statistical packages for doing
SEM. The first two are distributed by SPSS. LISREL popularized SEM in
sociology and the social sciences and is still the package of
reference in most articles about structural equation modeling. AMOS
(Analysis of MOment Structures) is a more recent package which,
because of its user-friendly graphical interface, has become popular
as an easier way of specifying structural models. AMOS also has a
BASIC programming interface as an alternative. See R. B. Kline (1998).
Software programs for structural equation modeling: AMOS, EQS, and
LISREL. Journal of Psychoeducational Assessment (16): 343-364.
. Indicators are observed variables, sometimes called manifest variables
or reference variables, such as items in a survey instrument. Four or
more is recommended, three is acceptable and common practice, two is
problematic, and with one measurement, error cannot be modeled. Models
using only two indicators per latent variable are more likely to be
underidentified and/or fail to converge, and error estimates may be
unreliable. By convention, indicators should have pattern coefficients
(factor loadings) of .7 or higher on their latent factors.
. Regression, path, and structural equation models. While SEM packages
are used primarily to implement models with latent variables (see
below), it is possible to run regression models or path models also.
In regression and path models, only observed variables are modeled,
and only the dependent variable in regression or the endogenous
variables in path models have error terms. Independents in regression
and exogenous variables in path models are assumed to be measured
without error. Path models are like regression models in having only
observed variables w/o latents. Path models are like SEM models in
having circle-and-arrow causal diagrams, not just the star design of
regression models. Using SEM packages for path models instead of doing
path analysis using traditional regression procedures has the benefit
that measures of model fit, modification indexes, and other aspects of
SEM output discussed below become available.
. Latent variables are the unobserved variables or constructs or factors
which are measured by their respective indicators. Latent variables
include both independent, mediating, and dependent variables.
"Exogenous" variables are independents with no prior causal variable
(though they may be correlated with other exogenous variables,
depicted by a double-headed arrow -- note two latent variables can be
connected by a double-headed arrow (correlation) or a single-headed
arrow (causation) but not both. Exogenous constructs are sometimes
denoted by the Greek letter ksi. "Endogenous" variables are mediating
variables (variables which are both effects of other exogenous or
mediating variables, and are causes of other mediating and dependent
variables), and pure dependent variables. Endogenous constructs are
sometimes denoted by the Greek letter eta. Variables in a model may be
"upstream" or "downstream" depending on whether they are being
considered as causes or effects respectively. The representation of