A Mixed Model for the Complete Three-Way Layout with Two Random-Effects Factors

Abstract

In the present paper the Mixed Model developed by Scheffe [10] for the complete two-way layout is extended to the complete three-way layout with two random-effects factors. The model involves three basic covariance matrices of unknown parameters in addition to the error variance and fixed effects. Assuming normality, tests of the usual statistical hypotheses, except that of no fixed main effects, are derived from the analysis of variance table. Those of no interaction between the fixed-effects and a random-effects factor are applicable only under a simplifying assumption. A reduced form of the model is derived which involves sets of independent identically distributed random vectors. These are used to obtain unbiased estimators of the basic covariance matrices and to construct a $T^2$ test of the hypothesis of no fixed main effects. This test involves nonoptimum estimators of the effects, but this is shown to result in general only in a small loss of power. Individual and simultaneous confidence intervals for the fixed main effects are obtained in terms of these nonoptimum estimators.