at least $1-\beta$ power of rejecting $H_0:\theta = 0$ for each alternative hypothesis $H_{Ak}:\theta = \theta_k$, where $\theta_{1} > \theta_{2} > \cdots > \theta_K > 0$. Specifically, $\{n_k\}$ and $\{c_k\}$ are identified recursively to satisfy $\alpha_{(k)} \le \alpha$ and under $\theta=\theta_k$, $1-\beta_{(k)}(\theta_k) = 1-\beta$. The first stage critical value $c_1$ and sample size $n_1$ are determined by $\alpha_1$ and by $\beta =\beta_1$ which depends on the first stage alternative hypothesis $\theta_1$. If the same alternative and sample size were used for stage 2, the power at stage 2 would be greater than $1-\beta$. A smaller $\theta_2<\theta_1$ and $\alpha_2$ are chosen to define a stage 2 sample size $n_2$ and the critical value $c_2$ that will keep the power at $1-\beta$. Additional decreasing hypotheses and $\alpha_k$ are used to determine samples sizes and critical values for following stages. Thus, this design is “flexible”. The total number of stages, $K$, does not need to be pre-determined. However, for ease of illustration, henceforth a fixed $K$ is considered.

                  Feature                                                                           Mathematical Definition
  ---------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------
          *overall type $1$ error*                                         $\alpha = \sum_{k=1}^K \alpha_k \prod_{j=1}^{k-1}\left(1-\alpha_j\right)$
       *stage-specific type 1 error*           $\alpha_k = \text{Pr}_{\theta}\left( Z_{(k)} > c_k \big| \cap_{j=1}^{k-1} \left\{ Z_{(j)}\le c_{j} \right\}, \theta = 0 \right)$
        *$\alpha$-spending 