understanding log-likelihood/model fit
On 20/08/2008, at 11:01 PM, Daniel Ezra Johnson wrote:
Everyone agrees about what happens here:
Nsubj <- 10
Ngrp <- 2
NsubjRep <- 5
set.seed(123)
test1s <- data.frame(subject = rep(seq(Nsubj * Ngrp), each =
NsubjRep),
response=500+c(rep(-100,Nsubj * NsubjRep),rep(100,Nsubj *
NsubjRep))+rnorm(Nsubj * Ngrp * NsubjRep, 0, 10),
fixed=(rep(c("A","B"),each=Nsubj * NsubjRep)))
null1 <- lmer(response~(1|subject),test1s)
fixed1 <- lmer(response~fixed+(1|subject),test1s)
I still have two questions which I'll try to restate. I should note
that I have attempted to understand the mathematical details of ML
mixed effect model fitting and it's a bit beyond me. But I hope that
someone can provide an answer I can understand.
Question 1: When you have an "outer" fixed effect and a "subject"
random effect in the same model, specifically why does the model
(apparently) converge in such a way that the fixed effect is maximized
and the random effect minimized? (Not so much why should it, as why
does it? This is the 'fixed1' case.)
Because your generated data has only a fixed effect, so the estimated random effect is zero. This is the same as if you generated data of the form y=x, it would be expected to fit with an intercept of close to zero. Someone else supplied an example of adding a random effect which, of course, will result in a fitted random effect of greater than zero.
Question 2: Take the fixed1 model from Question 1 and compare it to the null1 model, which has a random subject effect but no fixed effect. The predicted values of the two models -- the ones from fitted(), which include the ranefs -- are virtually the same. So why does fixed1 have a lower deviance, why is it preferred to null1 in a likelihood ratio test? (Again, I'm not asking why it's a better model. I'm asking questions about the software, estimation procedure, and/or theory of likelihood applied to such cases.)
The residual variance is slightly lower for the second model explaining the better fit. Knowing about the fixed effects helps but not much. The fitted are similar, but there is reasonable variation. It will always be better (in terms of likelihood and as a rule) to fit the correct model. What is nice is that the model with only a random effect gives sensible results, so in many situations I don't need to know why the clusters vary and departures from normality don't seem to matter much. Ken
D
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