Testing Lack-of-Fit of Generalized Linear Models via Laplace Approximation
In this study we develop a new method for testing the null hypothesis that the predictor function in a canonical link regression model has a prescribed linear form. The class of models, which we will refer to as canonical link regression models, constitutes arguably the most important subclass of generalized linear models and includes several of the most popular generalized linear models. In addition to the primary contribution of this study, we will revisit several other tests in the existing literature. The common feature among the proposed test, as well as the existing tests, is that they are all based on orthogonal series estimators and used to detect departures from a null model. Our proposal for a new lack-of-fit test is inspired by the recent contribution of Hart and is based on a Laplace approximation to the posterior probability of the null hypothesis. Despite having a Bayesian construction, the resulting statistic is implemented in a frequentist fashion. The formulation of the statistic is based on characterizing departures from the predictor function in terms of Fourier coefficients, and subsequent testing that all of these coefficients are 0. The resulting test statistic can be characterized as a weighted sum of exponentiated squared Fourier coefficient estimators, whereas the weights depend on user-specified prior probabilities. The prior probabilities provide the investigator the flexibility to examine specific departures from the prescribed model. Alternatively, the use of noninformative priors produces a new omnibus lack-of-fit statistic. We present a thorough numerical study of the proposed test and the various existing orthogonal series-based tests in the context of the logistic regression model. Simulation studies demonstrate that the test statistics under consideration possess desirable power properties against alternatives that have been identified in the existing literature as being important.