A composite likelihood approach to (co)variance components estimation

Abstract

Estimation of variance and covariance components has importance in various substantive fields such as animal breeding and evolutionary biology among others. The most popular methods of variance components estimation are maximum likelihood (ML), restricted maximum likelihood (REML), analysis of variance and covariance (ANOVA) and minimum quadratic norm (MINQUE). All these methods are computationally intensive. This computational barrier is particularly limiting in data obtained from large animal breeding experiments involving multiple traits. The purpose of this paper is to introduce a new method, which we call maximum composite likelihood (MCL), for the estimation of variance and covariance components. This method is as generally applicable as the method of maximum likelihood: to cases where designs are balanced or unbalanced, involving mixed effects and multiple traits or designs where random effects are correlated to each other. The MCL approach, in contrast to ML/REML or ANOVA, however, does not require inversion of matrices. As a consequence the computational burden is reduced from O( N 3) to O( N 2) where N denotes the total sample size. Moreover, and in contrast to the ML/REML estimating functions, the estimating functions obtained for MCL, after a minor modification, are shown to possess a unique solution thus guaranteeing convergence of the numerical optimization routine. Conditions are specified that assure consistency and asymptotic normality of these estimators. These results do not depend on the assumption of a Gaussian distribution of the random effects. Simulation study indicates that there is only a small loss of statistical efficiency in using MCL as compared to REML but a substantial gain in the computational efficiency.