integration, and their implementation in Bayesian inference 31 2.2 Markov chain Monte Carlo methods 2.2.1 The algorithm 36 2.2.2 Terminology and implementation details 2.3 Popular MCMC algorithms 2.3.1 The Metropolis-Hastings algorithm 2.3.2 Componentwise Metropolis-Hastings 2.3.3 The Gibbs sampler 2.3.4 Metropolis within Gibbs 2.3.5 The slice Gibbs sampler 2.3.6 A simple example using the slice sampler 2.4 Summary and closing remarks Problems 81 3 WinBUGS Software: Introduction, Setup, and Basic Analysis 83 275 8.1 Models with nonstandard distributions 8.1.1 Specification of arbitrary likelihood using the zeros-ones trick 8.12 The inverse Gaussian model 8.2 Models for positive continuous response variables 8.2.1 The gamma model 8.2.2 Other models 8.2.3 An example 8.3 Additional models for count data 8.3.1 The negative binomial model 8.3.2 The generalized Poisson model 8.3.3 Zero inflated models 8.3.4 The bivariate Poisson model 8.3.5 The Poisson difference model 8.4 Further GLM-based models and extensions 8.4.1 Survival analysis models 8.4.2 Multinomial models 8.4.3 Additional models and further reading Problems 9 Bayesian Hierarchical Models 305 9.1

Models based on the bivariate Poisson distribution are used for modelling sports data. Independent Poisson distributions are usually adopted to model the number of goals of two competing teams. We replace the independence assumption by considering a bivariate Poisson model and its extensions. The models proposed allow for correlation between the two scores, which is a plausible assumption in sports with two opposing teams competing against each other. The effect of introducing even slight correlation is discussed. Using just a bivariate Poisson distribution can improve model fit and prediction of the number of draws in football games. The model is extended by considering an inflation factor for diagonal terms in the bivariate joint distribution.This inflation improves in precision the estimation of draws and, at the same time, allows for overdispersed, relative to the simple Poisson distribution, marginal distributions. The properties of the models proposed as well as interpretation and estimation procedures are provided. An illustration of the models is presented by using data sets from football and water-polo.

The aim of this study within the Athens Study of Psychosis Proneness and Incidence of Schizophrenia (ASPIS) was the examination of the latent structure of schizotypal dimensions among a large population of young male conscripts in the Greek Air Force during their first week of military training. Confirmatory factor analysis (CFA) was conducted on 1,355 reliable responders to the self-rated Schizotypal Personality Questionnaire (SPQ), which covers all nine aspects of DSM-III-R schizotypal personality disorder (SPD). A four-factor model (cognitive/perceptual, paranoid, negative, and disorganization schizotypal dimensions) provided a better fit to the data than did other competing models (one-, two-, three-, four, and five-factor models). This result is in agreement with recent findings supporting the notion of a multidimensional construct of the schizotypy and related schizophrenia phenotype.

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Modelling football match outcomes is becoming increasingly popular nowadays for both team managers and betting funs. Most of the existing literature deals with modelling the number of goals scored by each team. In this paper, we work in a different direction. Instead of modelling the number of goals directly, we focus on the difference of the number of goals, i.e. the margin of victory. Modelling the differences instead of the scores themselves has some major advantages. Firstly, we eliminate correlation imposed by the fact that the two opponent teams compete each other, and secondly, we do not assume that the scored goals by each team are marginally Poisson distributed. Application of the Bayesian methodology for the Skellam's distribution using covariates is discussed. Illustrations using real data from the English Premiership for the season 2006-2007 are provided. The advantages of the proposed approach are also discussed.