Assessing the precision of estimates of variance components

			    Douglas Bates
		       Department of Statistics
		       University of Wisconsin

Good statistical practice suggests that we should not only provide
estimates of the parameters in a model but also provide a measure of
the precision of these estimates, typically in the form of a standard
error of the estimate.  Such a summary is meaningful if the estimator
is on a scale where an interval that is symmetric about the estimate
would be a suitable summary of the uncertainty.  A notable exception
to this practice of providing symmetric intervals is the confidence
interval on a population variance based on the $\chi^2$ distribution.
This interval recognizes that the distribution of the estimator of a
variance is quite asymmetric.  However, in much more complex models
using variance components or, more generally, linear mixed-effects
models most statistical software reverts to providing an estimate of a
variance component and a standard error of this estimate.  We discuss
why this is inappropriate and some alternatives based on profiling the
log-likelihood or using Markov-chain Monte Carlo simulation.