Leonhard Held

Gaussian Markov random fields : theory and applications

Gaussian Markov Random Field (GMRF) models are most widely used in spatial statistics, a very active area of research in which few up-to-date reference works are available. Gaussian Markov Random Field: Theory and Applications is the first book on the subject that provides a unified framework of GMRFs with particular emphasis on the computational aspects. The book includes extensive case studies and online a c-library for fast and exact simulation. With chapters contributed by leading researchers in the field, this volume is essential reading for statisticians working in spatial theory and its applications, as well as quantitative researchers in a wide range of science fields where spatial data analysis is important.

Bayesian auxiliary variable models for binary and multinomial regression

In this paper we discuss auxiliary variable approaches to Bayesian binary and multinomial regression. These approaches are ideally suited to automated Markov chain Monte Carlo simulation. In the first part we describe a simple technique using joint updating that improves the performance of the conventional probit regression algorithm. In the second part we discuss auxiliary variable methods for inference in Bayesian logistic regression, including covariate set uncertainty. Finally, we show how the logistic method is easily extended to multinomial regression models. All of the algorithms are fully automatic with no user set parameters and no necessary Metropolis-Hastings accept/reject steps.

Predictive Model Assessment for Count Data

We discuss tools for the evaluation of probabilistic forecasts and the critique of statistical models for count data. Our proposals include a nonrandomized version of the probability integral transform, marginal calibration diagrams, and proper scoring rules, such as the predictive deviance. In case studies, we critique count regression models for patent data, and assess the predictive performance of Bayesian age-period-cohort models for larynx cancer counts in Germany. The toolbox applies in Bayesian or classical and parametric or nonparametric settings and to any type of ordered discrete outcomes.

Redefine Statistical Significance

We propose to change the default P-value threshold for statistical significance from 0.05 to 0.005 for claims of new discoveries.

Towards joint disease mapping

This article discusses and extends statistical models to jointly analyse the spatial variation of rates of several diseases with common risk factors. We start with a review of methods for separate analyses of diseases, then move to ecological regression approaches, where the rates from one of the diseases enter as surrogate covariates for exposure. Finally, we propose a general framework for jointly modelling the variation of two or more diseases, some of which share latent spatial fields, but with possibly different risk gradients. In our application, we consider mortality data on oral, oesophagus, larynx and lung cancers for males in Germany, which all share smoking as a common risk factor. Furthermore, the first three cancers are also known to be related to excessive alcohol consumption. An empirical comparison of the different models based on a formal model criterion as well as on the posterior precision of the relative risk estimates strongly suggests that the joint modelling approach is a useful and valuable extension over individual analyses.

A statistical framework for the analysis of multivariate infectious disease surveillance counts

A framework for the statistical analysis of counts from infectious disease surveillance databases is proposed. In its simplest form, the model can be seen as a Poisson branching process model with immigration. Extensions to include seasonal effects, time trends and overdispersion are outlined. The model is shown to provide an adequate fit and reliable one-step-ahead prediction intervals for a typical infectious disease time series. In addition, a multivariate formulation is proposed, which is well suited to capture space-time dependence caused by the spatial spread of a disease over time. An analysis of two multivariate time series is described. All analyses have been done using general optimization routines, where ML estimates and corresponding standard errors are readily available.

Posterior and Cross-validatory Predictive Checks: A Comparison of MCMC and INLA

Model criticism and comparison of Bayesian hierarchical models is often based on posterior or leave-one-out cross-validatory predictive checks. Cross-validatory checks are usually preferred because posterior predictive checks are difficult to assess and tend to be too conservative. However, techniques for statistical inference in such models often try to avoid full (manual) leave-one-out cross-validation, since it is very time-consuming. In this paper we will compare two approaches for estimating Bayesian hierarchical models: Markov chain Monte Carlo (MCMC) and integrated nested Laplace approximations (INLA). We review how both approaches allow for the computation of leave-one-out cross-validatory checks without re-running the model for each observation in turn. We then empirically compare the two approaches in an extensive case study analysing the spatial distribution of bovine viral diarrhoe (BVD) among cows in Switzerland.

Sensitivity analysis in Bayesian generalized linear mixed models for binary data

Generalized linear mixed models (GLMMs) enjoy increasing popularity because of their ability to model correlated observations. Integrated nested Laplace approximations (INLAs) provide a fast implementation of the Bayesian approach to GLMMs. However, sensitivity to prior assumptions on the random effects precision parameters is a potential problem. To quantify the sensitivity to prior assumptions, we develop a general sensitivity measure based on the Hellinger distance to assess sensitivity of the posterior distributions with respect to changes in the prior distributions for the precision parameters. In addition, for model selection we suggest several cross-validatory techniques for Bayesian GLMMs with a dichotomous outcome. Although the proposed methodology holds in greater generality, we make use of the developed methods in the particular context of the well-known salamander mating data. We arrive at various new findings with respect to the best fitting model and the sensitivity of the estimates of the model components.

Improved auxiliary mixture sampling for hierarchical models of non-Gaussian data

The article considers Bayesian analysis of hierarchical models for count, binomial and multinomial data using efficient MCMC sampling procedures. To this end, an improved method of auxiliary mixture sampling is proposed. In contrast to previously proposed samplers the method uses a bounded number of latent variables per observation, independent of the intensity of the underlying Poisson process in the case of count data, or of the number of experiments in the case of binomial and multinomial data. The bounded number of latent variables results in a more general error distribution, which is a negative log-Gamma distribution with arbitrary integer shape parameter. The required approximations of these distributions by Gaussian mixtures have been computed. Overall, the improvement leads to a substantial increase in efficiency of auxiliary mixture sampling for highly structured models. The method is illustrated for finite mixtures of generalized linear models and an epidemiological case study.

Bayesian Variable Selection for Detecting Adaptive Genomic Differences Among Populations

We extend an Fst-based Bayesian hierarchical model, implemented via Markov chain Monte Carlo, for the detection of loci that might be subject to positive selection. This model divides the Fst-influencing factors into locus-specific effects, population-specific effects, and effects that are specific for the locus in combination with the population. We introduce a Bayesian auxiliary variable for each locus effect to automatically select nonneutral locus effects. As a by-product, the efficiency of the original approach is improved by using a reparameterization of the model. The statistical power of the extended algorithm is assessed with simulated data sets from a Wright–Fisher model with migration. We find that the inclusion of model selection suggests a clear improvement in discrimination as measured by the area under the receiver operating characteristic (ROC) curve. Additionally, we illustrate and discuss the quality of the newly developed method on the basis of an allozyme data set of the fruit fly Drosophila melanogaster and a sequence data set of the wild tomato Solanum chilense. For data sets with small sample sizes, high mutation rates, and/or long sequences, however, methods based on nucleotide statistics should be preferred.