Overlapping-sample Mendelian randomisation with multiple exposures: a Bayesian approach

BACKGROUND: Mendelian randomization (MR) has been widely applied to causal inference in medical research. It uses genetic variants as instrumental variables (IVs) to investigate putative causal relationship between an exposure and an outcome. Traditional MR methods have mainly focussed on a two-sample setting in which IV-exposure association study and IV-outcome association study are independent. However, it is not uncommon that participants from the two studies fully overlap (one-sample) or partly overlap (overlapping-sample). METHODS: We proposed a Bayesian method that is applicable to all the three sample settings. In essence, we converted a two- or overlapping- sample MR to a one-sample MR where data were partly unmeasured. Assume that all study individuals were drawn from the same population and unmeasured data were missing at random. Then the missing data were treated au pair with the model parameters as unknown quantities, and thus, were imputed iteratively conditioning on the observed data and estimated parameters using Markov chain Monte Carlo. We generalised our model to allow for pleiotropy and multiple exposures and assessed its performance by a number of simulations using four metrics: mean, standard deviation, coverage and power. We also compared our method with classic MR methods. RESULTS: In our proposed method, higher sample overlapping rate and instrument strength led to more precise estimated causal effects with higher power. Pleiotropy had a notably negative impact on the estimates. Nevertheless, the coverages were high and our model performed well in all the sample settings overall. In comparison with classic MR, our method provided estimates with higher precision. When the true causal effects were non-zero, power of their estimates was consistently higher from our method. The performance of our method was similar to classic MR in terms of coverage. CONCLUSIONS: Our model offers the flexibility of being applicable to any of the sample settings. It is an important addition to the MR literature which has restricted to one- or two- sample scenarios. Given the nature of Bayesian inference, it can be easily extended to more complex MR analysis in medical research.