F was accomplished as follows: For each nX:nY ratio, the first-stage R2 was held constant (to a value that produced an average first-stage F of 20 when nY = 10,000) and nY was varied from 100 to 10,000, with the value nX determined by the nX:nY ratio. This approach allowed us to assess weak IV biases for a wide spectrum of values for F and nX:nY. We also evaluated bias for weak IVs by varying nX:nY (from 0.1 to 1.0) and R2 (from 0.001 to 0.03), with βXY set to 0.1 and nY set to 10,000, 3,000, or 1,000 (simulation 4). Similar simulations were also conducted for 2-sample IV estimators, where the first-stage sample (nX) and the reduced-form sample (nY) consisted of independent sets of participants. Confounder effects βUX and βUY were set to 0.2 in simulations 1 and 2, leading to positive confounding. βUX and βUY were set to 0.3 in simulations 3 and 4 to better demonstrate weak IV bias. Simulations were repeated in the absence of confounding, although it is known that X-Y confounding does not produce substantial bias for traditional IV analyses when IVs are strong (7).
