No abstract

Cost data that arise in the evaluation of health care technologies usually exhibit highly skew, heavy-tailed and, possibly, multi-modal distributions. Distribution-free methods for analysing these data, such as the bootstrap, or those based on the asymptotic normality of sample means, may often lead to inefficient or misleading inferences. On the other hand, parametric models that fit the data (or a transformation of the data) equally well can produce very different answers. We consider a Bayesian approach, and model cost data with a distribution composed of a piecewise constant density up to an unknown endpoint, and a generalized Pareto distribution for the remaining tail.

Calculation of a suitable Bayes factor is required for Bayesian model comparison. The fractional Bayes factor is one of several alternative Bayes factors that have been introduced in recent years to address the problem of sensitivity of the usual Bayes factor when prior information is weak. Sensitivity of the fractional Bayes factor with respect to prior distributions is easy to assess when these are proper. On the other hand, when the priors are improper, most methods lead to trivial answers. Also, earlier work on fractional Bayes factors has assumed that sensitivity will be reduced if the training fraction, b, is increased, but this has only been justified by appeal to heuristic reasoning and simple examples. In this paper we derive a measure of the sensitivity of the fractional Bayes factor with respect to improper priors, and prove that it is a decreasing function of b in a class of problems

We consider the problem of assessing new and existing technologies for their cost-effectiveness in the case where data on both costs and effects are available from a clinical trial, and we address it by means of the cost-effectiveness acceptability curve. The main difficulty in these analyses is that cost data usually exhibit highly skew and heavy-tailed distributions so that it can be extremely difficult to produce realistic probabilistic models for the underlying population distribution, and in particular to model accurately the tail of the distribution, which is highly influential in estimating the population mean. Here, in order to integrate the uncertainty about the model into the analysis of cost data and into cost-effectiveness analyses, we consider an approach based on Bayesian model averaging: instead of choosing a single parametric model, we specify a set of plausible models for costs and estimate the mean cost with a weighted mean of its posterior expectations under each model, with weights given by the posterior model probabilities. The results are compared with those obtained with a semi-parametric approach that does not require any assumption about the distribution of costs.

The classical chi-square test of goodness of fit compares the hypothesis that data arise from some parametric family of distributions, against the nonparametric alternative that they arise from some other distribution. However, the chi-square test requires continuous data to be grouped into arbitrary categories, and, being based upon an approximation, it can only be used if there is sufficient data. In many practical situations these requirements are wasteful of information and overly restrictive respectively. We explore the use of the fractional Bayes factor to obtain a Bayesian alternative to the chi-square test when no specific prior information is available, including consideration of the extent to which it can handle small data sets and continuous data without arbitrary grouping.