= Conditional selection bias after futility continuation
{title2=$E(S\mid X\ge c)-\delta$}
For <independent random variables>[independent] stage <mean>[means] $X,Y\sim N(\delta,s^2)$ and combined <mean> $S=(X+Y)/2$, conditioning on the first-stage <sample mean> $X$ exceeding $c$ raises that <mean>. In this equal-stage setting, 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.
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