A Bayesian method of sample size determination with practical applications

Summary.  The problem motivating the paper is the determination of sample size in clinical trials under normal likelihoods and at the substantive testing stage of a financial audit where normality is not an appropriate assumption. A combination of analytical and simulation-based techniques within the Bayesian framework is proposed. The framework accommodates two different prior distributions: one is the general purpose fitting prior distribution that is used in Bayesian analysis and the other is the expert subjective prior distribution, the sampling prior which is believed to generate the parameter values which in turn generate the data. We obtain many theoretical results and one key result is that typical non-informative prior distributions lead to very small sample sizes. In contrast, a very informative prior distribution may either lead to a very small or a very large sample size depending on the location of the centre of the prior distribution and the hypothesized value of the parameter. The methods that are developed are quite general and can be applied to other sample size determination problems. Some numerical illustrations which bring out many other aspects of the optimum sample size are given.     

A Weibull Regression Model with Gamma Frailties for Multivariate Survival Data

Frequently in the analysis of survival data, survival times within the same group are correlated due to unobserved co-variates. One way these co-variates can be included in the model is as frailties. These frailty random block effects generate dependency between the survival times of the individuals which are conditionally independent given the frailty. Using a conditional proportional hazards model, in conjunction with the frailty, a whole new family of models is introduced. By considering a gamma frailty model, often the issue is to find an appropriate model for the baseline hazard function. In this paper a flexible baseline hazard model based on a correlated prior process is proposed and is compared with a standard Weibull model. Several model diagnostics methods are developed and model comparison is made using recently developed Bayesian model selection criteria. The above methodologies are applied to the McGilchrist and Aisbett (1991) kidney infection data and the analysis is performed using Markov Chain Monte Carlo methods. This revised version was published online in July 2006 with corrections to the Cover Date.     

A comparison of centring parameterisations of Gaussian process-based models for Bayesian computation using MCMC

Markov chain Monte Carlo (MCMC) algorithms for Bayesian computation for Gaussian process-based models under default parameterisations are slow to converge due to the presence of spatial- and other-induced dependence structures. The main focus of this paper is to study the effect of the assumed spatial correlation structure on the convergence properties of the Gibbs sampler under the default non-centred parameterisation and a rival centred parameterisation (CP), for the mean structure of a general multi-process Gaussian spatial model. Our investigation finds answers to many pertinent, but as yet unanswered, questions on the choice between the two. Assuming the covariance parameters to be known, we compare the exact rates of convergence of the two by varying the strength of the spatial correlation, the level of covariance tapering, the scale of the spatially varying covariates, the number of data points, the number and the structure of block updating of the spatial effects and the amount of smoothness assumed in a Matérn covariance function. We also study the effects of introducing differing levels of geometric anisotropy in the spatial model. The case of unknown variance parameters is investigated using well-known MCMC convergence diagnostics. A simulation study and a real-data example on modelling air pollution levels in London are used for illustrations. A generic pattern emerges that the CP is preferable in the presence of more spatial correlation or more information obtained through, for example, additional data points or by increased covariate variability.     

A Comparison of Frailty and Other Models for Bivariate Survival Data

Multivariate survival data arise when eachstudy subject may experience multiple events or when study subjectsare clustered into groups. Statistical analyses of such dataneed to account for the intra-cluster dependence through appropriatemodeling. Frailty models are the most popular for such failuretime data. However, there are other approaches which model thedependence structure directly. In this article, we compare thefrailty models for bivariate data with the models based on bivariateexponential and Weibull distributions. Bayesian methods providea convenient paradigm for comparing the two sets of models weconsider. Our techniques are illustrated using two examples.One simulated example demonstrates model choice methods developedin this paper and the other example, based on a practical dataset of onset of blindness among patients with diabetic Retinopathy,considers Bayesian inference using different models.     

Fusing point and areal level space–time data with application to wet deposition

Summary.  Motivated by the problem of predicting chemical deposition in eastern USA at weekly, seasonal and annual scales, the paper develops a framework for joint modelling of point- and grid-referenced spatiotemporal data in this context. The hierarchical model proposed can provide accurate spatial interpolation and temporal aggregation by combining information from observed point-referenced monitoring data and gridded output from a numerical simulation model known as the 'community multi-scale air quality model'. The technique avoids the change-of-support problem which arises in other hierarchical models for data fusion settings to combine point- and grid-referenced data. The hierarchical space–time model is fitted to weekly wet sulphate and nitrate deposition data over eastern USA. The model is validated with set-aside data from a number of monitoring sites. Predictive Bayesian methods are developed and illustrated for inference on aggregated summaries such as quarterly and annual sulphate and nitrate deposition maps. The highest wet sulphate deposition occurs near major emissions sources such as fossil-fuelled power plants whereas lower values occur near background monitoring sites.     

A new class of multivariate skew distributions with applications to bayesian regression models

Abstract: The authors develop a new class of distributions by introducing skewness in multivariate elliptically symmetric distributions. The class, which is obtained by using transformation and conditioning, contains many standard families including the multivariate skew‐normal and t distributions. The authors obtain analytical forms of the densities and study distributional properties. They give practical applications in Bayesian regression models and results on the existence of the posterior distributions and moments under improper priors for the regression coefficients. They illustrate their methods using practical examples.     

Updating Schemes, Correlation Structure, Blocking and Parameterization for the Gibbs Sampler

In this paper many convergence issues concerning the implementation of the Gibbs sampler are investigated. Exact computable rates of convergence for Gaussian target distributions are obtained. Different random and non-random updating strategies and blocking combinations are compared using the rates. The effect of dimensionality and correlation structure on the convergence rates are studied. Some examples are considered to demonstrate the results. For a Gaussian image analysis problem several updating strategies are described and compared. For problems in Bayesian linear models several possible parameterizations are analysed in terms of their convergence rates characterizing the optimal choice.