On Block Updating in Markov Random Field Models for Disease Mapping

Gaussian Markov random field (GMRF) models are commonly used to model spatial correlation in disease mapping applications. For Bayesian inference by MCMC, so far mainly single-site updating algorithms have been considered. However, convergence and mixing properties of such algorithms can be extremely poor due to strong dependencies of parameters in the posterior distribution. In this paper, we propose various block sampling algorithms in order to improve the MCMC performance. The methodology is rather general, allows for non-standard full conditionals, and can be applied in a modular fashion in a large number of different scenarios. For illustration we consider three different applications: two formulations for spatial modelling of a single disease (with and without additional unstructured parameters respectively), and one formulation for the joint analysis of two diseases. The results indicate that the largest benefits are obtained if parameters and the corresponding hyperparameter are updated jointly in one large block. Implementation of such block algorithms is relatively easy using methods for fast sampling of Gaussian Markov random fields ( Rue, 2001 ). By comparison, Monte Carlo estimates based on single-site updating can be rather misleading, even for very long runs. Our results may have wider relevance for efficient MCMC simulation in hierarchical models with Markov random field components.     

Nonparametric smoothing in the analysis of air pollution and respiratory illness

While most of epidemiology is observational, rather than experimental, the culture of epidemiology is still derived from agricultural experiments, rather than other observational fields, such as astronomy or economics. The mismatch is made greater as focus has turned to continue risk factors, multifactorial outcomes, and outcomes with large variation unexplainable by available risk factors. The analysis of such data is often viewed as hypothesis testing with statistical control replacing randomization. However, such approaches often test restricted forms of the hypothesis being investigated, such as the hypothesis of a linear association, when there is no prior empirical or theoretical reason to believe that if an association exists, it is linear. In combination with the large nonstochastic sources of error in such observational studies, this suggests the more flexible alternative of exploring the association. Conclusions on the possible causal nature of any discovered association will rest on the coherence and consistency of multiple studies. Nonparametric smoothing in general, and generalized additive models in particular, represent an attractive approach to such problems. This is illustrated using data examining the relationship between particulate air pollution and daily mortality in Birmingham, Alabama; between particulate air pollution, ozone, and SO₂ and daily hospital admissions for respiratory illness in Philadelphia; and between ozone and particulate air pollution and coughing episodes in children in six eastern U.S. cities. The results indicate that airborne particles and ozone are associated with adverse health outcomes at very low concentrations, and that there are likely no thresholds for these relationships.     

Physiologically Based Liver Modeling and Risk Assessment

Because of the inherent complexity of biological systems, there is often a choice between a number of apparently equally applicable physiologically based models to describe uptake and metabolism processes in toxicology or risk assessment. These models may fit the particular data sets of interest equally well, but may give quite different parameter estimates or predictions under different (extrapolated) conditions. Such competing models can be discriminated by a number of methods, including potential refutation by means of strategic experiments, and their ability to suitably incorporate all relevant physiological processes. For illustration, three currently used models for steady-state hepatic elimination--the venous equilibration model, the parallel tube model, and the distributed sinusoidal perfusion model--are reviewed and compared with particular reference to their application in the area of risk assessment. The ability of each of the models to describe and incorporate such physiological processes as protein binding, precursor-metabolite relations and hepatic zones of elimination, capillary recruitment, capillary heterogeneity, and intrahepatic shunting is discussed. Differences between the models in hepatic parameter estimation, extrapolation to different conditions, and interspecies scaling are discussed, and criteria for choosing one model over the others are presented. In this case, the distributed model provides the most general framework for describing physiological processes taking place in the liver, and has so far not been experimentally refuted, as have the other two models. These simpler models may, however, provide useful bounds on parameter estimates and on extrapolations and risk assessments.     

ON THE INDEPENDENCE OF THE SAMPLE MEAN AND TRANSLATION-INVARIANT STATISTICS FOR MATRIX NORMAL DISTRIBUTIONS

We present an explicit characterization of the joint dependency structure of an n×p matrix normal random matrix such that the p-dimensional sample mean vector is independent of all translation invariant statistics.     

Bayesian validation assessment of multivariate computational models

Multivariate model validation is a complex decision-making problem involving comparison of multiple correlated quantities, based upon the available information and prior knowledge. This paper presents a Bayesian risk-based decision method for validation assessment of multivariate predictive models under uncertainty. A generalized likelihood ratio is derived as a quantitative validation metric based on Bayes’ theorem and Gaussian distribution assumption of errors between validation data and model prediction. The multivariate model is then assessed based on the comparison of the likelihood ratio with a Bayesian decision threshold, a function of the decision costs and prior of each hypothesis. The probability density function of the likelihood ratio is constructed using the statistics of multiple response quantities and Monte Carlo simulation. The proposed methodology is implemented in the validation of a transient heat conduction model, using a multivariate data set from experiments. The Bayesian methodology provides a quantitative approach to facilitate rational decisions in multivariate model assessment under uncertainty.     

Bayesian analysis of directed graphs data with applications to social networks

Summary.  A fully Bayesian analysis of directed graphs, with particular emphasis on applica- tions in social networks, is explored. The model is capable of incorporating the effects of covariates, within and between block ties and multiple responses. Inference is straightforward by using software that is based on Markov chain Monte Carlo methods. Examples are provided which highlight the variety of data sets that can be entertained and the ease with which they can be analysed.     

New imperfect rankings models for ranked set sampling

Ranked set sampling is a sampling approach that leads to improved statistical inference in situations where the units to be sampled can be ranked relative to each other prior to formal measurement. This ranking may be done either by subjective judgment or according to an auxiliary variable, and it need not be completely accurate. In fact, results in the literature have shown that no matter how poor the quality of the ranking, procedures based on ranked set sampling tend to be at least as efficient as procedures based on simple random sampling. However, efforts to quantify the gains in efficiency for ranked set sampling procedures have been hampered by a shortage of available models for imperfect rankings. In this paper, we introduce a new class of models for imperfect rankings, and we provide a rigorous proof that essentially any reasonable model for imperfect rankings is a limit of models in this class. We then describe a specific, easily applied method for selecting an appropriate imperfect rankings model from the class.     

D-efficient window experimental designs

In the paper we present a new method of calculating sampling intervals, so-called windows, allowing an experimenter some flexibility in timing the sample collection, while a minimum required design efficiency for parameter estimation is assured. The method is based on the Equivalence Theorem for D-optimality what makes the length of each window related to the parameter sensitivities. An example of calculating the windows in a pharmacokinetic study is presented. Some other methods of calculating efficient sampling windows are briefly discussed.     

The evaluation of DNA evidence in pedigrees requiring population inference

Summary. The evaluation of nuclear DNA evidence for identification purposes is performed here taking account of the uncertainty about population parameters. Graphical models are used to detail the hypotheses being debated in a trial with the aim of obtaining a directed acyclic graph. Graphs also clarify the set of evidence that contributes to population inferences and they also describe the conditional independence structure of DNA evidence. Numerical illustrations are provided by re-examining three case-studies taken from the literature. Our calculations of the weight of evidence differ from those given by the authors of case-studies in that they reveal more conservative values.     

An affine‐invariant multivariate sign test for cluster correlated data

The author presents a multivariate location model for cluster correlated observations. He proposes an affine‐invariant multivariate sign statistic for testing the value of the location parameter. His statistic is an adaptation of that proposed by Randles (2000). The author shows, under very mild conditions, that his test statistic is asymptotically distributed as a chi‐squared random variable under the null hypothesis. In particular, the test can be used for skewed populations. In the context of a general multivariate normal model, the author obtains values of his test's Pitman asymptotic efficiency relative to another test based on the overall average. He shows that there is an improvement in the relative performance of the new test as soon as intra‐cluster correlation is present Even in the univariate case, the new test can be very competitive for Gaussian data. Furthermore, the statistic is easy to compute, even for large dimensional data. The author shows through simulations that his test performs well compared to the average‐based test. He illustrates its use with real data.