The three-fold nested random effects model

Estimation of the variance components and the mean of the balanced and unbalanced threefold nested design is considered. The relative merits of the following procedures are evaluated: Analysis of variance (ANOVA), maximum likelihood (ML), restricted maximum likelihood (REML), and minimum variance quadratic unbiased estimator (MIVQUE). A new procedure called the weighted analysis of means (WAM) estimator which utilizes prior information on the variance components is proposed. It is found to have optimum properties similar to the REML and MIVQUE, and it is also computationally simpler. For the mean, the overall sample average, grand mean, unweighted mean, and generalized least-squares (GLS) estimator with its weights obtained from the above estimators for the variance components are considered. Comparisons of the above procedures for the variance components and the mean are made from exact expressions for the biases and mean square errors (MSEs) of the estimators and from empirical investigations.