Date: Tue, 29 Jan 2008 08:37:53 -0700
I'm thinking of doing a somewhat formal analysis of the meaning of a failed covariance step. Some years ago Stu Beal explained that (as I recall), if the covariance step fails you cannot be sure that the minimum isn't a saddle point, which makes sense to me, and is consistent (I think), with the common message from NONMEM R MATRIX ALGORITHMICALLY SINGULAR AND ALGORITHMICALLY NON-POSITIVE-SEMIDEFINITE R MATRIX IS OUTPUT 0COVARIANCE STEP ABORTED I'm also finding one in NONMEM VI that I don't recall from NONMEM V, and I don't know what it means: ERROR RMATX- 1 Then there are messages! that seem to be related to conditional estimates: NUMERICAL HESSIAN OF OBJ. FUNC. FOR COMPUTING CONDITIONAL ESTIMATE IS NON POSITIVE DEFINITE MESSAGE ISSUED FROM COVARIANCE STEP and version VI of NONMEM will refuse to even try the covariance step for various reasons: PARAMETER ESTIMATE IS NEAR ITS BOUNDARY THIS MUST BE ADDRESSED BEFORE THE COVARIANCE STEP CAN BE IMPLEMENTED even, it seems when the parameter estimate is no where near the boundary. I'm thinking of looking at these various reasons that the covariance step fails, and seeing if any of them mean anything WRT whether the model is "good", by some objective measure (PPC, NPDE, predictive check). My question is, is there any way to formally test whether the failure is due to a saddle point in the objective function surface? My understanding of the current search algorithm used by NONMEM is that it is very, very robust WRT saddle points. So, I suspect that the vast majority of the failures are not due to a saddle, but rather just a fairly flat surface, with near 0 first and second derivatives, causing numerical problems inverting it, rather than actually being a saddle point. If the surface is just fairly flat, not a saddle, then I think that the answer is not "wrong", just not especially good, therefore other simulation based tests of "goodness" might be just fine. I suspect that you could test whether it is a saddle point by trying a slightly different value for the parameter (e.g., "minimum" is 10, so try 9.9 and 10.1 and see if the OBJ is better, in each dimension. Would this work? thanks Mark Mark Sale MD Next Level Solutions, LLC www.NextLevelSolns.com 919-846-9185 |