Model choice is a fundamental and much discussed activity in the analysis of data sets. Hierarchical models introducing random effects can not be handled by classical methods. Bayesian approaches using predictive distributions can, though the formal solution, which includes Bayes factors as a special case, can be criticized. We propose a predictive criterion where the goal is good prediction of a replicate of the observed data but tempered by fidelity to the observed values. We obtain this criterion by minimizing posterior loss for a given model and then, for models under consideration, select the one which minimizes this criterion. For a broad range of losses, the criterion emerges approximately as a form partitioned into a goodness-of-fit term and a penalty term. In the context of generalized linear mixed effects models we obtain a penalized deviance criterion comprised of a piece which is a Bayesian deviance measure and a piece which is a penalty for model complexity. We illustrate ...

this paper, we describe full Bayesian inference for generalised linear models where uncertainty

Recent computational advances have made it feasible to fit hierarchical models in a wide range of serious applications. If one entertains a collection of such models for a given data set, the problems of model adequacy and model choice arise. We focus on the former. While model checking usually addresses the entire model specification, model failures can occur at each hierarchical stage. Such failures include outliers, mean structure errors, dispersion misspecification, and inappropriate exchangeabilities. We propose another approach which is entirely simulation based. It only requires the model specification and that, for a given data set, one be able to simulate draws from the posterior under the model. By replicating a posterior of interest using data obtained under the model we can "see" the extent of variability in such a posterior. Then, we can compare the posterior obtained under the observed data with this medley of posterior replicates to ascertain whether the former is in agr...

SUMMARY. The generalized linear mixed model (GLMM), which extends the generalized linear model (GLM) to incorporate random effects characterizing heterogeneity among subjects, is widely used in analyzing correlated and longitudinal data. Although there is often interest in identify-ing the subset of predictors that have random effects, random effects selection can be challenging, particularly when outcome distributions are non-normal. This article proposes a fully Bayesian approach to the problem of simultaneous selection of fixed and random effects in GLMMs. Inte-grating out the random effects induces a covariance structure on the multivariate outcome data, and an important problem which we also consider is that of covariance selection. Our approach relies on variable selection-type mixture priors for the components in a special LDU decomposition of the random effects covariance. A stochastic search MCMC algorithm is developed, which relies on Gibbs sampling, with Taylor series expansions used to approximate intractable integrals. Simu-lated data examples are presented for different exponential family distributions, and the approach is applied to discrete survival data from a time-to-pregnancy study.

Abstract. A variety of simulation-based techniques have been proposed for detection of divergent behaviour at each level of a hierarchical model. We investigate a diagnostic test based on measuring the conflict between two independent sources of evidence regarding a parameter: that arising from its predictive prior given the remainder of the data, and that arising from its likelihood. This test gives rise to a p-value that exactly matches or closely approximates a cross-validatory predictive comparison, and yet is more widely applicable. Its properties are explored for normal hierarchical models and in an application in which divergent surgical mortality was suspected. Since full cross-validation is so computationally demanding, we examine full-data approximations which are shown to have only moderate conservatism in normal models. A second example concerns criticism of a complex growth curve model at both observation and parameter levels, and illustrates the issue of dealing with multiple p-values within a Bayesian framework. We conclude with the proposal of an overall strategy to detecting divergent behaviour in hierarchical models.

In analyzing longitudinal or clustered data with a mixed effects model (Laird and Ware, 1982), one may be concerned about violations of normality. Such violations can potentially impact subset selection for the fixed and random effects components of the model, inferences on the heterogeneity structure, and the accuracy of predictions. This article focuses on Bayesian methods for subset selection in nonparametric random effects models in which one is uncertain about the predictors to be included and the distribution of their random effects. We characterize the unknown distribution of the individual-specific regression coefficients using a weighted sum of Dirichlet process (DP)-distributed latent variables. By using carefully-chosen mixture priors for coefficients in the base distributions of the component DPs, we allow fixed and random effects to be effectively dropped out of the model. A stochastic search Gibbs sampler is developed for posterior computation, and the methods are illustrated using simulated data and real data from a multi-laboratory bioassay study.

This paper is an introduction to model selection intended for nonspecialists who have knowledge of the statistical concepts covered in a typical first

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This paper considers the development of envelope methods as a tool for simulation. Envelope methods are based on the construction of simple envelopes to functions. The proposed envelopes are general, require little input from the user and are based on the concavity structure of the function or some transformation of the function. The construction of these envelopes facilitates variate generation using the adaptive rejection algorithm.