Estimation of components of variance for partially cross-classified data in a factorial arrangement of treatments

A procedure is presented for obtaining estimates of components of variance for the main effects and interaction in a two-way factorial arrangement of treatments with only partially cross-classified data (sometimes referred to as unbalanced data in a two-way crossed classification model with interaction). The procedure entails the use of two hierarchical analyses of variance in which each main effect is first treated as classes and then as subclasses. The estimation of the variance component for each main effect results from the solution of two simultaneous equations with values used obtained directly from the hierarchical analyses. The interaction component of variance can be obtained from the variance components for main effects. An example is presented for the purpose of illustrating the procedure. Interpretation of the interaction variance component estimated by the method is limited due to the absence of some of the subclasses.