Fig 1.
A causal diagram illustrating the three assumptions on a valid IV gi.
Dashed lines (which are marked with a red ‘cross’) correspond to violations of assumptions 2 and 3.
Fig 2.
Simulation results with directional pleiotropy and InSIDE satisfied.
A: Empirical type-I error; B: Power with sample size n = 50 000. Each row corresponds to m = 10, 30, 100 SNPs and each column corresponds to 0, 30%, 50%, 70%, 100% invalid IVs.
Fig 3.
Simulation results with directional pleiotropy and InSIDE satisfied.
Empirical distributions of the estimates of the causal effect θ by the methods with n = 50000 and 70% invalid IVs. A: θ = 0. B: θ = 0.2.
Fig 4.
Estimates of the proportion of invalid IVs by mixIE-MA with n = 50 000 under directional pleiotropy and InSIDE satisfied.
The upper row corresponds to θ = 0 and the lower one to θ = 0.2; each column corresponds to m = 10, 30, 100 respectively.
Fig 5.
A simulated data example with n = 50 000, m = 30, θ = 0, p_invalid = 1 under directional pleiotropy and InSIDE satisfied.
A: Causal estimate and identified invalid IVs by mixIE-MA. B: Causal estimate and identified invalid IVs by mixIE-MA-DP. C: Posterior probability of each IV being invalid. D: Histogram of from 200 perturbations.
Fig 6.
Simulation results with directional pleiotropy and InSIDE violated.
A: Empirical type-I error; B: Power with sample size n = 50 000 and m = 30. Each column corresponds to b = 0.1, 0.4, 0.7 and each row corresponds to 30%, 50%, 70% invalid IVs.
Fig 7.
Simulation results with directional pleiotropy and InSIDE violated.
Empirical distributions of the estimates of the causal effect θ by the methods with n = 50 000, θ = 0.2 and 70% invalid IVs. A: m = 30. B: m = 100. Each column corresponds to b = 0.1, 0.4, 0.7.
Fig 8.
Simulation results for GOF testing with directional pleiotropy and InSIDE violated (while the plurality assumption for other three methods holding).
θ = 0.2 and n = 50 000. The y-axis gives the rejection rate that the results from two methods were consistent, while the x-axis gives the increasing degree of InSIDE being violated. A: m = 30. B: m = 100. The two columns correspond to 30% and 70% invalid IVs respectively.
Fig 9.
Simulation results with many invalid IVs having weak pleiotropic effects.
Empirical type-I error (for θ = 0) and power (for θ ≠ 0) curves with sample size n = 20000 and hy = 0.1.
Table 1.
Genome wide association studies for 4 common diseases and 12 risk factors.
Fig 10.
Results of various methods to detect causal relationships among 48 risk factor-disease pairs.
Table 2.
Numbers of significant pairs among 48 risk factor-disease pairs at the significance cutoff of p-value < 0.001.
Fig 11.
Q-Q plots for 53 (likely) null trait-pairs in the secondary real data examples.
Left panel: mixIE-MA; right panel: mixIE-MA-DP.