Tim B. Swartz

Bayesian Analysis of Dyadic Data

SYNOPTIC ABSTRACT This paper considers the Bayesian analysis of dyadic data with particular emphasis on applications in social psychology. Various existing models are extended and unified under a class of models where a single value is elicited to complete the prior specification. Certain situations which have sometimes been problematic (e.g. incomplete data, non-standard covariates, missing data, unbalanced data) are easily handled under the proposed class of Bayesian models. Inference is straightforward 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 in which they can now be analyzed.

Random Variable Generation Using Concavity Properties of Transformed Densities

Abstract Algorithms are developed for constructing random variable generators for families of densities. The generators depend on the concavity structure of a transformation of the density. The resulting algorithms are rejection algorithms and the methods of this article are concerned with constructing good rejection algorithms for general densities.

Statistical analyses for round robin interaction data

This paper considers the analysis of round robin interaction data whereby individuals from a group of subjects interact with one another, producing a pair of outcomes, one for each individual. The authors provide an overview of the various analyses applied to these types of data and extend the work in several directions. In particular, they provide a fully Bayesian analysis for such data and use a real data example for illustration purposes.

A simulation-intensive approach for checking hierarchical models

Recent computational advances have made it feasible to fit hierarchical models in a wide range of serious applications. In the process, the question of model adequacy arises. While model checking usually addresses the entire model specification, model failures can occur at each hierarchical stage. Such failures include outliers, mean structures errors, dispersion misspecification, and inappropriate exchangeabilities. We propose an approach which is entirely simulation based. Given a model specification and a dataset, we need only be able to simulate draws from the resultant posterior. By replicating a posterior of interest using data obtained under the model we can “see” the extent of variability in such a posterior. Then, we can compare the posterior obtained under the observed data with this medley of posterior replicates to ascertain whether the former is in agreement with them and accordingly, whether it is plausible that the observed data came from the proposed model. Many such comparisons can be run, each focusing on a different potential model failure. Focusing on generalized linear mixed models, we explore the questions of when hierarchical model stages are separable and checkable and illustrate the approach with both real and simulated data.

Optimal batting orders in one-day cricket

This paper concerns the search for optimal or nearly optimal batting orders in one-day cricket. A search is conducted over the space of permutations of batting orders where simulated annealing is used to explore the space. A non-standard aspect of the optimization is that the objective function (which is the mean number of runs per innings) is unavailable and is approximated via simulation. The simulation component generates runs ball by ball during an innings taking into account the state of the match and estimated characteristics of individual batsmen. The methods developed in the paper are applied to the national team of India based on their performance in one-day intemational cricket matches.

Bayesian identifiability and misclassification in multinomial data

The authors consider the Bayesian analysis of multinomial data in the presence of misclassi- fication. Misclassification of the multinomial cell entries leads to problems of identifiability which are categorized into two types. The first type, referred to as the permutation-type nonidentifiab ilities, may be handled with constraints that are suggested by the structure of the problem. Problems of identifiab ility of the second type are addressed with informative prior information via Dirichlet distributions. Computations are carried out using a Gibbs sampling algorithm.

Bayesian Analysis of Stochastically Ordered Distributions of Categorical Variables

Abstract This article considers a finite set of discrete distributions all having the same finite support. The problem of interest is to assess the strength of evidence produced by sampled data for a hypothesis of a specified stochastic ordering among the underlying distributions and to estimate these distributions subject to the ordering. We present a Bayesian approach that is an alternative to using the posterior probability of the hypothesis and the Bayes factor in favor of the hypothesis. We develop computational methods for the implementation of Bayesian analyses. We analyze examples to illustrate inferential and computational developments. The methodology used for testing a hypothesis is seen to apply to a wide class of problems in Bayesian inference and has some distinct advantages.

Applications: Estimation of the Magnitude of Victory in One‐day Cricket RMIT University, Mayo Clinic Rochester and Simon Fraser University

The Duckworth–Lewis method is steadily becoming the standard approach for resetting targets in interrupted one-day cricket matches. In this paper we show that a modification of the Duckworth–Lewis resource table can be used to quantify the magnitude of a victory in one-day matches. This simple and direct application is particularly useful in breaking ties in tournament standings and in quantifying team strength.

Duckworth–Lewis and Twenty20 Cricket

Originally designed for 1-day cricket, this paper considers the use of the Duckworth–Lewis method as an approach to resetting targets in interrupted Twenty20 cricket matches. The Duckworth–Lewis table is reviewed and an alternative resource table is presented. The alternative table is constructed using observed scoring rates from international Twenty20 matches. A desideratum of a resource table is monotonicity in both the rows and columns corresponding to wickets and overs respectively. Consequently, a Gibbs sampling scheme related to isotonic regression is applied to the observed scoring rates to provide a non-parametric resource table. Taking into account the more aggressive batting style of Twenty20 compared to 1-day cricket, the resultant resource table is seen to possess sensible features. A discussion is provided concerning the use of the Duckworth–Lewis method applied to Twenty20.

Distribution theory and inference for polynomial-normal densities

This paper considers a class of densities formed by taking the product of nonnegative polynomials and normal densities. These densities provide a rich class of distributions that can be used in modelling when faced with non-normal characteristics such as skewness and multimodality. In this paper we address inferential and computational issues arising in the practical implementation of this parametric family in the context of the linear model. Exact results are recorded for the conditional analysis of location-scale models and an importance sampling algorithm is developed for the implementation of a conditional analysis for the general linear model when using polynomial-normal distributions for the error.