The analysis of categorical data from mixed models

Abstract

This paper is concerned with contingency tables which are analogous to the well-known mixed model in analysis of variance. The corresponding experimental situation involves exposing each of n subjects to each of the d levels of a given factor and classifying the d responses into one of r categories. The resulting data are represented in an r r $cdots$ r contingency table of d dimensions. The hypothesis of principal interest is equality of the one-dimensional marginal distributions. Alternatively, if the r categories may be quantitatively scaled, then attention is directed at the hypothesis of equality of the mean scores over the d first order marginals. Test statistics are developed in terms of minimum Neyman $chi^2$ or equivalently weighted least squares analysis of underlying linear models. As such, they bear a strong resemblance to the Hotelling T$^2$ procedures used with continuous data in mixed models. Several numerical examples are given to illustrate the use of the various methods discussed.