Maximum-likelihood estimation for the mixed analysis of variance model

TLDR: A procedure is developed for the maximum-likelihood estimation of the unknown constants and variances included in the general mixed analysis of variance model, involving fixed and random factors and interactions, and applies to all cases where the design matrices satisfy certain conditions.

Abstract: SUMMARY A procedure is developed for the maximum-likelihood estimation of the unknown constants and variances included in the general mixed analysis of variance model, involving fixed and random factors and interactions. The method applies to all cases where the design matrices satisfy certain conditions. The consistency and asymptotic efficiency of the estimates are discussed. Tests of hypotheses and confidence regions are derived. In this paper we develop a procedure for maximum-likelihood estimation for the general mixed analysis of variance model, defined in (1) below, involving any number of fixed and random factors and possibly interactions of any order. We do not specify 'equal numbers' or indeed any other experimental balance for our procedure, but we do require that our design matrices satisfy certain conditions of estimability for the parameters. In the case of balanced designs the estimation problem for the constants and variances involved in the linear model has been extensively treated: confining ourselves to just one reference on variance estimation, optimality properties of the classical analysis of variance procedures have already been demonstrated for various balanced designs (e.g. Graybill, 1961). However, results for unbalanced factorial and nested data are much more restricted: Henderson (1953) has suggested a method of unbiased estimation of variance components for the unbalanced two-way classification but his method is computationally cumbersome for a mixed model and when the number of classes is large. Searle & Henderson (1961) have suggested a simpler method also for the unbalanced two-way classification with one fixed factor containing a moderate number of levels and a random factor permitted to have quite a large number of levels. Bush & Anderson (1963) have investigated for the two-way classification random model the relative efficiency of Henderson's (1953) method and two other methods, A and B, based on the respective methods of fitting constants and weighted squares of means described by Yates (1934) for experiments based on a fixed effects model which also provide unbiased estimates of variance components. Possibilities of generalizations are indicated. In all the above methods the estimates of any constants in the model are computed from the 'Aitken Type' weighted least squares estimators based on the exact variance-covariance matrix of the experimental responses which involves the unknown variance ratios. The estimation of the latter is then based on various unbiased procedures so that little is known about any optimality properties of any of the resulting estimators. However, all these methods reduce to the well-known procedures based on minimal sufficient statistics in the special cases of balanced designs. The method of maximum-likelihood estimation here developed differs from the above in that maximum-likelihood equations are used and solved for both the estimates of constants