Newsgroups: bit.listserv.stat-l
Date:         Mon, 31 Aug 1992 15:52:48 -0400
From:         TBN@UTXVM.BITNET
Subject:      communalities > 1

 Jaap Hartog < ABRAMSE@HROEUR1.BITNET > writes

" Dear Networkers,
I am doing a factor analysis on 4 variables with SPSS PC+.
One loading proved to be greater than 1 (1.01 in fact). May be
rounding difficulties. So I tried an other extraction method
(PAF). Extraction stopped because of communalities greater than
one. So I tried another method (ALPHA). Again a loading greater
than one.
Is there someone among you who can shed some light on this
dark matter.
                              Jaap Hartog  "

A question very similar to this one was sent to the SPSSX-L bulletin
board, and I have included the text of my letter on the subject below.
One quick thing you may want to try would be to use a / Diagonal =
50 .50 .50 .50 subcommand followed by / Extraction = PAF. This
will give you a principal-axis factor analysis with .5 as the
initial starting value used in computing the estimates of
communality. As I've noted in the letter below, I've had some
luck with using .5 as a starting value, particularly in the case of
factor analysis models.

In my quick glance through the SPSS Base System User's Guide I did
not come across any references to Heywood cases or how to deal with
them. As I note below, SAS allows a user to arbitrarily (for better
or worse) enforce a communality less-than-or-equal-to-one rule, which
allows a factor analysis program to continue running when it would
have crashed otherwise. I am curious to hear what SPSS users do when
they encounter Heywood cases.

Here is the similar letter posted to SPSSX-L and my response to it.
Doubtless  there will be other responses to it, and you may wish
to monitor that bulletin board as well as this one for the many
useful suggestions which will no doubt follow your and Susan
Kashubek's postings....

***************************************************************


To: spssx-l, sk7811R@acad.drake.edu
Subject: Heywood Cases

Susan Kashubek (SK7811R@ACAD.DRAKE.EDU) writes the following:


" HELP!
For those of you who use Lisrel 7:

I am using Lisrel 7 to do some structual equation modeling and am having
trouble with a recurring error message.  This message states:
	Warning: Theta Eps is not positive definite

I have played around some with the way the model is specified (adding or
deleting variables, fixing or freeing paths) and sometimes get the same
warning, other times the Theta Delta matrix or the Psi matrix will not
be positive definite.  The result is that the modification indices, t-values,
residuals, etc. can't be computed, and I assume that the final parameter
estimates are somewhat arbitrary.

My sample size is about 190 subjects, with 23 measured variables (though I
have tried deleting 4 or 5 to see if that helps).  I did get the model to
"run" once without any errors or warnings at all, but when I attempted to
modify the model based on the results in the modification indices, I ended
up getting more warnings that the Theta Eps matrix wasn't positive definite.

Any ideas on why I am getting this message?  Do I have too many variables?
Are there common strategies for dealing with this problem?  Any help would
be greatly appreciated, as there are few, if any, people here at Drake
who are familiar with Lisrel.

Many thanks!

Susan Kashubeck
Drake University
SK7811R@ACAD.DRAKE.EDU  "

What Ms. Kashubeck has encountered is what is popularly called a
Heywood Case in the factor analysis literature. Heywood Cases arise
in two situations in structural modeling. The first occurs when
the unique properties of the correlation matrix which is being
analyzed by LISREL lead LISREL to estimate a path value (or factor
loading in the case of a confirmatory factor analysis model) that is
larger that 1.00 (If you are using a standarized solution, the
presence of such an estimate may be checked on the LISREL output).

The result of a parameter estimate that is larger than 1.00 means
that "negative error variance" exists--there is no positive variance
"left over" to fill one of the error matrices. You will recall that
LISREL has a total of eight matrices. The Theta Epsilon (Theta EPS)
is the error matrix associated with Y-residuals (i.e., downstream
variable residuals). I would recommend that you check the LY (Lamba Y)
matrix
parameter estimates and try to determine if any of them exceed 1.00
in a standarized solution. LISREL is expecting some left over error
variance that it can use to fashion a "gramian" (i.e., positive
definite matrix) Theta Epsilon
matrix, and if it cannot do so then it announces that Theta Epsilon
is "not positive definite".

What can one do in this situation? One quick solution is to try
setting different starting values other than LISREL's default.
I've had moderate success with using / ST .5 ALL. Another option is
to move from a Maximum-Likelihood-based solution to a ordinary least-
squares or a generalized least-squares solution, since the former
method seems to be particularly prone to Heywood cases and the latter
two methods much less so. I believe that ordinary least-squares and
generalized least-squares solutions are available by typing
UL or GLS on the / OU line, though I am not certain of this. ML
, which is the default, will give you the maximum-likelihood solution.

A second way you can obtain a Heywood case is called "Empirical
Underidentification". If you have a situation, say, where you have
two measurement variables leading into one latent variable on
a model with two latent variables joined by one correlation, with
3 manifest variables attached to the other latent variable by one path
each, the situation looks fine for a good solution, but
there may be an infinite number of solutions possible for the path
values (parameter estimates). This may be particularly true when the
correlation or covariance which links the latent variables is
small. To determine if this type of situation is contributing to
a non-positive definite matrix, I would suggest checking the
standard error of the estimates in the output. If these values are
large, then this situation may be the cause of your difficulty.
You can try to get around this difficulty by setting your residuals
equal to one another with / EQ statements.

Finally, if worse comes to worse, you could try to figure out which
of the individual residual matrix elements (i.e., estimates in the
THETA EPS) matrix are less than 0 and fix them to some value greater
than zero, but I would try to exercise all other options before
undertaking such a radical venture. I'm sure other SPSS-L readers
will have ready solutions for you to try because this is not an
uncommon problem for structural equation modelers to contend with.

Heywood cases are definitely a royal pain in the statistical
posterior. SAS in its Proc Calis has even gone so far as to add an
option that allows users to quickly and simply specify and option
which tells SAS not to let any parameter estimates exceed 1.00, and
therefore all of the error matrices are kept positive definite.
LISREL contains no such luxury, however, but the LISREL user can do
the same thing via a little more work and patience.


Tor Neilands, Systems Analyst
Statistical and Mathematical Support Services
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Disclaimer: All views expressed above and herein are only those of
the author and do not necessarily represent the views of the University
of Texas, the UT Comp Center, and anyone else I've forgotten to include
in this disclaimer.
Comments: Gated by NETNEWS@AUVM.AMERICAN.EDU