Bayesian Mendelian Randomization with Study Heterogeneity and Data Partitioning for Large Studies

Background: Mendelian randomization (MR) is a useful approach to causal inference from observational studies when randomised controlled trials are not feasible. However, study heterogeneity of two association studies required in MR is often overlooked. When dealing with large studies, recently developed Bayesian MR can be computationally challenging, and sometimes even prohibitive.

Methods: We addressed study heterogeneity by proposing a random effect Bayesian MR model with multiple exposures and outcomes. For large studies, we adopted a subset posterior aggregation method to overcome the problem of computational expensiveness of Markov chain Monte Carlo. In particular, we divided data into subsets and combined estimated causal effects obtained from the subsets. The performance of our method was evaluated by a number of simulations, in which exposure data was partly missing.

Results: Random effect Bayesian MR outperformed conventional inverse-variance weighted estimation, whether the true causal effects were zero or non-zero. Data partitioning of large studies had little impact on variations of the estimated causal effects, whereas it notably affected unbiasedness of the estimates with weak instruments and high missing rate of data. For the cases being simulated in our study, the results have indicated that the "divide (data) and combine (estimated subset causal effects)" can help improve computational efficiency, for an acceptable cost in terms of bias in the causal effect estimates, as long as the size of the subsets is reasonably large.

Conclusions: We further elaborated our Bayesian MR method to explicitly account for study heterogeneity. We also adopted a subset posterior aggregation method to ease computational burden, which is important especially when dealing with large studies. Despite the simplicity of the model we have used in the simulations, we hope the present work would effectively point to MR studies that allow modelling flexibility, especially in relation to the integration of heterogeneous studies and computational practicality.