Fitting mixed-effects models using efficient EM-type algorithms

In recent years numerous advances in EM methodology have led to algorithms which can be very efficient when compared with both their EM predecessors a nd other numerical methods (e.g., algorithms based on Newton-Raphson). This article combines several of these new methods to develop a set of mode-fin ding algorithms for the popular mixed-effects model which are both fast and more reliable than such standard algorithms as proc mixed in SAS. We prese nt efficient algorithms for maximum likelihood (ML), restricted maximum lik elihood (REML), and computing posterior modes with conjugate proper and imp roper priors. These algorithms are not only useful in their own right, but also illustrate how parameter expansion, conditional data augmentation, and the ECME algorithm can be used in conjunction to form efficient algorithms . In particular, we illustrate a difficulty in using the typically very eff icient PXEM (parameter-expanded EM) for posterior calculations, but show ho w algorithms based on conditional data augmentation can be used. Finally, w e present a result that extends Hobert and Casella's result on the propriet y of the posterior for the mixed-effects model under an improper prior, an important concern in Bayesian analysis involving these models that when not properly understood has lead to difficulties in several applications.