 Solution

Conditioning on continuation selects the upper part of the stage-1 estimator's distribution, since $C=\{{\hat{\delta }}_1\ge f_1/\sqrt{I_1}\}$. Thus ${\mathbb{E}}_{\delta }({\hat{\delta }}_1\mid C)>\delta$ for every finite threshold. The fresh stage-2 estimate ${\hat{\delta }}_2^{\ast }$ is independent of $C$ and remains unbiased for $\delta$. The pooled estimator retains a positive weight on the selected stage-1 data, givingConditional on full enrollment, the ordinary pooled MLE overestimates the treatment effect. This is conditional selection bias after futility continuation. Randomization of treatment assignments does not undo selection on the interim outcome.