A Two-stage Linear Mixed Model (TS-LMM) for Summary-data-based Multivariable Mendelian Randomization.

Multivariable Mendelian randomization (MVMR) methods provide a strategy for applying genome-wide summary statistics to assess simultaneous causal effects of multiple risk factors on a disease outcome. In contrast to univariate MR methods that assumes no horizonal pleiotropy (genetic variants only associate with one risk factor), MVMR allows for genetic variants associate with multiple risk factors and models pleiotropy by including summary statistics with risk factors as multiple variables into the regression model. Here, we propose a two-stage linear mixed model (TS-LMM) for MVMR that accounts for variance of summary statistics not only in outcome, but also in all of the risk factors. In stage I, we apply linear mixed model to treat variance in summary statistics of disease as fixed-/random-effects, while accounting for covariance between genetic variants due to linkage disequilibrium (LD). Particularly, we use an iteratively re-weighted least squares algorithm to obtain estimates for the random-effects. In stage II, we account for variance in summary statistics of multiple risk factors simultaneously by applying measurement error correction methods that take into consideration LD between genetic variants and correlation between summary statistics of risk factors. We compared our MVMR approach to other approaches in a simulation study. When most of the instrumental variables (IVs) were strong, our model generated the highest coverage of true causal associations, the highest power of detecting significant causal associations, and the lowest false positive rate of identifying null causal effect for a range of scenarios that varied correlation (weak, strong) between summary statistics of risk factors and LD among genetic variants (weak LD [γ 2 ≤0.1], moderate LD [0.1< γ 2 ≤0.5]). When the proportion of strong IVs was reduced, our model showed performances comparable to MVMR-Egger and MVMR-IVW. The more accurate inference of our model in the presence of correlation among risk factors supports potential wide application to -omics data that are commonly multi dimensional and correlated, as shown in application to determinants of longevity, where our method nominated a specific significant lipoprotein subfraction for causal association from a panel of 10 lipoprotein cholesterol measures. The robustness of our model to correlation structure suggests that in practice we can allow moderate LD in selection of IVs, thereby potentially leveraging genome-wide summary data in a more effective manner. Our model is implemented in 'TS_LMM' macro in R.