Spencer Graves <[EMAIL PROTECTED]> writes: > Am I correct that changing the parameterization should NOT change > the estimates of the variance components, because you are minimizing > essentially the same objective function over the same subspace? The > only thing that changes is the logdet(X'WX) term mentioned by I White? > Moreover, letting X = QR, and using the fact that det(AB) = > det(A)*det(B) if they are both square, we get det(R'Q'WQR) = > det(Q'WQ)*det(R)^2. Thus, the change in parameterization affects only > R, not Q, which means that it can't affect det(Q'WQ). > > Is this accurate? > > As a simple sanity check, I ran a very simple model with the same > fixed effects in different parameterizations, and got the same > estimates for the variance components under REEL but different > "log-restricted-likelihood" (see below). > > Comments?
Yes, that is accurate. Basically, you are optimizing two functions that differ by a constant, namely logdet(M). And the expected values are also identical; if the parameter estimates were betahat, after reparametrization they become inv(M)betahat. > >The REML loglikelihood includes a term -(1/2)logdet(X'WX) where X is the > >design matrix for the fixed effects and W is the inverse covariance matrix > >for the observations. Under reparametrisation, X becomes XM with M a > >non-singular matrix, and the REML loglikelihood changes by logdet(M). -- O__ ---- Peter Dalgaard Blegdamsvej 3 c/ /'_ --- Dept. of Biostatistics 2200 Cph. N (*) \(*) -- University of Copenhagen Denmark Ph: (+45) 35327918 ~~~~~~~~~~ - ([EMAIL PROTECTED]) FAX: (+45) 35327907 ______________________________________________ R-help@stat.math.ethz.ch mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html
