Approximately exact calculations for linear mixed models

This paper is about computations for linear mixed models having two variances, σ²_e for residuals and σ²_e 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 RL(σ²_e, σ²_s) or the joint posterior π(σ²_e, σ²_s | y) or their logarithms. Both logRL and log π 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 RL or π, we show how to find a box B in the (σ²_e, σ²_s) plane such that 1. all local and global maxima of log f lie within B; 2. (Equation presented) − M for a M > 0; and prespecified M and 3. log f can be estimated to within a prespecified tolerance ε everywhere in B with no danger of missing regions where log f is large. Taken together these conditions imply that the (σ²_e, σ²_s) plane can be divided into two parts: B, where we know logf as accurately as we wish, and B^c, where logf 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.