Testing for misspecification in generalized linear mixed models

Abstract

Generalized linear mixed models have become a frequently used tool for the analysis of non-Gaussian longitudinal data. Estimation is often based on maximum likelihood theory, which assumes that the underlying probability model is correctly specified. Recent research shows that the results obtained from these models are not always robust against departures from the assumptions on which they are based. Therefore, diagnostic tools for the detection of model misspecifications are of the utmost importance. In this paper, we propose 2 diagnostic tests that are based on 2 equivalent representations of the model information matrix. We evaluate the power of both tests using theoretical considerations as well as via simulation. In the simulations, the performance of the new tools is evaluated in many settings of practical relevance, focusing on misspecification of the random-effects structure. In all the scenarios, the results were encouraging, however, the tests also exhibited inflated Type I error rates when the sample size was small or moderate. Importantly, a parametric bootstrap version of the tests seems to overcome this problem, although more research in this direction may be needed. Finally, both tests were also applied to analyze a real case study in psychiatry.

Analysis of cluster randomised stepped wedge trials with repeated cross-sectional samples

“…This cannot and should not be taken as a generalisable result – we expect model misspecification to have a large impact in some situations. Indeed, model misspecification in generalised linear mixed models is known to be important, as others have demonstrated in clustered data [35]. Misspecification can arise because of either misspecification of the model for the mean or misspecification of the correlation structure.…”

Asymptotic Properties and Information Criteria for Misspecified Generalized Linear Mixed Models

“…This assumption is known as the 'correctly specified assumption' (White, 1982). However, from both theoretical and practical perspectives, this assumption is restrictive and sometimes difficult to verify (Abad et al, 2010;Kyung et al, 2010;McCulloch and Neuhaus, 2011). In fact, model misspecification is unavoid-able in practice (Lv and Liu, 2014).…”

“…a goodness-of-fit test) to check the appropriateness of the model. Specifically, Abad et al (2010) proposed two diagnostic tests for GLMMs based on the model information matrix, and the tests can be adopted to check for departure from the distributional assumptions on which maximum likelihood estimators are based. Yin and Ma (2013) proposed a bootstrapbased modification of the Pearson test statistic to recover the χ 2 -distribution and constructed the model checking instrument for fixed effects generalized linear models.…”

Pairwise residuals and diagnostic tests for misspecified dependence structures in models for binary longitudinal data

“…None of these contributions provide general diagnostics for the dependence structure within a likelihood framework. However, generic misspecification tests for any kind of model misspecification have been proposed by Alonso et al 14 and Alonso Abad et al 15 and applied by these authors to the problem of detecting misspecified dependence structures. The tests are modified versions of the information matrix test 16 which compares a model-based estimate of the covariance matrix of the parameter estimates with a robust estimate (sandwich estimator).…”

“…The tests are modified versions of the information matrix test 16 which compares a model-based estimate of the covariance matrix of the parameter estimates with a robust estimate (sandwich estimator). We will compare the performance of the sandwich estimator test (SET) and the modified information matrix test (MIMT) of Alonso Abad et al 15 with our proposed tests.…”