In particular, how should the standard errors of the fixed effects in a linear mixed effects model be calculated (in a frequentist sense)?
I have been lead to believe that the typical estimates (${\rm Var}(\hat\beta)=(X'VX)^{-1}$), such as those presented in Laird and Ware [1982] will give SE's that are underestimated in size because the estimated variance components are treated as though they are the true values.
I have noticed that the SE's produced by the lme
and summary
functions in the nlme
package for R are not simply equal to the square root of the diagonals of the variance-covariance matrix given above. How are they calculated?
I am also under the impression that Bayesians use inverse gamma priors for the estimation of variance components. Do these give the same results (in the right setting) as lme
?
Best Answer
My initial thought was that, for ordinary linear regression, we just plug in our estimate of the residual variance, $\sigma^2$, as if it were the truth.
However, take a look at McCulloch and Searle (2001) Generalized, linear and mixed models, 1st edition, Section 6.4b, "Sampling variance". They indicate that you can't just plug in the estimates of the variance components:
They go on to explain $T$.
So this answers the first part of your question and indicates that your intuition was correct (and mine was wrong).