power conditional on a historical control

The work of Korn and Freidlin was a good summary prior to the causal inference era. The paper recommended unconditional power vs (plug in) conditional power, which was nice (because the conditional power formulus was less intuitive). The authors also showed (fig 1) that, for normal endpoints,

one sample test power > plug in conditional power > unconditional power (which suggests unconditional power is conservative as it requires a larger N)

conditional power varies as the function of the difference between the observed mean in the historical control and the true control mean (which was eventually addressed in the causal inference literature)

The work of Dixon and Simon because a special situation in Korn's discussion. As an old paper, it still had some interesting small features to be remembered.

log hazard calculation under exponential assumption (formula 1)

The powers based on conditional formula were indeed smaller than those from unconditional approach. code and output are here.