Hierarchical testing of parametric models using the power-divergence family of test statistics
Bishop, Fienberg, and Holland (1975, Chapters 4 and 14) describe a method for testing hierarchical parametric models on contingency tables using maximum likelihood estimation and the loglikelihood ratio statistic G . We extend these ideas to the power-divergence family of test statistics (Cressie and Read, 1984). This is a one-parameter family of goodness-of-fit statistics that includes the loglikelihood ratio statistic G 2 , Pearson's X2 , the Freeman-Tukey statistic, the modified loglikelihood ratio statistic, and the Neyman-modified chi-squared statistic. In addition, we show that under Birch's conditions (Birch, 1964) an analysis of divergence is possible with the power-divergence family, analogous to the usual partitioning of G . Further, we give an algorithm, similar to iterative proportional fitting, for finding cell probability estimates. Finally, to illustrate these ideas, we fit loglinear models to several data sets and carry out analyses of divergence.