Re: Why does covariance fail?

Hi Mark I am pretty confident (although I do not have a proof) that all (or all reasonably good, meaning that you invested some time trying to make them good) models with failed covariance steps are not the saddle points but over-parametrized models with degenerate direction(s). Some exception could be related to the problems with the odd-type data where I have less experience, so let's restrict the discussion by continuous-type data. The check should be pretty easy. I would just evaluate OF in 100-1000 random points in the vicinity of the solution (automated similar to the bootstrap runs). It is better be random rather than univariate (parameter by parameter) to investigate all possible parameter-space directions. If you have bootstrap results, it could be used instead of this new check since it is very unlikely that all bootstrap runs would stick to the saddle-point rather than move along the gradient to the lower minimum. Answer seems obvious to me (not a saddle points) but it would be interesting to see a more definite results. Please, update if you see anything interesting Thanks Leonid P.S. As far as I remember, this message: PARAMETER ESTIMATE IS NEAR ITS BOUNDARY THIS MUST BE ADDRESSED BEFORE THE COVARIANCE STEP CAN BE IMPLEMENTED can be given if some of the OMEGA or SIGMA elements (including the off-diagonal terms) are close to zero. You can block this case, ICON distributed the patch for it, see archives. see also http://www.cognigencorp.com/nonmem/current/2007-July/0335.html -------------------------------------- Leonid Gibiansky, Ph.D. President, QuantPharm LLC web: www.quantpharm.com e-mail: LGibiansky at quantpharm.com tel: (301) 767 5566 Mark Sale - Next Level Solutions wrote: > 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 http://www.NextLevelSolns.com > 919-846-9185