Derivative-intense Restricted Maximum Likelihood Estimation of Covariance Components for Animal Models

INTRODUCTION Estimation of covariance components by Restricted Maximum Likelihood (REML) fitting an animal model is widely carried out using a derivative-free (DF) algorithm to locate the maximum of the likelihood function. While such algorithms have proven robust and easy to use, they are generally slow to converge, often requiring many likelihood evaluations, in particular for multivariate analyses fitting several random factors. As outlined by Graser et al. (1987), however, they only require factorization of a large matrix rather than its inverse and can be implemented efficiently using sparse matrix techniques for analyses involving tens of thousands of animals. Although there has been some use of algorithms using derivatives of the likelihood function, i.e. ExpectationMaximization type or even Method of Scoring procedures, for large scale animal model analyses, they have involved the use of a supercomputer or some approximation of the inverse of the coefficient matrix required (Ducrocq, 1993; Misztal, 1990; Misztal et al., 1992). This paper describes the calculation of first and second derivatives of the REML (log) likelihood (logL) using a simple extension of the large matrix factorization required to evaluate logL (for a DF algorithm) only, and illustrates their use in estimating covariance components using a Newton-Raphson algorithm.