Group sequential design

A group sequential design analyzes accumulating data at prespecified interim analyses and may stop early for efficacy or futility.

A futility boundary specifies an interim result below which a clinical trial stops without rejecting the null hypothesis. Removing only potential rejection paths cannot increase the Type I error. With a finite boundary and a nonsingular joint normal test-statistic distribution, it strictly reduces that error relative to the corresponding fixed-final-boundary test.

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.

The conditional error principle permits adaptation of a testing design if, conditional on the observed interim information, the modified future rule's Type I error does not exceed the original rule's conditional future rejection probability. Averaging then preserves the original unconditional error bound. For a completed single-look test that has not rejected, there is no future conditional error budget under that original design.

An alpha-spending function assigns the cumulative Type I error budget to successive information fractions in a group sequential design. Its total at the final analysis is the overall significance level. The rejection boundaries must be calibrated using the joint distribution of the sequential test statistics; independent fixed-level tests do not automatically spend the assigned budget correctly.

An interim analysis examines data before a study reaches its planned maximum sample size.

A sequential stopping boundary specifies values of an accumulating test statistic that trigger early stopping for efficacy, futility, or harm.

At information levels $I_1<\cdots <I_J$, standardized efficient test statistics have unit variances, means proportional to $\sqrt{I_j}$, and correlations $\mathrm{Corr}(Z_j,Z_k)=\sqrt{I_j/I_k}$ for $j\le k$.