On the choice of parameterisation and priors for the Bayesian analyses of Mendelian randomisation studies

Mendelian randomisation is a form of instrumental variable analysis that estimates the causal effect of an intermediate phenotype or exposure on an outcome or disease in the presence of unobserved confounding, using a genetic variant as the instrument. A Bayesian approach allows current knowledge to be incorporated into the analysis in the form of informative prior distributions, and the unobserved confounder can be modelled explicitly. We consider Bayesian methods for Mendelian randomisation in the case where all relationships are linear and there are no interactions. A 'full' model in which the unobserved confounder is included explicitly is not completely identifiable, although the causal parameter can be estimated. We compare inferences from this general but non-identified model with a reduced parameter model that is identifiable. We show that, theoretically, additional information about the causal parameter can be obtained by using the non-identifiable full model, rather than the identifiable reduced model, but that this is advantageous only when realistically informative priors are used and when the instrument is weak or the sample size is small. Furthermore, we consider the impact of using 'vague' versus 'informative' priors.