Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 32 / 3 / a

A group sequential design can stop early when the treatment is insufficiently promising, reducing expected recruitment, patient exposure and cost when the effect is poor. It also allows a planned decision before maximum enrollment.

A disadvantage is that the analysis is selected by the interim results: estimation and uncertainty must account for stopping, and power and operational planning depend on the stopping rule. The following parts quantify the resulting conditional estimation bias. If a design also allows repeated efficacy tests, its rejection boundaries must control overall Type I error. In the present futility-only design, stopping without rejection need not inflate the error rate of a single prespecified final test; it can make that test conservative. Earlier decisions are valuable, but the ordinary fixed-sample analysis is not automatically appropriate after selection.

At stage 1, the maximum-likelihood estimators of the two normal arm means are their sample means. Their difference estimates the treatment effect:Each arm has $n_1$ patients and variance ${\sigma }^2/n_1$, so independence across arms givesThe quantity $I_1$ is the Fisher information for the mean difference when the common variance is known. The PDF's preliminary expression uses $n_i$ for the variance denominator, although $i$ indexes arms and $n_0$ is not defined. The stage-specific sample size is $n_j$; the calculation uses this intended reading, consistent with the specified $n_1,n_2$ per-arm recruitment and the later weighted estimator. Fresh stage-2 patients are independent of stage-1 patients in the usual trial model; the cumulative stage means themselves are not independent across stages.

The Wald statistic has law $Z_1={\hat{\delta }}_1\sqrt{I_1}\sim N(\delta \sqrt{I_1},1)$. Write $C=\{Z_1\ge f_1\}$ for continuation. By the normal cumulative distribution function,The final sample size per arm is $N=n_1+n_2^{\ast }{1}_C$, soIt lies between $n_1$ and $n_2=n_1+n_2^{\ast }$, increasing with the treatment effect: a more promising treatment is more likely to reach full enrollment. Total expected recruitment over both arms is twice this expression. Equality at the futility boundary has probability zero under the normal model.