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How does lmer obtain ML estimates that are not stationary points?
2 messages · Asaf Weinstein, Ben Bolker
Asaf Weinstein <asafw.at.wharton at ...> writes:
Hi, I am looking at a two-way random-effects ANOVA layout as a particular case of the general mixed-model, y|b ~ N(X beta + Zb, sigsq I) b ~ N(0,sigsq Gamma_theta ) [Gamma diagonal], I am trying to compute ML estimates for theta under a KNOWN sigsq (i.e., error variance is known). I derived my own ML estimates since lmer() estimates sigsq rather than assuming it is known. As long as the variance components (or theta's) are "big" (far from zero), the output of my algorithm is consistent with the output of lmer (when plugging in the estimated sigsq into my functions); but I run into problems when the ML estimates for at least one of the theta's is close zero. Before i roll up my sleeves, I would like to know how lmer handles the case in which the value of theta which nullifies the derivative is negative (or does not exist at all), i.e., the ML estimate cannot be obtained by searching for a root of the derivative of the profile likelihood. (If my point was not clear, I refer to the case similar to what happens in the balanced two-way ANOVA when the row (or column) sum of squares minus the estimated error component is negative..).
lmer fits models on a constrained space where the variances are not allowed to be negative. So it would give the best fit on the boundary of the feasible space (although I would be very slightly suspicious of the results in this case; it is easy to misconverge / run into optimization difficulties when the results are on the boundary, although I don't have any concrete examples of where lmer gets this wrong). With the development version of R, you can get the deviance function and evaluate it yourself over a range of parameter values to see what it does.