 Conditional selection bias after futility continuation

In a two-stage normal trial, conditioning on a first-stage sample mean exceeding $c$ raises that mean. With stage standard error $s$, mean parameter $\delta$, and combined mean $S$, the estimator bias after continuation is $E(S\mid X\ge c)-\delta =(s/2)\lambda ((c-\delta )/s)$, where $\lambda$ is the upper-tail Inverse Mills ratio. It is positive and tends to zero as continuation becomes virtually certain.