Linear mixed model, negative information criteria values and Hessian matrix not positive definite

Question

I am analyzing (in SPSS 19) the data from a field experiment using a Linear Mixed Model with Repeated Measures. When checking the model, I always obtain negative values in the information criteria table. I know the rule of thumb “smaller is better”, but how should these negative values be interpreted? As example, when I select a type of covariance “Unstructured” the Bayesian information criterion (BIC) reads $-642.868$ and with a “Compound symmetry” covariance it reads $-497.270$. So, which type of covariance, “Unstructured” or “Compound symmetry”, is better?

The second question is that I receive the following warning message: "The Final Hessian matrix is not positive definite although all convergence criteria are satisfied. The MIXED procedure continues despite the warning. Validity of subsequent cannot be ascertained". May I ignore it or I have a real trouble? Is there some way to fix it?

Answer 1

The negative sign matters, so -642 is smaller than -497; by that criteria the unstructured model is better. (Nice job paying attention to the sign, by the way; many people make the mistake of mentally taking the absolute value, which is not correct.)

The second question is more difficult to answer; in rough terms, this probably means that the optimization is at a saddle point instead of a true maximum, but exactly what to do about it and how much it matters is unclear. It's probably a good idea to try to simplify your model to reduce the computational complexity and see if you can get it to go away; if the results give you the same qualitative answer, that's probably a good thing. If not, it could mean that the warning really mattered, or, it could just mean you've simplified it too far. I'll be curious what others recommend.