Bayesian credible intervals for population attributable risk from case–control, cohort and cross-sectional studies

Population attributable risk (PAR) and population attributable fraction (PAF) are used in epidemiology to predict the impact of removing a risk factor from the population. Until recently, no standard approach for calculating confidence intervals or the variance for PAR in particular was available in the literature. Previously we outlined a fully Bayesian approach to provide credible intervals for the PAR and PAF from a cross-sectional study, where the data was presented in the form of a 2×2 table. However, extensions to cater for other frequently used study designs were not provided. In this paper we provide methodology to calculate credible intervals for the PAR and PAF for case–control and cohort studies. Additionally, we extend the cross-sectional example to allow for the incorporation of uncertainty that arises when an imperfect diagnostic test is used. In all these situations the model becomes over-parameterised, or non-identifiable, which can result in standard ‘off-the-shelf’ Markov Chain Monte Carlo (MCMC) updaters taking a long time to converge or even failing altogether. We adapt an importance sampling methodology to overcome this problem, and propose some novel MCMC samplers that take into consideration the shape of the posterior ridge to aid in the convergence of the Markov chain.