The Estimated G matrix not positive definite message issued by PROC MIXED in the log usually indicates that one or more variance components in the RANDOM statement are estimated to be zero. These are likely to be negative estimates but by default PROC MIXED constrains all variance component estimates to be non-negative and therefore sets them equal to zero.

What to do about negative variance component estimates remains an unsolved controversy in mixed models analysis. Some analysts suggest removing the random effects associated with negative variance estimates and rerunning the analysis. This approach is preferred in observational studies where no experimental design is associated with the model. Others argue that depending on the nature of your experimental design, a negative variance component might indicate negative correlation in the data and therefore you might want to keep the associated random effects in the model. However, they do acknowledge the awkwardness in interpreting a negative variance estimate and suggest respecifying the model using the appropriate REPEATED statement if possible. For example, if a variance component model is fit with a RANDOM statement and results in a negative variance estimate for that random effect, you might consider using a different syntax to fit a statistically equivalent model. Note that the following two model specifications result in the same marginal model:

random int / subject=A;

and

repeated / subject=A type=CS;

The analysis using the REPEATED statement is likely to produce a negative covariance estimate which can be easily interpreted.

For more information on the equivalence between the random intercept model and the REPEATED / TYPE=CS model, see "Mixed Models Theory" in the Details section of the PROC MIXED documentation. References that discuss zero or negative variance components are Searle, Casella, and McCulloch (1992) and Stroup (2012).

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