= \alpha_0$ leading to $$\alpha = \sum_{k=1}^K \alpha_0 \prod_{j=1}^{k-1} \left(1-\alpha_0\right)= 
\alpha_0 \sum_{k=1}^K \left(1-\alpha_0\right)^{k-1} = \alpha_0\left(1-\left(1-\alpha_0\right)^K\right),\label{alphak2}$$ where the last equality follows from a property of geometric sequences. Thus, for a fixed $K$, stage-specific critical values are defined via . For a two-stage design with $\alpha = 0.05$, $\alpha_0\approx 0.0257$; at $K=3$, $\alpha_0 \approx 0.0172$; at $K=4$, $\alpha_0 \approx 0.0144$.

The numerical example that follows shows how stage-specific power requirements can be determined from the overall power $1-\beta$ at $\theta = \theta_k$ requirement and sample sample sizes can be found that satisfy these requirements.

### A numeric example at $K=3$.

Consider $H_{A1}:\theta = 0.3$, $H_{A2}:\theta = 0.2$, and $H_{A3}:\theta = 0.1$. Stage-specific $n_k$ and $c_k$ $(k=1,2,3)$ are found from the following system of (nonlinear) equations: $$\begin{aligned}
\label{GSnew}
\text{Pr}_{\theta}\left(Z_{(1)}>c_1\big|\theta = 0\right) &=&  \alpha_0 \notag \\
\text{Pr}_{\theta}\left(Z_{(2)}>c_2\big|Z_{(1)}\le c_1,\theta = 0\right) &=&  \alpha_0\notag \\
\text{Pr}_{\theta}\left(Z_{(3)}>c_3\big|Z_{(2)}\le c_2, Z_{(1)}\le c_1,\theta = 0\right) &=&  \alpha_0\notag \\
\text{Pr}_{\theta}\left(Z_{(1)}>c_1\big|\theta = 0.3 \right) &=&  1-\beta \notag \\
\text{Pr}_{\theta}\left(Z_{(1)}>c_1\big|\theta = 0.2 \right) + \text{Pr}_{\theta}\left(\left\{Z_{(2)}>c_2\right\}\cap \left\{Z_{(1)}\le c_1\right\}\vert \theta = 0.2\right) &=&  1-\beta \notag \\
\sum_{k=1}^3 \text{Pr}_{\theta}\left(
\left\{Z_{(k)}>c_k \right\} \cap \left\{\cap_{1\le j \le k-1} \left\{ Z_{(j)}\le c_{j} \right\} \right\}\big| \theta = 0.1 \right) &=& 1-\beta\end{aligned}$$ Recall that $Z_{(1)}$, $Z_{(2)}$ and $Z_{(3)}$ depend on the 