Summary. We consider the problem of comparing complex hierarchical models in which the number of parameters is not clearly defined. Using an information theoretic argument we derive a measure pD for the effective number of parameters in a model as the difference between the posterior mean of the deviance and the deviance at the posterior means of the parameters of interest. In general pD approximately corresponds to the trace of the product of Fisher's information and the posterior covariance, which in normal models is the trace of the ‘hat’ matrix projecting observations onto fitted values. Its properties in exponential families are explored. The posterior mean deviance is suggested as a Bayesian measure of fit or adequacy, and the contributions of individual observations to the fit and complexity can give rise to a diagnostic plot of deviance residuals against leverages. Adding pD to the posterior mean deviance gives a deviance information criterion for comparing models, which is related to other information criteria and has an approximate decision theoretic justification. The procedure is illustrated in some examples, and comparisons are drawn with alternative Bayesian and classical proposals. Throughout it is emphasized that the quantities required are trivial to compute in a Markov chain Monte Carlo analysis.

/pdf/bayesian-measures-of-model-complexity-and-fit-1ouzmtfu7x.pdf

Bayesian measures of model complexity and fit

Missing data in clinical trials can have a major effect on the validity of the inferences that can be drawn from the trial. This article reviews methods for preventing missing data and, failing that, dealing with data that are missing.

/pdf/the-prevention-and-treatment-of-missing-data-in-clinical-gvpp8p81b1.pdf

The Prevention and Treatment of Missing Data in Clinical Trials

Subjects often drop out of longitudinal studies prematurely, yielding unbalanced data with unequal numbers of measures for each subject. Modern software programs for handling unbalanced longitudinal data improve on methods that discard the incomplete cases by including all the data, but also yield biased inferences under plausible models for the drop-out process. This article discusses methods that simultaneously model the data and the drop-out process within a unified model-based framework. Models are classified into two broad classes—random-coefficient selection models and random-coefficient pattern-mixture models—depending on how the joint distribution of the data and drop-out mechanism is factored. Inference is likelihood-based, via maximum likelihood or Bayesian methods. A number of examples in the literature are placed in this framework, and possible extensions outlined. Data collection on the nature of the drop-out process is advocated to guide the choice of model. In cases where the drop-...

Modeling the Drop-Out Mechanism in Repeated-Measures Studies

This article provides an overview of generalized linear mixed models (GLMMs), how they are fit to data, and the inferences possible when using them. GLMMs are a class of statistical models that handle a wide variety of distributions for the outcome, accommodate nonlinear models, and model correlated data. As regression methods, they are not only capable of estimation and testing of covariate effects but also can be used to draw inferences about correlation structures in the data and are able to calculate predicted values that take into account not only covariates but also observed outcomes. We briefly describe software available for fitting GLMMs.

Generalized Linear Mixed Models

Markov chain Monte Carlo (MCMC) is a technique for estimating by simulation the expectation of a statistic in a complex model. Successive random selections form a Markov chain, the stationary distribution of which is the target distribution. It is particularly useful for the evaluation of posterior distributions in complex Bayesian models. In the Metropolis–Hastings algorithm, items are selected from an arbitrary “proposal” distribution and are retained or not according to an acceptance rule. The Gibbs sampler is a special case in which the proposal distributions are conditional distributions of single components of a vector parameter. Various special cases and applications are considered.


Keywords:

Metropolis–Hastings;
Gibbs sampler;
stationary distribution;
proposal distribution;
hybrid chain;
reversible jump;
hierarchical;
missing data

Markov Chain Monte Carlo

Analysis of longitudinal studies is often complicated through differences amongst individuals in the number and spacing of observations. Laird and Ware (1982, Biometrics 38, 963-974) proposed a linear random-effects model to deal with this problem. We propose a generalisation of this model to accommodate multiple random effects, and show how Gibbs sampling can be used to estimate it. We illustrate the methodology with an analysis of long-term response to hepatitis B vaccination, and demonstrate that the methodology can be easily and effectively extended to deal with censoring in the dependent variable.

Yvonnet B

Papers

Random-effects models, for longitudinal data using Gibbs sampling.