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Normal approximation for two-stage winner design. (English) Zbl 1158.62050
Hsiung, Agnes Chao (ed.) et al., Random walk, sequential analysis and related topics. A Festschrift in honor of Yuan-Shih Chow. Papers presented at the international conference, Shanghai, China, July 18–19, 2004. Hackensack, NJ: World Scientific (ISBN 981-270-355-1/hbk). 28-43 (2006).
From the introduction: We consider a study where several potential new treatments are compared to a control (standard) treatment $$C$$. In stage I of the study, a winner $$W$$ among the new treatments is determined based on the treatment effect compared to $$C$$. In stage II, the losers in stage I are dropped and additional patients are allocated only to treatment arms $$W$$ and $$C$$. At the end of the study, the data of $$W$$ and $$C$$ patients in stages I and II combined are compared to determine whether $$W$$ is better than $$C$$. This design is called “two-stage winner design”.
We start with the special case where there are only two new treatments, $$A$$ and $$B$$. To our pleasant surprise, the distribution of the test statistic comparing $$W$$ and $$C$$ is extremely close to the normal. Thus, we propose a normal approximation approach in dealing with the type I error rate and power associated with the Winner design for normally distributed data. An almost exact method is also proposed. The normal approximation is extremely convenient, especially when an integration program is not handy. For almost all practical situations, it provides a simple way to understand the insights in designing such studies when the two treatment effects are similar. In more general settings when the effects are different, we provide a simple table to facilitate the evaluation of the sample size for the desired power levels. The case of unknown variance and the extension to more than two treatments are also discussed.
The paper is organized as follows: In Section 2, the Winner design and its test statistic are introduced; in Section 3, we consider tail probabilities using exact calculations, normal approximations, and an almost exact approach; in Section 4, the power is evaluated; in Section 5, a small sample case with unknown variance is considered; and finally we complete this article with discussions in Section 6.
For the entire collection see [Zbl 1130.60007].

##### MSC:
 62L05 Sequential statistical design 62E17 Approximations to statistical distributions (nonasymptotic) 62P10 Applications of statistics to biology and medical sciences; meta analysis