Diagnosing misspecification of the random-effects distribution in mixed models

Drikvandi, Reza ORCID: https://orcid.org/0000-0002-7245-9713, Verbeke, Geert and Molenberghs, Geert (2017) Diagnosing misspecification of the random-effects distribution in mixed models. Biometrics, 73 (1). pp. 63-71. ISSN 0006-341X

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Abstract

© 2016, The International Biometric Society. It is traditionally assumed that the random effects in mixed models follow a multivariate normal distribution, making likelihood-based inferences more feasible theoretically and computationally. However, this assumption does not necessarily hold in practice which may lead to biased and unreliable results. We introduce a novel diagnostic test based on the so-called gradient function proposed by Verbeke and Molenberghs (2013) to assess the random-effects distribution. We establish asymptotic properties of our test and show that, under a correctly specified model, the proposed test statistic converges to a weighted sum of independent chi-squared random variables each with one degree of freedom. The weights, which are eigenvalues of a square matrix, can be easily calculated. We also develop a parametric bootstrap algorithm for small samples. Our strategy can be used to check the adequacy of any distribution for random effects in a wide class of mixed models, including linear mixed models, generalized linear mixed models, and non-linear mixed models, with univariate as well as multivariate random effects. Both asymptotic and bootstrap proposals are evaluated via simulations and a real data analysis of a randomized multicenter study on toenail dermatophyte onychomycosis.