Robust inference of bi-directional causal relationships in presence of correlated pleiotropy with GWAS summary data

To infer a causal relationship between two traits, several correlation-based causal direction (CD) methods have been proposed with the use of SNPs as instrumental variables (IVs) based on GWAS summary data for the two traits; however, none of the existing CD methods can deal with SNPs with correlated pleiotropy. Alternatively, reciprocal Mendelian randomization (MR) can be applied, which however may perform poorly in the presence of (unknown) invalid IVs, especially for bi-directional causal relationships. In this paper, first, we propose a CD method that performs better than existing CD methods regardless of the presence of correlated pleiotropy. Second, along with a simple but yet effective IV screening rule, we propose applying a closely related and state-of-the-art MR method in reciprocal MR, showing its almost identical performance to that of the new CD method when their model assumptions hold; however, if the modeling assumptions are violated, the new CD method is expected to better control type I errors. Notably bi-directional causal relationships impose some unique challenges beyond those for uni-directional ones, and thus requiring special treatments. For example, we point out for the first time several scenarios where a bi-directional relationship, but not a uni-directional one, can unexpectedly cause the violation of some weak modeling assumptions commonly required by many robust MR methods. We also offer some numerical support and a modeling justification for the application of our new methods (and more generally MR) to binary traits. Finally we applied the proposed methods to 12 risk factors and 4 common diseases, confirming mostly well-known uni-directional causal relationships, while identifying some novel and plausible bi-directional ones such as between body mass index and type 2 diabetes (T2D), and between diastolic blood pressure and stroke.


S1 Fig:
When both X and Y are continuous, θ XY = 0 and ξ = 0, the proportions of significant simulation results obtained by the methods for direction X → Y (left column) and Y → X (right column). The first row shows results for four main methods: MR-cML-DP-S, CD-cML-DP-S, CD-Ratio-S, and CD-Egger-S; the second row shows results for four methods without screening: MR-cML-DP, CD-cML-DP, CD-Ratio, and CD-Egger; the third row shows results for other five methods. 2), the proportions of significant simulation results obtained by the methods for direction X → Y (left column) and Y → X (right column). The first row shows results for four main methods: MR-cML-DP-S, CD-cML-DP-S, CD-Ratio-S, and CD-Egger-S; the second row shows results for four methods without screening: MR-cML-DP, CD-cML-DP, CD-Ratio, and CD-Egger; the third row shows results for other five methods. 2), the proportions of significant simulation results obtained by the methods for direction X → Y (left column) and Y → X (right column). The first row shows results for four main methods: MR-cML-DP-S, CD-cML-DP-S, CD-Ratio-S, and CD-Egger-S; the second row shows results for four methods without screening: MR-cML-DP, CD-cML-DP, CD-Ratio, and CD-Egger; the third row shows results for other five methods. 2), the proportions of significant simulation results obtained by the methods for direction X → Y (left column) and Y → X (right column). The first row shows results for four main methods: MR-cML-DP-S, CD-cML-DP-S, CD-Ratio-S, and CD-Egger-S; the second row shows results for four methods without screening: MR-cML-DP, CD-cML-DP, CD-Ratio, and CD-Egger; the third row shows results for other five methods. 2), the proportions of significant simulation results obtained by the methods for direction X → Y (left column) and Y → X (right column). The first row shows results for four main methods: MR-cML-DP-S, CD-cML-DP-S, CD-Ratio-S, and CD-Egger-S; the second row shows results for four methods without screening: MR-cML-DP, CD-cML-DP, CD-Ratio, and CD-Egger; the third row shows results for other five methods. 3 and ξ = 0, the proportions of significant simulation results obtained by the methods for direction X → Y (left column) and Y → X (right column). The first row shows results for four main methods: MR-cML-DP-S, CD-cML-DP-S, CD-Ratio-S, and CD-Egger-S; the second row shows results for four methods without screening: MR-cML-DP, CD-cML-DP, CD-Ratio, and CD-Egger; the third row shows results for other five methods.   The first row shows results for four main methods: MR-cML-DP-S,  CD-cML-DP-S, CD-Ratio-S, and CD-Egger-S; the second row shows results for four methods  without screening: MR-cML-DP, CD-cML-DP, CD The first row shows results for four main methods: MR-cML-DP-S, CD-cML-DP-S, CD-Ratio-S, and CD-Egger-S; the second row shows results for four methods without  screening: MR-cML-DP, CD-cML-DP, CD The first row shows results for four main methods: MR-cML-DP-S,  CD-cML-DP-S, CD-Ratio-S, and CD-Egger-S; the second row shows results for four methods  without screening: MR-cML-DP, CD-cML-DP, CD The first row shows results for four main methods: MR-cML-DP-S, CD-cML-DP-S, CD-Ratio-S, and CD-Egger-S; the second row shows results for four methods without  screening: MR-cML-DP, CD-cML-DP, CD The first row shows results for four main methods: MR-cML-DP-S,  CD-cML-DP-S, CD-Ratio-S, and CD-Egger-S; the second row shows results for four methods  without screening: MR-cML-DP, CD-cML-DP, CD The first row shows results for four main methods: MR-cML-DP-S, CD-cML-DP-S, CD-Ratio-S, and CD-Egger-S; the second row shows results for four methods without  screening: MR-cML-DP, CD-cML-DP, CD The first row shows results for four main methods: MR-cML-DP-S,  CD-cML-DP-S, CD-Ratio-S, and CD-Egger-S; the second row shows results for four methods  without screening: MR-cML-DP, CD-cML-DP, CD  X is binary, Y is continuous, θ XY = 0.2 and ξ = 0, the proportions of significant simulation results obtained by the methods for direction X → Y (left column) and Y → X (right column). The first row shows results for four main methods: MR-cML-DP-S, CD-cML-DP-S, CD-Ratio-S, and CD-Egger-S; the second row shows results for four methods without screening: MR-cML-DP, CD-cML-DP, CD-Ratio, and CD-Egger; the third row shows results for other five methods.  2), the proportions of significant simulation results obtained by the methods for direction X → Y (left column) and Y → X (right column). The first row shows results for four main methods: MR-cML-DP-S, CD-cML-DP-S, CD-Ratio-S, and CD-Egger-S; the second row shows results for four methods without screening: MR-cML-DP, CD-cML-DP, CD-Ratio, and CD-Egger; the third row shows results for other five methods.  3 and ξ = 0, the proportions of significant simulation results obtained by the methods for direction X → Y (left column) and Y → X (right column). The first row shows results for four main methods: MR-cML-DP-S, CD-cML-DP-S, CD-Ratio-S, and CD-Egger-S; the second row shows results for four methods without screening: MR-cML-DP, CD-cML-DP, CD-Ratio, and CD-Egger; the third row shows results for other five methods.  2), the proportions of significant simulation results obtained by the methods for direction X → Y (left column) and Y → X (right column). The first row shows results for four main methods: MR-cML-DP-S, CD-cML-DP-S, CD-Ratio-S, and CD-Egger-S; the second row shows results for four methods without screening: MR-cML-DP, CD-cML-DP, CD-Ratio, and CD-Egger; the third row shows results for other five methods.  X is continuous, Y is binary, θ XY = 0 and ξ = 0, the proportions of significant simulation results obtained by the methods for direction X → Y (left column) and Y → X (right column). The first row shows results for four main methods: MR-cML-DP-S, CD-cML-DP-S, CD-Ratio-S, and CD-Egger-S; the second row shows results for four methods without screening: MR-cML-DP, CD-cML-DP, CD-Ratio, and CD-Egger; the third row shows results for other five methods.  2), the proportions of significant simulation results obtained by the methods for direction X → Y (left column) and Y → X (right column). The first row shows results for four main methods: MR-cML-DP-S, CD-cML-DP-S, CD-Ratio-S, and CD-Egger-S; the second row shows results for four methods without screening: MR-cML-DP, CD-cML-DP, CD-Ratio, and CD-Egger; the third row shows results for other five methods.  X is continuous, Y is binary, θ XY = 0.02 and ξ = 0, the proportions of significant simulation results obtained by the methods for direction X → Y (left column) and Y → X (right column). The first row shows results for four main methods: MR-cML-DP-S, CD-cML-DP-S, CD-Ratio-S, and CD-Egger-S; the second row shows results for four methods without screening: MR-cML-DP, CD-cML-DP, CD-Ratio, and CD-Egger; the third row shows results for other five methods.  2), the proportions of significant simulation results obtained by the methods for direction X → Y (left column) and Y → X (right column). The first row shows results for four main methods: MR-cML-DP-S, CD-cML-DP-S, CD-Ratio-S, and CD-Egger-S; the second row shows results for four methods without screening: MR-cML-DP, CD-cML-DP, CD-Ratio, and CD-Egger; the third row shows results for other five methods.  X is continuous, Y is binary, θ XY = 0.1 and ξ = 0, the proportions of significant simulation results obtained by the methods for direction X → Y (left column) and Y → X (right column). The first row shows results for four main methods: MR-cML-DP-S, CD-cML-DP-S, CD-Ratio-S, and CD-Egger-S; the second row shows results for four methods without screening: MR-cML-DP, CD-cML-DP, CD-Ratio, and CD-Egger; the third row shows results for other five methods. 2), the proportions of significant simulation results obtained by the methods for direction X → Y (left column) and Y → X (right column). The first row shows results for four main methods: MR-cML-DP-S, CD-cML-DP-S, CD-Ratio-S, and CD-Egger-S; the second row shows results for four methods without screening: MR-cML-DP, CD-cML-DP, CD-Ratio, and CD-Egger; the third row shows results for other five methods.
When X is continuous, Y is binary, θ XY = 0.2 and ξ = 0, the proportions of significant simulation results obtained by the methods for direction X → Y (left column) and Y → X (right column). The first row shows results for four main methods: MR-cML-DP-S, CD-cML-DP-S, CD-Ratio-S, and CD-Egger-S; the second row shows results for four methods without screening: MR-cML-DP, CD-cML-DP, CD-Ratio, and CD-Egger; the third row shows results for other five methods. 2), the proportions of significant simulation results obtained by the methods for direction X → Y (left column) and Y → X (right column). The first row shows results for four main methods: MR-cML-DP-S, CD-cML-DP-S, CD-Ratio-S, and CD-Egger-S; the second row shows results for four methods without screening: MR-cML-DP, CD-cML-DP, CD-Ratio, and CD-Egger; the third row shows results for other five methods.
3 and ξ = 0, the proportions of significant simulation results obtained by the methods for direction X → Y (left column) and Y → X (right column). The first row shows results for four main methods: MR-cML-DP-S, CD-cML-DP-S, CD-Ratio-S, and CD-Egger-S; the second row shows results for four methods without screening: MR-cML-DP, CD-cML-DP, CD-Ratio, and CD-Egger; the third row shows results for other five methods. 2), the proportions of significant simulation results obtained by the methods for direction X → Y (left column) and Y → X (right column). The first row shows results for four main methods: MR-cML-DP-S, CD-cML-DP-S, CD-Ratio-S, and CD-Egger-S; the second row shows results for four methods without screening: MR-cML-DP, CD-cML-DP, CD-Ratio, and CD-Egger; the third row shows results for other five methods.
When both X and Y are binary, θ XY = 0 and ξ = 0, the proportions of significant simulation results obtained by the methods for direction X → Y (left column) and Y → X (right column). The first row shows results for four main methods: MR-cML-DP-S, CD-cML-DP-S, CD-Ratio-S, and CD-Egger-S; the second row shows results for four methods without screening: MR-cML-DP, CD-cML-DP, CD-Ratio, and CD-Egger; the third row shows results for other five methods. 2), the proportions of significant simulation results obtained by the methods for direction X → Y (left column) and Y → X (right column). The first row shows results for four main methods: MR-cML-DP-S, CD-cML-DP-S, CD-Ratio-S, and CD-Egger-S; the second row shows results for four methods without screening: MR-cML-DP, CD-cML-DP, CD-Ratio, and CD-Egger; the third row shows results for other five methods.  When both X and Y are binary, θ XY = 0.02 and ξ = 0, the proportions of significant simulation results obtained by the methods for direction X → Y (left column) and Y → X (right column). The first row shows results for four main methods: MR-cML-DP-S, CD-cML-DP-S, CD-Ratio-S, and CD-Egger-S; the second row shows results for four methods without screening: MR-cML-DP, CD-cML-DP, CD-Ratio, and CD-Egger; the third row shows results for other five methods.  2), the proportions of significant simulation results obtained by the methods for direction X → Y (left column) and Y → X (right column). The first row shows results for four main methods: MR-cML-DP-S, CD-cML-DP-S, CD-Ratio-S, and CD-Egger-S; the second row shows results for four methods without screening: MR-cML-DP, CD-cML-DP, CD-Ratio, and CD-Egger; the third row shows results for other five methods.  1 and ξ = 0, the proportions of significant simulation results obtained by the methods for direction X → Y (left column) and Y → X (right column). The first row shows results for four main methods: MR-cML-DP-S, CD-cML-DP-S, CD-Ratio-S, and CD-Egger-S; the second row shows results for four methods without screening: MR-cML-DP, CD-cML-DP, CD-Ratio, and CD-Egger; the third row shows results for other five methods.  2), the proportions of significant simulation results obtained by the methods for direction X → Y (left column) and Y → X (right column). The first row shows results for four main methods: MR-cML-DP-S, CD-cML-DP-S, CD-Ratio-S, and CD-Egger-S; the second row shows results for four methods without screening: MR-cML-DP, CD-cML-DP, CD-Ratio, and CD-Egger; the third row shows results for other five methods.  2 and ξ = 0, the proportions of significant simulation results obtained by the methods for direction X → Y (left column) and Y → X (right column). The first row shows results for four main methods: MR-cML-DP-S, CD-cML-DP-S, CD-Ratio-S, and CD-Egger-S; the second row shows results for four methods without screening: MR-cML-DP, CD-cML-DP, CD-Ratio, and CD-Egger; the third row shows results for other five methods.  2), the proportions of significant simulation results obtained by the methods for direction X → Y (left column) and Y → X (right column). The first row shows results for four main methods: MR-cML-DP-S, CD-cML-DP-S, CD-Ratio-S, and CD-Egger-S; the second row shows results for four methods without screening: MR-cML-DP, CD-cML-DP, CD-Ratio, and CD-Egger; the third row shows results for other five methods.  3 and ξ = 0, the proportions of significant simulation results obtained by the methods for direction X → Y (left column) and Y → X (right column). The first row shows results for four main methods: MR-cML-DP-S, CD-cML-DP-S, CD-Ratio-S, and CD-Egger-S; the second row shows results for four methods without screening: MR-cML-DP, CD-cML-DP, CD-Ratio, and CD-Egger; the third row shows results for other five methods.

S2.3 Links to GWAS Summary Datasets
We downloaded the GWAS summary datasets from the IEU GWAS database [1], which are the same as the data included in R package TwoSampleMR. The links are shown in S10 Table. S10 Proof. First, we show P B XY (m 0 XY ) = B 0 XY → 1, which is equivalent to show for any B 1 ⊆ {1, · · · , m} such that ≤ P( miñ . Note that, for g ∈ (B 0 XY ) c , r Xg −ρ Xg SE(r Xg ) ∼ N(0, 1) and N(0, 1). So for any ε > 0, there exists C > 0 such that (1) And we have After profiling outρ Xg 's, we get We have SE(r Y g ) 2 +K 2 SE(r Xg ) 2 follows non-central χ 2 distribution with degrees of freedom (m − m 0 XY ) and non-centrality parameter λK depending onK With Assumption 2, we get With Assumption 1, we know here v is a constant. This is because, with Assumption 1, there is noK making K XY · ρ Xg + b XY g −K · ρ Xg = 0 for all g ∈ B c 1 simultaneously. So we have minK λK ≥ N 2 · v. Then as N 2 large enough, we have Combining (1) and (2), we get P(B XY (m 0 XY ) = B 0 XY ) → 1 as N 1 , N 2 → ∞.
Similarly, since m 1 < m 0 XY , with Assumption 2 we have λK ≥ N 2 · v for some constant v, so for any |B| = m 1 , we get This gives us P(m I = m 1 ) → 0 for any m 1 < m 0 XY . For any m 1 > m 0 XY , we have is a central χ 2 distribution with degrees of freedom 2(m − m 0 XY ), we get P(m I = m 1 ) → 0 for any m 1 > m 0 XY . So we have P(m I = m 0 XY ) → 1 as N 1 , N 2 → ∞.
As P(B XY (m I ) = B 0 XY ) → 1, we could consistently select all invalid IVs. Following Theorem 3.2 in [4], we have In our model ρ Xg 's and ρ Y g 's are fixed constants, σ 2 Xg 's and σ 2 Y g 's are O(1/n), so we have V /V 1 → 1, and