Fig. 2: Sensitivity contour plot with benchmark bounds.
The horizontal axis shows the partial R2 of unobserved variables W with the genetic instrument; this corresponds to the percentage of residual variation of the genetic instrument explained by W. The vertical axis shows the partial R2 of W with the trait of interest, which can be either the exposure trait or the outcome trait; again, this stands for the percentage of residual phenotypic variance explained by W. Given any pair of partial R2 values, the contour lines show the t-value that one would have obtained for testing the significance of the genetic association with the (exposure/outcome) trait, had a W with such strengths been included in the analysis. The point represented by a black triangle (left lower corner) shows the t-value of a traditional MR study (i.e., t = 13)—note it assumes exactly zero biases due to unobserved variables W. As we move along both axes, the biases due to W get worse, and can eventually be strong enough to reduce the t-value below a chosen critical level t*, shown in the red dashed line (e.g., t* ≈ 2 for a significance level of α = 5%). Unobserved variables W with strength below the critical red line are not strong enough to change the conclusions of the original MR study; on the other hand, unobserved variables W with strength above the critical red line are strong enough to be problematic. The point represented by a red diamond bounds the maximum strength of W if it were as strong as observed genomic principal components (1× PCs). They show the maximum bias caused by residual population stratification, if it had the same explanatory power as the PCs in explaining genetic and phenotypic variation. In this example, the plot reveals that residual population stratification as strong as the first genomic principal components would not be sufficient to make the genetic association statistically insignificant (i.e., the adjusted t-value accounting for a W with such strength is 7.88, which is still above the critical threshold of t* ≈ 2). Finally, we note that if the unobserved variable W is a singleton, then all the sensitivity analysis results are exact. If W consists of multiple variables, then all sensitivity analysis results are conservative, meaning that this is the worst bias that a multivariate W could cause if it had such strengths46.
