Goodness of Fit of Product Multinomial Regression Models to Sparse Data

Abstract Tests of goodness of fit of sparse multinomial models with non-canonical links is proposed by using approximations to the first three moments of the conditional distribution of a modified Pearson Chi-square statistic. The modified Pearson statistic is obtained using a supplementary estimating equation approach. Approximations to the first three conditional moments of the modified Pearson statistic are derived. A simulation study is conducted to compare, in terms of empirical size and power, the usual Pearson Chi-square statistic, the standardized modified Pearson Chi-square statistic using the first two conditional moments, a method using Edgeworth approximation of the p-values based on the first three conditional moments and a score test statistic. There does not seems to be any qualitative difference in size of the four methods. However, the standardized modified Pearson Chi-square statistic and the Edgeworth approximation method of obtaining p-values using the first three conditional moments show power advantages compared to the usual Pearson Chi-square statistic, and the score test statistic. In some situations, for example, for small nominal level, the standardized modified Pearson Chi-square statistic shows some power advantage over the method using Edgeworth approximation of the p-values using the first three conditional moments. Also, the former is easier to use and so is preferable. Two data sets are analyzed and a discussion is given.