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I am using LISREL 8 to do some structural equation modeling and am having trouble with a recurring error message. This message states: WARNING: THETA EPS NOT 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. Is there any solution to this?
LISREL's Theta Epsilon (Theta EPS) matrix is the error matrix associated with Y-residuals (i.e., downstream variable residuals). A negative variance estimate in this matrix makes it "not positive definite"; the variance estimate for the measurement error is negative. Negative variance estimates are a result of a communality (squared correlation between the latent variable and a measurement variable) estimate larger than 1.00.
This situation is called a Heywood case in the factor analysis literature. Heywood cases have many possible causes, including insufficient data, bad prior estimates, and a poorly specified model. Thus, possible solutions include collecting more data, making better prior estimates, and specifying a better-fitting model.
The possible solutions related to computer code include a) providing better prior estimates, and b) using different methods of solving for estimates.
A) Replace the default starting values: some users have had
moderate success with using ST .5 ALL.
B) Replace the default maximum-likelihood-based solution with
an ordinary least-squares or a generalized least-squares solution,
since the former method is particularly prone to produce
Heywood cases . Ordinary least-squares and generalized least-squares solutions are available by typing UL or GLS
on the OU line, respectively.
Poor model specification can also produce Heywood cases; one example is called "empirical under-identification". This occurs when there are an infinite number of solutions possible for the path values (parameter estimates). This is particularly likely when the correlation or covariance matrix linking the latent variables to the measured variables has a small number of entries (e.g., only one or two measured variables per latent variable). To determine if this situation is present, check whether the standard error of the estimates is large. You can try to get around this difficulty by setting your residuals equal to one another with EQ statements.
Heywood cases are a problem for any software. SAS has added the HEYWOOD option to PROC CALIS that sets communality estimates greater than 1.0 to 1. LISREL contains no such luxury, but with a little more work the LISREL user can do the same thing through the use of the techniques described above.
If you have further questions, send E-mail to stats@ssc.utexas.edu.