Bayesian Mendelian randomization with an interval causal null hypothesis: ternary decision rules and loss function calibration

We enhance the Bayesian Mendelian Randomization (MR) framework of Berzuini et al. (Biostatistics 21(1):86-101, 2018) by allowing for interval null causal hypotheses, where values of the causal effect parameter that fall within a user-specified interval of "practical equivalence" (ROPE) (Kruschke, Adv Methods Pract Psychol Sci 1(2):270-80, 2018) are regarded as equivalent to "no effect". We motivate this move in the context of MR analysis. In this approach, the decision over the hypothesis test is taken on the basis of the Bayesian posterior odds for the causal effect parameter falling within the ROPE. We allow the causal effect parameter to have a mixture prior, with components corresponding to the null and the alternative hypothesis. Inference is performed via Markov chain Monte Carlo (MCMC) methods. We speed up the calculations by fitting to the data a simpler model than the intended, "true", one. We recover a set of samples from the "true" posterior distribution by weighted importance resampling of the MCMC-generated samples. From the final samples we obtain a simulation consistent estimate of the desired posterior odds, and ultimately of the Bayes factor for the interval-valued null hypothesis, [Formula: see text], vs [Formula: see text]. In those situations where the posterior odds is neither large nor small enough, we allow for an uncertain outcome of the test decision, thereby moving to a ternary decision logic. Finally, we present an approach to calibration of the proposed method via loss function. We illustrate the method with the aid of a study of the causal effect of obesity on risk of juvenile myocardial infarction based on a unique prospective dataset.