In the traditional complete-data setting, weak IV bias can be explained as resulting from a correlation between the 2 terms in the Wald estimator: the first-stage and reduced-form estimates (14). In the 2-sample setting, these estimates are uncorrelated, since they are derived from different data sources. In this case, imprecision in the estimation of the G-X and G-Y associations is analogous to nondifferential measurement error in an observational estimate and results in bias towards the null similar to regression dilution bias (15). In the subsample setting, the bias towards the null and the bias in the direction of the observational association (which is usually in the same direction as the causal effect when this is present) can balance each other out, as demonstrated in the left-hand panel of Figure 3 when the ratio nX:nY is 0.5. The precise ratio required to give unbiased estimates is likely to depend on the characteristics of a given example rather than to be a generalizable result.
