Restricted Maximum Likelihood (REML) Estimation of Variance Components in the Mixed Model

The maximum likelihood (ML) procedure of Hartley aud Rao 2 is modified by adapting a transformation from Pattersou and Thompson 7 which partitions the likelihood render normality into two parts, one being free of the fixed effects. Maximizing this part yields what are called restricted maximum likelihood (REML) estimators. As well as retaining the property of invariance under translation that ML estimators have, the REML estimators have the additional property of reducing to the analysis variance (ANOVA) estimators for many, if not all, cases of balanced data (equal subclass numbers). A computing algorithm is developed, adapting a transformation from Hemmerle and Hartley 6, which reduces computing requirements to dealing with matrices having order equal to the dimension of the parameter space rather than that of the sample space. These same matrices also occur in the asymptotic sampling variances of the estimators.