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

This warning is not uncommon with structural equation modelling. Sometimes it can be triggered by eigenvalues that are 0 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. PROC CALIS checks to see if a central model matrix has negative eigenvalues (but it does not actually compute the eigenvalues). Other SAS procedures (and other structural equation software packages) do not necessarily perform this check. Hence, the warning does not imply that there is a problem with PROC CALIS but not with other routines. CALIS is possibly providing additional information.

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

 EstParms#1=X

where X is a new data set name.

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

1) use the SAS/IML command VAL=EIGVAL(X) to compute the vector VAL of eigenvalues of matrix X.

Or,

2) convert the matrix to a TYPE=COV data set and use it as input to PROC PRINCOMP which will compute the eigenvalues of the covariance matrix. (See SAS/STAT User's Guide, "Secial SAS Data Sets" for details on creating TYPE=COV data sets.)

If the computed eigenvalues are indeed small but not negative, it is possible that the warning can be ignored. If negative eigenvalues are found, it is possible that the model may need to be changed; or, if the model fits well and is deemed substantively meaningful, negative eigenvalues may have been due to some uncontrollable stochastic factors. In this latter case, the model possibly still provides 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/or the 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 on the PROC CALIS statement will create modification indices, and the researcher chooses to accept or reject the changes based on content knowledge.

For additional information see:

 SAS/STAT User's Guide
  The CALIS Procedure
   Details
    Computational Problems
     Central Matrices with Negative Eigenvalues

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