Analysis of Categorical Data by Linear Models

Assume there are ni., 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 {7rij: i = 1, 2, * , s; j = 1, 2, * , r, where E jrij l 1 } that have derivatives up to the second order with respect to 7rij, 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 7rin are developed in detail, and some examples of how the general approach can be used to analyze various types of categorical data are presented.