Evaluation of frequentist test statistics using constrained statistical inference in the context of the generalized linear model

When faced with a binary or count outcome, informative hypotheses can be tested in the generalized linear model using the distance statistic as well as modified versions of the Wald, the Score and the likelihood-ratio test (LRT). In contrast to classical null hypothesis testing, informative hypotheses allow to directly examine the direction or the order of the regression coefficients. Since knowledge about the practical performance of informative test statistics is missing in the theoretically oriented literature, we aim at closing this gap using simulation studies in the context of logistic and Poisson regression. We examine the effect of the number of constraints as well as the sample size on type I error rates when the hypothesis of interest can be expressed as a linear function of the regression parameters. The LRT shows the best performance in general, followed by the Score test. Furthermore, both the sample size and especially the number of constraints impact the type I error rates considerably more in logistic compared to Poisson regression. We provide an empirical data example together with R code that can be easily adapted by applied researchers. Moreover, we discuss informative hypothesis testing about effects of interest, which are a non-linear function of the regression parameters. We demonstrate this by means of a second empirical data example.