1 - Polytomous Responses

Abstract

The analysis of data by log-linear models involves several distinct stages. In the first stage, a plausible model is proposed for the data under study. In the second stage, unknown parameters in the model are estimated from the data, generally by the method of maximum likelihood. This methods yields estimates available in closed form, as in the equiprobability model, or estimates computed by a version of the Newton–Raphson algorithm with analogies to weighted regression analysis, as in the log-linear time-trend model. In the third stage, these parameter estimates are used in the statistical tests of the model's adequacy. More specific insight into deviations between model and data are provided by the analysis of adjusted residuals and of selected linear combinations of frequencies. This chapter discusses maximum-likelihood equations and the Newton–Raphson algorithm for general log-linear models for both multinominal and Poisson data. The general analogies with weighted regression analysis are used to describe the Newton-Raphson algorithm and to develop large-sample approximations to the distributions of parameter estimates, residuals, and chi-square statistics.