RE: posthoc step

From: "Kowalski, Ken" Ken.Kowalski@pfizer.com Subject: RE: [NMusers] posthoc step Date: Tue, December 7, 2004 5:17 pm Pravin and Nick, I think Nick's objective function below is essentially correct when using an additive error model and a diagonal Omega with the exception that (THETAhatk - THETAki) in the second term of his expression for SS needs to be squared and the i index he uses for the second term is for subjects while the i in the first term indexes observations. If we assume additive errors for the ETAs for ease of exposition such that THETAki = THETAhatk + ETAki and use j as an index for observations then we can re-write the MAPB objective function as: SSi = sum j=1 to Nobsi [(Yobsij - Yhatij)^2/SIGMA^2] + sum k=1 to Npar [ETAki^2/OMEGAhatk^2] and we minimize SSi for subject i with respect to the ETAki (i.e., find the values of ETAki, k=1,...,Npar that minimize SSi). I use an expression like this to help illustrate the shrinkage estimation properties of MAPB estimates. That is, the second term of the objective function will dominate when Nobsi is small (sparse data) and if you take it to the extreme when Nobsi=0 then the second term and hence SSi is minimized when ETAki=0 for all k=1,...,Npar. That is, in absence of any data for a new individual the best estimate is the typical individual prediction we obtain from the population parameter estimates (THETAhatk) where ETAki=0 for all k. Ken