Analysis of Categorical Data Under Log-Linear Models

Abstract

This chapter considers analysis of categorical data from complex surveys using log-linear models for cell probabilities ${\varvec{\pi }}$ in contingency tables. Noting that appropriate ML equations for the model parameter $\theta$ and hence of ${\varvec{\pi }}(\theta )$ are difficult to obtain for general survey designs, ‘pseudo-MLE’s have been used to estimate the cell probabilities. The asymptotic distributions of goodness-of-fit (G-o-F) statistic $X_P^2$, and likelihood ratio (LR) statistic $G^2$ have been derived and these test statistics have been modified using Rao-Scott (J Amer Stat Assoc 76: 221–230, 1981, Ann Stat 12: 46–60, 1984) first- and second-order corrections, F-based corrections, and Fellegi’s (J Amer Stat Assoc 75: 261–268, 1980) correction. Wald’s test statistic has been looked into. All these modified statistics have been examined in G-o-F tests, homogeneity tests, and independence tests. Fay’s Jackknifed versions to these statistics have been considered. Brier’s model has also been looked into. Lastly, nested models have been considered and all the above results have been examined in its light.