PX × AI : algorithmics for better convergence in restricted maximum likelihood estimation

INTRODUCTION Maximising the (log) likelihood (logL) in restricted maximum likelihood (REML) estimation of variance components almost invariably represents a constrained optimisation problem. Iterative algorithms available to solve this problem differ substantially in computational resources needed, ease of implementation, sensitivity to choice of starting values and rates of convergence. One of the most widely used methods is the ‘average information’ (AI) algorithm, which “often converges in a few rounds of iteration” (Thompson et al. 2005). However, there have been some, albeit mainly anecdotal reports of the AI algorithm failing to converge, in particular for analyses involving multiple random effects, numerous traits or ‘bad’ starting values. A popular alternative are expectation-maximisation (EM) algorithms. While these are guaranteed to increase logL in each iterate, they are often painfully slow to converge. Recently, Foulley and van Dyk (2000) considered the ‘parameter expanded’ (PX) variant of the EM algorithm for mixed model REML, and demonstrated dramatically improved convergence compared to standard EM. Yet, there has been little use of the PX-EM algorithm. No comparisons between AI and PX-EM algorithms are available. This paper compares convergence rates of standard EM, PX-EM and AI algorithms for some practical examples of analyses of beef cattle data.