Improved transformed deviance statistic for testing a logistic regression model

In logistic regression models, we consider the deviance statistic (the log likelihood ratio statistic) DD as a goodness-of-fit test statistic. In this paper, we show the derivation of an expression of asymptotic expansion for the distribution of DD under a null hypothesis. Using the continuous term of the expression, we obtain a Bartlett-type transformed statistic D̃ that improves the speed of convergence to the chi-square limiting distribution of DD. By numerical comparison, we find that the transformed statistic D̃ performs much better than DD. We also give a real data example of D̃ being more reliable than DD for testing a hypothesis.