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.

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.

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 is limited by its computational expensiveness.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 tackle the problem of computation. In particular, we divided data into subsets and combine estimated subset causal effects obtained from the subsets". The performance of our method was evaluated by a number of simulations, in which part of exposure data was missing.Results: Random effect Bayesian MR outperformed conventional inverse-variance weighted estimation, whether the true causal effects are 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. Our simulation results indicate that data partitioning is a good way of improving computational efficiency, for little cost of decrease in unbiasedness of the estimates, as long as the sample size of subsets is reasonably large. Conclusions:We have further advanced Bayesian MR by including random effects to explicitly account for study heterogeneity. We also adopted a subset posterior aggregation method to address the issue of computational expensiveness of MCMC, which is important especially when dealing with large studies. Our proposed work is likely to pave the way for more general model settings, as Bayesian approach itself renders great flexibility in model constructions.

Our approach to Mendelian Randomization (MR) analysis is designed to increase reproducibility of causal effect "discoveries" by: (i) using a Bayesian approach to inference; (ii) replacing the point null hypothesis with a region of practical equivalence consisting of values of negligible magnitude for the effect of interest (Kruschke [1]), while exploiting the ability of Bayesian analysis to quantify the evidence of the effect falling inside/outside the region; (iii) rejecting the usual binary decision logic in favour of a ternary logic where the hypothesis test may result in either an acceptance or a rejection of the null, while also accommodating an "uncertain" outcome. We present an approach to calibration of the proposed method via loss function, which we use to compare our approach with a frequentist one. We illustrate the method with the aid of a study of the causal effect of obesity on risk of juvenile myocardial infarction.

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 dominantly 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 method that is applicable to all the three sample settings. In essence, we converted a two-or overlapping-sample problem to a one-sample problem where data of some or all of the individuals were incomplete. Assume that all individuals were drawn from the same population and unmeasured data were missing at random. Then the unobserved data were treated au pair with the model parameters as unknown quantities, and thus, could be 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.Results: Higher sample overlapping rate and stronger instruments led to estimates with higher precision and power. Pleiotropy had a notably negative impact on the estimates. Nevertheless, overall the coverages were high and our model performed well in all the sample settings.Conclusions: Our model offers the flexibility of being applicable to any of the sample settings, which 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.