PARTIAL AND CONDITIONAL MAXIMUM-LIKELIHOOD FOR VARIANCE-COMPONENT ESTIMATION

PATTERSON and THOMPSON's idea of 'error contrasts' (or restricted maximum likelihood) (1971) was extended to multiple sets of linear contrasts for variance component estimtion. The error contrasts were established in such a way that only errors are retained in the model. The error variance was then estimated by maximizing the likelihood function obtained from the error contrasts. More sets of linear contrasts were then progressively established such that each set of linear contrasts contains only one class of random effects and the errors. A likelihood function was constructed and maximized for each variance of random effects given the error variance held at its estimated value. The likelihood function for estimating the covariance component between two classes of random effects was established such that all other random effects are treated as fixed effects. The likelihood function was then maximized with respect to the covariance given the two variance components fixed at their estimated values. The multidimensional optimization problem in the traditional restricted maximum-likelihood problem was then turned into several one-dimensional optimization problems by using this technique. Inasmuch as the error variance was estimated using a partial likelihood function and the other variance components are estimated using likelihood functions conditional on the estimated error variance, the method is referred to as partial and conditional maximum likelihood (PCML).