Approximately exact calculations for linear mixed models

Abstract

This paper is about computations for linear mixed models having two variances, $\sigma^{2}_{e}$ for residuals and $\sigma^{2}_{s}$ for random effects, though the ideas can be extended to some linear mixed models having more variances. Researchers are often interested in either the restricted (residual) likelihood $\text{RL}(\sigma_{e}^{2},\sigma_{s}^{2})$ or the joint posterior $\pi(\sigma_{e}^{2},\sigma_{s}^{2}\,|\,y)$ or their logarithms. Both $\log\text{RL}$ and $\log\pi$ can be multimodal and computations often rely on either a general purpose optimization algorithm or MCMC, both of which can fail to find regions where the target function is high. This paper presents an alternative. Letting $f$ stand for either $\text{RL}$ or $\pi$, we show how to find a box $B$ in the $(\sigma_{e}^{2},\sigma_{s}^{2})$ plane such that

1. all local and global maxima of $\log f$ lie within $B$;

2. $\sup_{(\sigma_{e}^{2},\sigma_{s}^{2})\in B^{c}}\log f(\sigma_{e}^{2},\sigma_{s}^{2})\le \sup_{(\sigma_{e}^{2},\sigma_{s}^{2})\in B}\log f(\sigma_{e}^{2},\sigma_{s}^{2})-M$ for a prespecified $M>0$; and

3. $\log f$ can be estimated to within a prespecified tolerance $\epsilon$ everywhere in $B$ with no danger of missing regions where $\log f$ is large.

Taken together these conditions imply that the $(\sigma_{e}^{2},\sigma_{s}^{2})$ plane can be divided into two parts: $B$, where we know $\log f$ as accurately as we wish, and $B^{c}$, where $\log f$ is small enough to be safely ignored. We provide algorithms to find $B$ and to evaluate $\log f$ as accurately as desired everywhere in $B$.