Bayesian Modelling of Inseparable Space-Time Variation in Disease Risk

This paper proposes a unified framework for a Bayesian analysis of incidence or mortality data in space and time. We introduce four different types of prior distributions for space $\times$ time interaction in extension of a model with only main effects. Each type implies a certain degree of prior dependence for the interaction parameters, and corresponds to the product of one of the two spatial with one of the two temporal main effects. The methodology is illustrated by an analysis of Ohio lung cancer data 1968-88 via Markov chain Monte Carlo simulation. We compare the fit and the complexity of several models with different types of interaction by means of quantities related to the posterior deviance. Our results confirm an epidemiological hypothesis about the temporal development of the association between urbanization and risk factors for cancer

Prognosis of Lung Cancer Mortality in West Germany: A Case Study in Bayesian Prediction. (REVISED, January 2000)

We apply a generalized Bayesian age-period-cohort (APC) model to a dataset on lung cancer mortality in West Germany, 1952-1996. Our goal is to predict future deaths rates until the year 2010, separately for males and females. Since age and period is not measured on the same grid, we propose a generalized APC-model where consecutive cohort parameters represent strongly overlapping birth cohorts. This approach results in a rather large number of parameters, where standard algorithms for statistical inference by Markov chain Monte Carlo (MCMC) methods turn out to be computationally intensive. We propose a more efficient implementation based on ideas of block sampling from the time series literature. We entertain two different formulations, penalizing either first or second differences of age, period and cohort parameters. To assess the predictive quality of both formulations, we first forecast the rates for the period 1987-1996 based on data until 1986. A comparison with the actual observed rates is made based on quantities related to the predictive deviance. Predictions of lung cancer mortality until 2010 both for males and females are finally reported

Dynamic Rating of Sports Teams. (REVISED 1999)

We consider the problem of dynamically rating sports teams based on the categorical outcome of paired comparisons such as win, draw and loss in football. Our modelling framework is the cumulative link model for ordered response, where latent parameters represent the strength of each team. A dynamic extension of this model is proposed with close connections to nonparametric smoothing methods. As a consequence, recent results have more influence for estimating current abilities than results in the past. We highlight the importance of using a specific constrained random walk prior for time--changing abilities which guarantees an equal treatment of all teams. Estimation is done within an extended Kalman filter type approach. An additional hyperparameter which determines the temporal dynamic of the latent team abilities is chosen based on optimal one-step-ahead predictive power. Alternative estimation methods are also considered. We apply our method to the results from the German football league ``Bundesliga'' 1996/97 and to the results from the American National Basketball Association (NBA) 1996/97

Conditional Prior Proposals in Dynamic Models

Dynamic models extend state space models to non-normal observations. This paper suggests a specific hybrid Metropolis-Hastings algorithm as a simple, yet flexible and efficient tool for Bayesian inference via Markov chain Monte Carlo in dynamic models. Hastings proposals from the (conditional) prior distribution of the unknown, time-varying parameters are used to update the corresponding full conditional distributions. Several blocking strategies are discussed to ensure good mixing and convergence properties of the simulated Markov chain. It is also shown that the proposed method is easily extended to robust transition models using mixtures of normals. The applicability is illustrated with an analysis of a binomial and a binary time series, known in the literature

Dynamic discrete-time duration models. (REVISED)

Discrete-time grouped duration data, with one or multiple types of terminating events, are often observed in social sciences or economics. In this paper we suggest and discuss dynamic models for flexible Bayesian nonparametric analysis of such data. These models allow simultaneous incorporation and estimation of baseline hazards and time-varying covariate effects, without imposing particular parametric forms. Methods for exploring the possibility of time-varying effects, as for example the impact of nationality or unemployment insurance benefits on the probability of re-employment, have recently gained increasing interest. Our modelling and estimation approach is fully Bayesian and makes use of Markov Chain Monte Carlo (MCMC) simulation techniques. A detailed analysis of unemployment duration data, with full-time job, part-time job and other causes as terminating events, illustrates our methods and shows how they can be used to obtain refined results and interpretations

Dynamic and semiparametric models

This paper surveys dynamic or state space models and their relationship to non- and semiparametric models that are based on the roughness penalty approach. We focus on recent advances in dynamic modelling of non-Gaussian, in particular discrete-valued, time series and longitudinal data, make the close correspondence to semiparametric smoothing methods evident, and show how ideas from dynamic models can be adopted for Bayesian semiparametric inference in generalized additive and varying coefficient models. Basic tools for corresponding inference techniques are penalized likelihood estimation, Kalman filtering and smoothing and Markov chain Monte Carlo (MCMC) simulation. Similarities, relative merits, advantages and disadvantages of these methods are illustrated through several applications

Bayesian Detection of Clusters and Discontinuities in Disease Maps. (REVISED, February 1999)

An interesting epidemiological problem is the analysis of geographical variation in rates of disease incidence or mortality. One goal of such an analysis is to detect clusters of elevated (or lowered) risk in order to identify unknown risk factors regarding the disease. We propose a nonparametric Bayesian approach for the detection of such clusters based on Green's (1995) reversible jump MCMC methodology. The prior model assumes that geographical regions can be combined in clusters with constant relative risk within a cluster. The number of clusters, the location of the clusters and the risk within each cluster is unknown. This specification can be seen as a change-point problem of variable dimension in irregular, discrete space. We illustrate our method through an analysis of oral cavity cancer mortality rates in Germany and compare the results with those obtained by the commonly used Bayesian disease mapping method of Besag, York and Mollie (1991)

On block updating in Markov random field models for disease mapping. (REVISED, May 2001)

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 bad 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 models: 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. We apply the proposed algorithms to two datasets known from the literature. The results indicate that the largest benefits are obtained if parameters and the corresponding hyperparameter are updated jointly in one large block. In certain situations, even updating of all or nearly all parameters in one block may be necessary. Implementation of such block algorithms is surprisingly easy using methods for fast sampling of Gaussian Markov random fields (Rue, 2000). By comparison, estimates of the relative risk and related quantities, such as the posterior probability of an exceedence relative risk, 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

Efficient simulation of Bayesian logistic regression models

In this paper we highlight a data augmentation approach to inference in the Bayesian logistic regression model. We demonstrate that the resulting conditional likelihood of the regression coefficients is multivariate normal, equivalent to a standard Bayesian linear regression, which allows for efficient simulation using a block Gibbs sampler. We illustrate that the method is particularly suited to problems in covariate set uncertainty and random effects models

Markov Chain Monte Carlo Simulation in Dynamic Generalized Linear Mixed Models

Dynamic generalized linear mixed models are proposed as a regression tool for nonnormal longitudinal data. This framework is an interesting combination of dynamic models, by other name state space models, and mixed models, also known as random effect models. The main feature is, that both time- and unit-specific parameters are allowed, which is especially attractive if a considerable number of units is observed over a longer period. Statistical inference is done by means of Markov chain Monte Carlo techniques in a full Bayesian setting. The algorithm is based on iterative updating using full conditionals. Due to the hierarchical structure of the model and the extensive use of Metropolis-Hastings steps for updating this algorithm mainly evaluates (log-)likelihoods in multivariate normal distributed proposals. It is derivative-free and covers a wide range of different models, including dynamic and mixed models, the latter with slight modifications. The methodology is illustrated through an analysis of artificial binary data and multicategorical business test data