Transformed goodness-of-fit statistics for a generalized linear model of binary data

In a generalized linear model of binary data, we consider models based on a general link function including a logistic regression model and a probit model as special cases. For testing the null hypothesis H0H0 that the considered model is correct, we consider a family of ϕϕ-divergence goodness-of-fit test statistics CϕCϕ that includes a power divergence family of statistics RaRa. We propose a transformed CϕCϕ statistics that improves the speed of convergence to a chi-square limiting distribution and show numerically that the transformed RaRa statistic performs well. We also give a real data example of the transformed RaRa statistic being more reliable than the original RaRa statistic for testing H0H0.