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Analysis of categorical data by linear models. (English) Zbl 1149.62317
Summary: Assume there are $$n_i$$ $$(i=1,2,\cdots,s)$$ samples from $$s$$ multinomial distributions, each having $$r$$ categories of response. Then define any $$u$$ functions of the unknown true cell probabilities $$\{\pi_{ij}: i=1,2,\cdots,s; j=1,2,\cdots,r$$, where $$\sum_{i=1}^r\pi_{ij}=1\}$$ that have derivatives of order up to the second 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.

##### MSC:
 62J05 Linear regression; mixed models 62G10 Nonparametric hypothesis testing
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