Comments on: Goodness-of-fit tests in mixed modes

Abstract

The paper proposes several methods for testing the goodness-of-fit of the distributions of random effects and errors in mixed models. Indeed, the paper introduces two completely different approaches to achieve the goal. Their first approach (Sects. 5 and 6) is based on order selection tests and is valid for testing the goodness-of-fit of the random-effects distribution when the error distribution is known (Sect. 5.3) and for testing simultaneously the distributions of random effects and errors (Sect. 5.4). In case of rejecting the null hypothesis, the selected order provides an approximation of the true density. Their second approach (Sects. 7 and 8) is based on minimizing the discrepancy between the hypothesized parametric distribution and some nonparametric estimator of the true distribution which plays the role of the empirical distribution in classical distance-based goodness-of-fit tests. In Sect. 7.1 they propose a test of fit for the error distribution that does not depend on the random effects, and in Sect. 7.2 they propose a test for the random effects distribution irrespective of the error distribution. The problem of testing the goodness of fit in mixed models has received a very limited attention in the literature so far, and most of approaches test the overall goodnessof-fit of the response distribution without distinction on the distribution of random effects and errors. Moreover, normality is a common assumption when using mixed models due to the lack of testing procedures under these models and also due to the difficulty in fitting a mixed model under a different distribution for either the random effects and/or errors. The extension of mixed models to GLMMs is well studied in