Population‐based reversible jump Markov chain Monte Carlo methods for Bayesian variable selection and evaluation under cost limit restrictions

Summary. The measurement and improvement of the quality of health care are important areas of current research and development. A judgement of appropriateness of medical outcomes in hospital quality‐of‐care studies must depend on an assessment of patient sickness at admission to hospital. Indicators of patient sickness often must be abstracted from medical records, and some variables are more expensive to measure than others. Quality‐of‐care studies are frequently undertaken in an environment of cost restriction; thus any scale measuring patient sickness must simultaneously respect two optimality criteria: high predictive accuracy and low cost. Here we examine a variable selection strategy for construction of a scale of sickness in which predictive accuracy is optimized subject to a bound on cost. Conventional model search algorithms (such as those based on standard reversible jump Markov chain Monte Carlo (RJMCMC) sampling) in our setting will often fail, because of the existence of multiple modes of the criterion function with movement paths that are forbidden because of the cost restriction. We develop a population‐based trans‐dimensional RJMCMC (population RJMCMC) algorithm, in which ideas from the population‐based MCMC and simulated tempering algorithms are combined. Comparing our method with standard RJMCMC sampling, we find that the population‐based RJMCMC algorithm moves successfully and more efficiently between distant neighbourhoods of ‘good’ models, achieves convergence faster and has smaller Monte Carlo standard errors for a given amount of central processor unit time. In a case‐study of n=2532 pneumonia patients on whom p=83 sickness indicators were measured, with marginal costs varying from smallest to largest across the predictor variables by a factor of 20, the final model chosen by population RJMCMC sampling, on the basis of both highest posterior probability and specifying the median probability model, was clinically sensible for pneumonia patients and achieved good predictive ability while capping data collection costs.