It is well known that traditional “complete-data” IV estimators are biased towards the confounded association and that bias is most severe when the IV is weak (7, 14). In contrast, 2-sample IV estimates are known to be biased towards the null, even when the confounded estimate is biased away from the null (4, 5). In Figure 3 (left panel), we show that the direction of the weak IV bias for subsample MR analyses depends on the nX:nY ratio. If nX represents a small percentage of nY, bias moves towards the null as F decreases, similar to the 2-sample case. In contrast, if the nX:nY ratio is close to 1, bias moves towards the observational estimate as F decreases, similar to the complete-data scenario. The total number of participants (nY) is shown as a diagonal line, and the first-stage R2 is fixed for each ratio. Figure 3 (right panel) shows the same simulations conducted in the absence of confounding; hence, no bias towards the confounded association is observed. For weak IVs, bias towards the null is present for all subsample scenarios,
