Empirical Bayes Gibbs sampling

The wide applicability of Gibbs sampling has increased the use of more complex and multi-level hierarchical models. To use these models entails dealing with hyperparameters in the deeper levels of a hierarchy. There are three typical methods for dealing with these hyperparameters: specify them, estimate them, or use a 'flat' prior. Each of these strategies has its own associated problems. In this paper, using an empirical Bayes approach, we show how the hyperparameters can be estimated in a way that is both computationally feasible and statistically valid.     

Robust estimation of multivariate covariance components

In many settings, such as interlaboratory testing, small area estimation in sample surveys, and heritability studies, investigators are interested in estimating covariance components for multivariate measurements. However, the presence of outliers can seriously distort estimates obtained using standard procedures such as maximum likelihood. We propose a procedure based on M-estimation for robustly estimating multivariate covariance components in the presence of outliers; the procedure applies to balanced and unbalanced data. We present an algorithm for computing the robust estimates and examine the performance of the estimator through a simulation study. The estimator is used to find covariance components and identify outliers in a study of variability of egg length and breadth measurements of American coots.     

Spatio-temporal modeling of localized brain activity

Functional neuroimaging, including positron emission tomography (PET) and functional magnetic resonance imaging (fMRI), plays an important role in identifying specific brain regions associated with experimental stimuli or psychiatric disorders such as schizophrenia. PET and fMRI produce massive data sets that contain both temporal correlations from repeated scans and complex spatial correlations. Several methods exist for handling temporal correlations, some of which rely on transforming the response data to induce either a known or an independence covariance structure. Despite the presence of spatial correlations between the volume elements (voxels) comprising a brain scan, conventional methods perform voxel-by-voxel analyses of measured brain activity. We propose a two-stage spatio-temporal model for the estimation and testing of localized activity. Our second-stage model specifies a spatial auto-regression, capturing correlations within neural processing clusters defined by a data-driven cluster analysis. We use maximum likelihood methods to estimate parameters from our spatial autoregressive model. Our model protects against type-I errors, enables the detection of both localized and regional activations (including volume of interest effects), provides information on functional connectivity in the brain, and establishes a framework to produce spatially smoothed maps of distributed brain activity for each individual. We illustrate the application of our model using PET data from a study of working memory in individuals with schizophrenia.     

A case study on the choice,interpretation and checking of multilevel models for longitudinal binary outcomes

Recent advances in statistical software have led to the rapid diffusion of new methods for modelling longitudinal data. Multilevel (also known as hierarchical or random effects) models for binary outcomes have generally been based on a logistic-normal specification, by analogy with earlier work for normally distributed data. The appropriate application and interpretation of these models remains somewhat unclear, especially when compared with the computationally more straightforward semiparametric or 'marginal' modelling (GEE) approaches. In this paper we pose two interrelated questions. First, what limits should be placed on the interpretation of the coefficients and inferences derived from random-effect models involving binary outcomes? Second, what diagnostic checks are appropriate for evaluating whether such random-effect models provide adequate fits to the data? We address these questions by means of an extended case study using data on adolescent smoking from a large cohort study. Bayesian estimation methods are used to fit a discrete-mixture alternative to the standard logistic-normal model, and posterior predictive checking is used to assess model fit. Surprising parallels in the parameter estimates from the logistic-normal and mixture models are described and used to question the interpretability of the so-called 'subject-specific' regression coefficients from the standard multilevel approach. Posterior predictive checks suggest a serious lack of fit of both multilevel models. The results do not provide final answers to the two questions posed, but we expect that lessons learned from the case study will provide general guidance for further investigation of these important issues.     

Disease mapping and spatial regression with count data

In this paper, we provide critical reviews of methods suggested for the analysis of aggregate count data in the context of disease mapping and spatial regression. We introduce a new method for picking prior distributions, and propose a number of refinements of previously used models. We also consider ecological bias, mutual standardization, and choice of both spatial model and prior specification. We analyze male lip cancer incidence data collected in Scotland over the period 1975-1980, and outline a number of problems with previous analyses of these data. In disease mapping studies, hierarchical models can provide robust estimation of area-level risk parameters, though care is required in the choice of covariate model, and it is important to assess the sensitivity of estimates to the spatial model chosen, and to the prior specifications on the variance parameters. Spatial ecological regression is a far more hazardous enterprise for two reasons. First, there is always the possibility of ecological bias, and this can only be alleviated by the inclusion of individual-level data. For the Scottish data, we show that the previously used mean model has limited interpretation from an individual perspective. Second, when residual spatial dependence is modeled, and if the exposure has spatial structure, then estimates of exposure association parameters will change when compared with those obtained from the independence across space model, and the data alone cannot choose the form and extent of spatial correlation that is appropriate.