Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 207 / 1

Use the usual clinical trial sampling assumption that all patients' responses are independent, with fixed positive arm sizes and positive known variances. Marginal normal distributions alone would not specify the variance of a treatment contrast without this assumption. Put ${\overline{Y}}_i=n_i^{-1}{\sum }_j{Y}_{ij}$, $v_i={\sigma }_i^2/n_i$, and $s_i=\sqrt{v_i+v_0}$. The sample mean contrast estimates ${\delta }_i$ and has standard error $s_i$. Thus the one-sided Wald statistics areLarge positive values of these Wald test statistics provide evidence against the respective null hypotheses.

The three independent sample means satisfy ${\overline{Y}}_i\sim \mathcal{N}({\mu }_i,v_i)$. A linear transformation of their joint multivariate normal distribution is again normal. The two treatment contrasts have covarianceWriting $m_i={\delta }_i/s_i$ and $\rho =v_0/(s_1{s}_2)$, their joint distribution isThe positive correlation coefficient arises from the shared control arm. Under positive variances, $0<\rho <1$.

Let ${\varphi }_{\rho }$ denote the centered unit-variance bivariate normal distribution density,The complement of rejecting either null hypothesis is $W_1\le c,W_2\le c$. Therefore the probability of at least one rejection isWhen both null hypotheses are true, this is the familywise error rate. If one null hypothesis is false, a rejection of that hypothesis is not a type I error, so the displayed probability is then an any-rejection probability rather than the familywise error rate.

Couple the Wald test statistics as $W_i=Z_i+{\delta }_i/s_i$, where the joint centered bivariate normal distribution of $(Z_1,Z_2)$ is fixed. Increasing either ${\delta }_i$ enlarges, pointwise in this coupling, the event $\{W_1>c\text{ or }{W}_2>c\}$. Its probability is consequently nondecreasing in each treatment effect.

If both null hypotheses hold, then ${\delta }_1,{\delta }_2\le 0$, so this probability is maximal at the joint boundary $(0,0)$. To check the familywise error rate across all configurations, suppose only hypothesis $i$ is true. A type I error then occurs only if $W_i>c$, with probability at most $1-\Phi (c)$, where $\Phi$ is the standard normal distribution function. This is no larger than the any-rejection probability at $(0,0)$, since that joint event contains $\{Z_i>c\}$. If neither null hypothesis is true, the familywise error rate is zero. Hence the least-favourable null configuration for type I error iswhere ${\Phi }_2$ is the centered bivariate normal distribution function with correlation coefficient $\rho$.

Choose a group sequential design. Randomize patients among the three arms, schedule interim analyses after prespecified amounts of information, and calculate the two Wald test statistics at each look. Prespecify efficacy boundaries and a futility rule; allow early stopping of an ineffective experimental arm or of the trial when sufficiently convincing efficacy evidence is obtained. Calibrate the boundaries for both repeated looks and the two shared-control comparisons so that the familywise error rate remains controlled. An alpha-spending function is one way to allocate the error probability across looks.

Two potential benefits are lower expected recruitment and shorter duration when a treatment is clearly effective or futile, and earlier identification of benefit or lack of benefit, reducing continued exposure to an ineffective experimental treatment. These benefits concern expected performance; the maximum sample size may still be required.

Two drawbacks are the extra operational work needed for rapid, reliable outcomes and confidential interim analyses, and more complicated inference: repeated testing requires adjusted boundaries, and ordinary fixed-sample effect estimates and confidence intervals can be misleading after data-dependent stopping. Continuing two comparisons against a shared control also requires care if an arm is stopped.