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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. ispartof: Biometrics vol:73 issue:1 pages:63-71 ispartof: location:United States status: published

This article reviews various recently suggested approaches to estimate the random effects distribution in a linear mixed model, i.e. (1) the smoothing by roughening approach of Shen and Louis,(1) (2) the semi-non-parametric approach of Zhang and Davidian,(2) (3) the heterogeneity model of Verbeke and Lesaffre(3) and (4) a flexible approach of Ghidey et al.(4) These four approaches are compared via an extensive simulation study. We conclude that for the considered cases, the approach of Ghidey et al.(4) often shows to have the smallest integrated mean squared error for estimating the random effects distribution. An analysis of a longitudinal dental data set illustrates the performance of the methods in a practical example.

Medical research applications often involve hierarchical data structures such as patients grouped in hospitals or teeth clustered within mouths. In the modeling process, it is important to incorporate the correlation among observations belonging to the same cluster. Failing to do so, may lead to incorrect estimation of the variability in the data and misleading inference. The Bayesian approach offers tools to directly account for the hierarchical structure of the data via augmented latent variables. In addition, Bayesian methods allow us to fit flexible models for complex data structures, which may be difficult to estimate based on maximum likelihood. Moreover, the inclusion of tailored priors can solve identifiability issues in elaborate models. The increasing complexity in modern data sets motivates the development of novel models that can accurately describe the data. In this sense, we propose some extensions of Bayesian methods to model hierarchical data. We cover a broad spectrum of topics such as covariance matrix modeling for multivariate outcomes, model selection using the deviance information criterion, multi-group regression methods and factor analysis. The four topics are introduced throughout the chapters of this dissertation. These approaches are motivated in the context of medical sciences but they can be implemented in many other fields in general. We carry out exhaustive simulation studies to demonstrate the advantages of the proposed models compared to the existing methods in the literature. In addition, the procedures are illustrated using the RN4CAST project \citep{serme11} and other well-known data sets. We implement all the models in JAGS running interactively from R. The relevant code can be found online as supplementary material of the published papers. status: published