controls added to the baseline design. Compared to the baseline model (blue diamond), adding 500 public controls (pink curve) does not improve and may worsen power: showing equivalent power to the baseline model when R2 = 0.9 but steadily declining as imputation accuracy declines. Adding increasing numbers of public controls results in a marginal to substantial improvement in power. For example, adding 2,000 public controls (2,000 cases: 4,000 controls—green line) increases power to between 86 and 93 % when R2 is between 0.8 and 0.9.
