This work is based on hypothesis testing regarding efficacy of 3 independent groups of treatment -- experimental treatment (E), active reference (R) and placebo (P). The important assumptions here are homoscedasticity and normality. The observation for the $k$th patient in group $i$ is denoted as $X_{ik}$ and is distributed as $X_{ik} \sim N(\mu, \sigma^2)$, for $i = E, P, R$; $k=1, \ldots, n_i$. The symbol $n_i$ represents the number of patients in group $i$, $\delta_{ij} \geq 0$ the inferiority/superiority margin, for $i = E, P, R$; $i \neq j$ and $z_\alpha$ upper $\alpha$-percentile of standard normal distribution.
With the rejection of the principal null hypothesis $H_0$ we prove the superiority of experimental treatment (E) versus placebo (P) and, simultaneously, non-inferiority of experimental treatment (E) versus active reference (R). The asymptotic power for the rejection of $H_0$ equals
\begin{align*}
\Phi^{\Sigma} \Big(\frac{\mu_E - \mu_P - \delta_{EP}}{\sigma \sqrt{\frac{1}{n_E} + \frac{1}{n_P}}} - z_{\alpha}, \frac{\mu_E - \mu_R + \delta_{ER}}{\sigma \sqrt{\frac{1}{n_E} + \frac{1}{n_R}}} - z_{\alpha} \Big).
\end{align*}
With the rejection of the extended null hypothesis $\widetilde{H}_0$ we prove, in addition, the superiority of active reference (R) versus placebo (P). The asymptotic power for the rejection of $\widetilde{H}_0$ equals
\begin{align*}
\Phi^{\widetilde{\Sigma}} \Big( \frac{\mu_E - \mu_P - \delta_{EP}}{\sigma \sqrt{\frac{1}{n_E} + \frac{1}{n_P}}} - z_{\alpha}, \frac{\mu_E - \mu_R + \delta_{ER}}{\sigma \sqrt{\frac{1}{n_E} + \frac{1}{n_R}}} - z_{\alpha}, \frac{\mu_R - \mu_P - \delta_{RP}}{\sigma \sqrt{\frac{1}{n_R} + \frac{1}{n_P}}} - z_{\alpha} \Big).
\end{align*}
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