Analysis of categorical data by linear models

Abstract

Assume there are n$_i$., i = 1, 2, ..., s, samples from s multinomial distributions each having r categories of response. Then define any u functions of the unknown true cell probabilities {$pi_{ij}$ : i = 1, 2, ..., s; j = 1, 2, ..., r, where $sum^r_{j=1}pi_{ij}$ = 1} that have derivatives up to the second order with respect to $pi_{ij}$, and for which the matrix of first derivatives is of rank u. A general noniterative procedure is described for fitting these functions to a linear model, for testing the goodness-of-fit of the model, and for testing hypotheses about the parameters in the linear model. The special cases of linear functions and logarithmic functions of the $pi_{ij}$ are developed in detail, and some examples of how the general approach can be used to analyze various types of categorical data are presented.