Because the reduced-form power is the approximate upper limit for power, power for MR is most efficiently increased by increasing the size of the sample used for estimation of the reduced-form equation, rather than the first-stage equation (assuming exposure data are available for a sufficient subset of participants). This conclusion is somewhat intuitive because the reduced-form association (the numerator of the Wald estimator) is typically quite weak and difficult to estimate with statistical confidence, since the association between the IV and the outcome is mediated entirely through the exposure. In contrast, the first-stage association (the denominator) should be well-established and easily detectable in a large epidemiologic study.
