When fitting a structural equation model, PROC CALIS might produce the message:

WARNING: The central parameter matrix _PHI_ has probably 3 negative eigenvalue(s).

This warning is not uncommon with structural equation modelling. It can be triggered by eigenvalues that are zero or by very small positive values that appear negative because of numerical error or sampling variation. Currently, there is nothing to prevent this from happening during the optimization stage.

Conceptually, a covariance matrix cannot have negative eigenvalues, since a negative eigenvalue means that some linear combination of the variables has negative variance. Although negative eigenvalues cannot occur when you use real data for computing the sample covariance matrix, they could happen to the central model matrix due to the estimation of covariance structure parameters in PROC CALIS. Like the issue of negative variance estimates, negative eigenvalues in the central model matrix could weaken the validity of the model. Unlike negative variance estimates that could be examined directly from the estimation results, negative eigenvalues of the central model matrix are not displayed in the output. Therefore, PROC CALIS checks the eigenvalues in the background and issues warnings if negative eigenvalues of the central model matrix are found.

To save the matrix in a SAS data set for further analysis, specify the PRIMAT option in the PROC CALIS statement and use an ODS OUTPUT statement to save the EstParms data set.

Two of the methods that can be used to examine the eigenvalues are:

• use the EIGEN function in PROC IML (for example: VAL=EIGVAL(X); ) to compute the vector VAL containing eigenvalues of matrix X, or
• convert the matrix to a TYPE=COV data set and use it as input to PROC PRINCOMP which computes the eigenvalues of the covariance matrix. For details on creating TYPE=COV data sets, see "Special SAS Data Sets" in the SAS/STAT User's Guide.

If the computed eigenvalues are indeed small but not negative, it is possible that the warning can be ignored. If negative eigenvalues are found, a change in the model might be needed or, if the model fits well and is deemed substantively meaningful, negative eigenvalues might be due to some uncontrollable stochastic factors. In this case, the model might still be a good approximation. Which of these possibilities are statistically appropriate in any given situation is a decision made by the researcher taking into account the data and purpose of the analysis based on knowledge of the subject matter.

PROC CALIS can suggest model changes based on mathematics only (the nature of the data is ignored). The MOD option in the PROC CALIS statement provides modification indices and the researcher can choose to accept or reject the changes based on content knowledge.

For additional information see "Computational Problems: Central Matrices with Negative Eigenvalues" in the Details section of the CALIS documentation.