The Analysis of Unbalanced Cross-Classifications

the President Dr H. P. WYNN in the Chair) SUMMARY The problem of choosing appropriate models for multiply cross-classified data is examined. It is shown that a standard method of analysis used in many ANOVA programs, equivalent to Yates's method of weighted squares of means, may lead to inappropriate models. A simultaneous test procedure equivalent to one proposed in regression analysis is used to derive a class of acceptable models. The procedure is applied to an example. THIS paper presents a simple procedure for the analysis of multiply cross-classified data, for both normal and non-normal responses. The procedure is based on the hierarchical partition- ing of the total sum of squares or maximized likelihood, with a simultaneous test procedure, proposed in regression analysis, applied to the hierarchical partition to determine appropriate models for the data. This leads to a class of minimal adequate models; there may be several models in this class. The procedure is illustrated by applying it to a set of normal response data. Before discussing the general cross-classification, we consider some difficulties that have arisen in the treatment of the unbalanced two-way classification. 2. TIk TWO-WAY CLASSICATION 2.1. The2x2 Case Consider a 2 x 2 cross-classification with disproportionate subclass frequencies. We will be concerned with the case in which the disproportionality is considerable, so that approximate methods based on replacing the observed frequencies by proportional expected frequencies are inappropriate. Let A1, A2 and B1, B2 denote the levels of the cross-classifying factors, and let nU, j and ,uH be the sample size, and sample and population means in the (i,j) cell of the classifica- tion, and let n. and n 1 be the row and column marginal totals: B1 B2 B1 B2 B1 B2