Mendelian randomization (MR) is a powerful approach to examine the causal relationships between health risk factors and outcomes from observational studies. Due to the proliferation of genome-wide association studies (GWASs) and abundant fully accessible GWASs summary statistics, a variety of two-sample MR methods for summary data have been developed to either detect or account for horizontal pleiotropy, primarily based on the assumption that the effects of variants on exposure ({gamma}) and horizontal pleiotropy ({alpha}) are independent. This assumption is too strict and can be easily violated because of the correlated horizontal pleiotropy (CHP). To account for this CHP, we propose a Bayesian approach, MR-Corr2, that uses the orthogonal projection to reparameterize the bivariate normal distribution for {gamma} and {alpha}, and a spike-slab prior to mitigate the impact of CHP. We develop an efficient algorithm with paralleled Gibbs sampling. To demonstrate the advantages of MR-Corr2 over existing methods, we conducted comprehensive simulation studies to compare for both type-I error control and point estimates in various scenarios. By applying MR-Corr2 to study the relationships between pairs in two sets of complex traits, we did not identify the contradictory causal relationship between HDL-c and CAD. Moreover, the results provide a new perspective of the causal network among complex traits. The developed R package and code to reproduce all the results are available at https://github.com/QingCheng0218/MR.Corr2.
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