Conditional selection bias after futility continuation ($E(S\mid X\ge c)-\delta$)

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.

In a two-stage normal trial, invert the pooled estimator's conditional mean after continuation to correct its selection shift. The root is unique when the pooled stage-1 weight satisfies $0<w<1$, because its conditional-mean map has derivative between $1-w$ and one. This root also maximizes the corresponding conditional likelihood. It is distinct from exact conditional unbiasedness, which is supplied by a suitable Rao-Blackwell theorem construction.

For two independent stage means $X,Y\sim N(\delta ,s^2)$ and continuation $X\ge c$, let $S=(X+Y)/2$ and $v=s/\sqrt{2}$. The second-stage mean is conditionally unbiased. The Rao-Blackwell theorem gives $E(Y\mid S,X\ge c)=S-v\lambda ((c-S)/v)$, using $X\mid S\sim N(S,v^2)$ and truncation. This is a conditionally unbiased estimator using both stages through their combined sample mean.