RE: posthoc step

From: jerry.nedelman@pharma.novartis.com Subject: RE: [NMusers] posthoc step Date: Wed, December 8, 2004 11:08 pm Friends: Thanks to all for the interesting insights. I guess the bottom line is that something about how NONMEM handles the nonlinearity breaks down when the data are unlikely relative to the prior. For real examples, this might mean that posthoc estimates for outlying subjects are shrunk substantially more than they should be, as Leonid pointed out. There seem to be some issues with the PRIOR method, too, based on Nick's results. Steve shows that WinBUGS manages things OK. For those who thought the MAP estimate should have been close to the prior mean 10 because the prior mass was concentrated away from the sparse data, consider the linear case, where SAS and NONMEM agreed. The same prior and data are used there. And there, too the MAP estimate (1.1128) is close to the OLS estimate (1.1064) and far from the prior mean. One can actually find the MAP estimate for the linear case analytically. Let omega = prior variance (= 4 in the example) theta = prior mean (= 10 in the example) k_ols = OLS estimate of k (= 1.1064 in the example) v_ols = sampling variance of k_ols, i.e, the square of its standard error (= 0.04/(1^2 + 2^2 + 3^2) = 1/350 in the example) k_map = MAP estimate of k (= 1.1128 in the example) Then k_map = k_ols - v_ols*(k_ols - theta)/(v_ols + omega). The amount of shrinkage is determined by v_ols/(v_ols + omega) = (1/350) / ( (1/350) + 4 ) = 1/1401. Thus, even though the data were generated by a very unlikely value of k relative to the prior, the precision of the least squares estimate is so great, relative to the prior, that it dominates in the blending of the prior and the data to yield the posterior. The same thing should happen in the nonlinear case, and does happen with SAS and WinBUGS, but something breaks down with NONMEM's way of handling it. Jerry