Newsgroups: bit.listserv.stat-l
Date:         Fri, 13 Nov 1992 11:28:25 EST
From:         Walter Davis <WDAVIS@UNCVM1.BITNET>
Subject:      reply to LISREL questions

hi all,

Over the last few days, Heiyi Xie has aske a couple of questions
regarding latent variable models and LISREL in particular.

First, regarding the 'assumption of multinormality' and LISREL, let
me make a few clarifying points first.

The ML estimator in LISREL was originally derived under the assumption
of multinormality for all variables.  However, Browne (1974) shows that
only the assumption of no excessive multivariate kurtosis is necessary.
Moreover, regardless of the distribution of the variables, all the
LISREL estimators are consistent.  The distribution only affects
significance tests.

That said, there are some other cases in which the distribution of the
variables need not be multinormal nor have no excessive kurtosis.
If you have exogenous X's (observed variables), e.g. dummy variables,
and these are either fixed in repeated sampling or distributed
independently of the error terms, then you need only assume that
the error terms are multinormal.  Or if all variables have the
same kurtosis, then everything is fine.

Bollen (1989) and D'Agostino (1986) provide tests for
multinormality and multivariate kurtosis.  Bollen (1989) also
provides some results and citations to simulations investigating
how well the ML and GLS estimators perform under non-normality
(generally pretty well).

When all else fails, Browne's (1984) Arbitrary Distribution Function
(ADF, also known as WLS in LISREL and CALIS) provides consistent,
asymptotically efficient and unbiased estimation with standard
errors regardless of the distribution of the variables.  This
is a rather computationally intensive estimation and it requires
the asymptotic covariances of the covariances (which can be
calculated in PRELIS for LISREL, CALIS handles this automatically,
and I'm not sure how EQS performs this).  The question about
this estimator is how large does the sample need to be before
its nice properties hold.  From my experience, ML, GLS and WLS
will give very similar estimates and lead to the same substantive
conclusions.

I have written a SAS/IML macro for testing multivariate normality.
If anyone would like a copy, please send me a note and I'll
get it out to you as soon as possible.

He then asked whether he could simply add up the indicators to
form an index (adding also that he had a small sample and that
the CFA had given him unexpected results).  How well the additive
index would reflect the latent variable depends on how reliable
the various indicators are and whether their factor loadings
are similar.  Cronbach's alpha, a popular measure of reliability,
assumes that the indicators have equal loadings and equal error
variances.  If your indicators closely approximate this, then
Cronbach's alpha will be a fairly good estimate of how reliable
the index is.  If that reliability is high, you may be in good
shape.

For these questions, I strongly recommend that you read Ken
Bollen's _Structural Equations with Latent Variables_ (Wiley: 1989)
(disclaimer: he is my adviser).  This answers all of the questions
asked and a good deal more, including summaries of sample size
effects and distribution requirements.

Yesterday, Xie added a question about different packages.
Structural equation software (LISREL, EQS, CALIS, AMOS, etc.)
all do basically the same thing.  They differ slightly in their
options, but I believe they all provide the same estimators
(ML, GLS, WLS/ADF), give very similar if not identical results
(differences in my experience are usually due to different
technical defaults such as convergence criteria, etc).  Bollen
and Ting did a review of LISREL vs. EQS for American Statistician
within the last couple of years which highlights the similarities
and differences.  With a couple of exceptions (e.g. CALIS does
not do multi-group analysis), the problem being analyzed will
have no effect on your choice of software.  More specifically they
have the same small sample properties, etc.

LISREL7 and LISCOMP can also deal with categorical endogenous
variables.  I have no experience with LISCOMP, but the LISREL
package is relatively easy to use.  Unfortunately, the asymptotic
covariance matrix needed to estimate these models in LISREL is
mis-calculated.  See an article in _Journal of Marketing Research_
(1991 or 92) by Rigdon and Ferguson.  I understand that the problem
will be corrected in the new version and that you may be able to
get a corrected version for current stand-alone versions of
LISREL from Scientific Software.

My last point concerns Xie's question about the package LVPLS.
I have not heard of this before, but assume from its name that
it uses the Partial Least Squares (PLS) estimator.  This
estimator is an inconsistent estimator.  That is, even with an
infinite sample size, the estimator does not converge on the population
value (see an article and dissertation by Dijkstra (cite not handy)).
Given that, I know of no situations in which PLS can be recommended.

hope this helps,

Walter Davis <WDAVIS@UNCVM1>
Department of Sociology
Institute for Research in Social Science
UNC - Chapel Hill
Comments: Gated by NETNEWS@AUVM.AMERICAN.EDU