Assessing the convergence of Markov Chain Monte Carlo methods: an example from evaluation of diagnostic tests in absence of a gold standard.

The accessibility of Markov Chain Monte Carlo (MCMC) methods for statistical inference have improved with the advent of general purpose software. This enables researchers with limited statistical skills to perform Bayesian analysis. Using MCMC sampling to do statistical inference requires convergence of the MCMC chain to its stationary distribution. There is no certain way to prove convergence; it is only possible to ascertain when convergence definitely has not been achieved. These methods are rather subjective and not implemented as automatic safeguards in general MCMC software. This paper considers a pragmatic approach towards assessing the convergence of MCMC methods illustrated by a Bayesian analysis of the Hui-Walter model for evaluating diagnostic tests in the absence of a gold standard. The Hui-Walter model has two optimal solutions, a property which causes problems with convergence when the solutions are sufficiently close in the parameter space. Using simulated data we demonstrate tools to assess the convergence and mixing of MCMC chains using examples with and without convergence. Suggestions to remedy the situation when the MCMC sampler fails to converge are given. The epidemiological implications of the two solutions of the Hui-Walter model are discussed.