= r_{(k)}$, then equations (\[MLE\_k\]) and (\[tau\_n\]) lead to the standardized test statistic given on $D=k$: $$V_{(k)} = \sqrt{n_{(k)}} \sum_{j=1}^k \frac{n_j}{n_{(k)}} \left(\widehat\theta_j-\theta\right) =  
 \sum_{j=1}^k \sqrt{\frac{n_j}{n_{(k)}}} \sqrt{n_{j}} \left(\widehat\theta_j-\theta\right) \to \sum_{j=1}^k \frac{\xi_j}{\sqrt{r_{(k)j}}}.$$ Hence, the distribution of $V_{(D)}$ is $$\begin{aligned}
{\text{Pr}_{\theta}}\left(V_{(D)} < v \right) &\to& 
\sum_{k=1}^K {\text{Pr}_{\theta}}\left(r_{(D)}=r_{(k)}\right){\text{Pr}_{\theta}}\left(\sum_{j=1}^k \frac{\xi_j}{\sqrt{r_{(k)j}}}  < v \Big\vert r_{(D)}=r_{(k)} \right).\end{aligned}$$

Example
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To illustrate use of a sequence of ordered alternative hypotheses in designing a group sequential study, a dataset of $7,874$ persons available in Dr. Terry Therneau’s “Survival” R package is reanalyzed. Serum free light chain (FLC), age, sex, creatinine, survival status and a follow-up period are available in this dataset. This smaller dataset is a subset of a larger dataset of $15,759$ persons where relationship between FLC and survival was analyzed using a proportional hazards model \[@dispenzieri2012\].

A Cox proportional hazards regression model is used to analyze the effect of FLC on survival hazards controlling for age, sex and creatinine: $$\log h(t|FLC, A, M, C) = \log h_b(t) + \beta_1 FLC + \beta_2 A + \beta_2 M + \beta_3 C,$$ where $h_b(t)$ is the baseline survival hazard; $A$ is “Age”; $M$ is an indicator of male sex, and $C$ is a creatinine value. When this model is applied 