Bayesian Mendelian Randomization

Our Bayesian approach to Mendelian Randomisation uses multiple instruments to assess the putative causal effect of an exposure on an outcome. The approach is robust to violations of the (untestable) Exclusion Restriction condition, and hence it does not require instruments to be independent of the outcome conditional on the exposure and on the confounders of the exposure-outcome relationship. The Bayesian approach offers a rigorous handling of the uncertainty (e.g. about the estimated instrument-exposure associations), freedom from asymptotic approximations of the null distribution and the possibility to elaborate the model in any direction of scientific relevance. We illustrate the last feature with the aid of a study of the metabolic mediators of the disease-inducing effects of obesity, where we elaborate the model to investigate whether the causal effect of interest interacts with a covariate. The proposed model contains a vector of unidentifiable parameters, $\beta$, whose $j$th element represents the pleiotropic (i.e., not mediated by the exposure) component of the association of instrument $j$ with the outcome. We deal with the incomplete identifiability by assuming that the pleiotropic effect of some instruments is null, or nearly so, formally by imposing on $\beta$ Carvalho's horseshoe shrinkage prior, in such a way that different components of $\beta$ are subjected to different degrees of shrinking, adaptively and in accord with the compatibility of each individual instrument with the hypothesis of no pleiotropy. This prior requires a minimal input from the user. We present the results of a simulation study into the performance of the proposed method under different types of pleiotropy and sample sizes. Comparisons with the performance of the weighted median estimator are made. Choice of the prior and inference via Markov chain Monte Carlo are discussed.