在醫學領域裡，醫生常常要針對不同病況而施予不同藥物作為治療，而選擇適合藥物是一個很重要議題。Cheng-Shiun Leu與 Bruce Levin 提出一個選擇藥物之實驗程序，名為adaptive design in phase-II clinical trials，主要針對在第二期臨床試驗中，如何有效率地選出有較佳藥效之藥物，減少受試者人數，降低實驗成本。他們把服藥後的治療結果視為服從Bernoulli分布，透過逐次程序去選出較好的藥物。而逐次程序分為兩種，non-adaptive（無調適）及adaptive（調適）LRL逐次程序。其中，使用LRL逐次程序之實驗，能滿足正確選對藥品之一下界，同時能減少較多受試者人數。而有些藥物的治療結果並不只有成功或失敗，像降血壓之類藥物，其反應是一個連續變數。故本文針對服藥後之反應服從常態分布之情況，提出逐次實驗程序，並比較使用無調適及調適程序之實驗結果。最後經過電腦模擬驗證，證實在藥品反應服從常態分布下，使用調適程序比使用無調適程序之實驗，更能夠減少受試者人數。

In the medical study, selecting the appropriate medicines is a very important issue. Cheng-Shiun Leu and Bruce Levin have designed a procedure, for selecting medicines in phase-II, which is called adaptive design in phase-II clinical trials. It focuses on how to select efficiently the medicines with better efficacy, and also focus on reducing the sample size and costs of the experiments. The responses they consider are Bernoulli distributed. Through the sequential procedures, we can select the best medicines. Two sequential procedures are considered, one is non-adaptive and the other is adaptive. By using these procedures, the probability of correct selection has a low bound. Also, the number of testers can be reduced. But there are drugs, like antihypertensive drugs, their responses are continuous variables. In this thesis, we study those drugs with Normal distributed responses, and provide corresponding sequential procedures (non-adaptive and adaptive). We compare the results of these procedures. By simulations, we can verify that adaptive procedure reduces more testers than non-adaptive one does.

[1] Bechhofer, R.E., Kiefer, J. and Sobel, M. (1968). Sequential Identication and Ranking Procedures. University of Chicago Press, Chicago.
[2] Chiu, W.K. (1974). A generalized selection goal for a sequential ranking procedure. Nanta Math. 7, 42-46.
[3] Leu, C.S. and Levin, B. (1999a). On the probability of correct selection in the Levin-Robbins sequential elimination procedure. Statistica Sinica 9, 879-91.
[4] Leu, C.S. and Levin, B. (1999b). Proof of a lower bound formula for the expected reward in the Levin-Robbins sequential elimination procedure. Sequential Analysis 18, 81-105.
[5] Leu, C.S. and Levin, B. (2004). Selecting the best subset of b out of c coins with the Levin-Robbins sequential elimination procedure: Proof of the lower bound formula for the probability of correct selection in the case b = 2, c = 4. Technical Report #B-91, Department of Biostatistics, Columbia University.
[6] Leu, C.S. and Levin, B. (2008). A generalization of the Levin-Robbins procedure for binomial subset selection and recruitment problems. Statistica Sinica 18, 203-218.
[7] Leu, C.S. and Levin, B. (2013). On two lemmas used to establish a key inequality that implies the lower bound formula for the probability of correct selection in the Levin-Robbins-Leu family of sequential binomial subset selection procedures. Technical Report #B-148, Department of Biostatistics, Columbia University.
[8] Levin, B. (1984). On a sequential selection procedure of Bechhofer, Kiefer, and Sobel. Statist. Probab. Lett. 2, 91-94.
[9] Levin, B. and Robbins, H. (1981). Selecting the highest probability in binomial or multinomial trials. Proc. Natl. Acad. Sci. USA 78, 4663-4666.
[10] Zybert, P. and Levin, B. (1987). Selecting the highest of three binomial probabilities. Proc. Natl. Acad. Sci. USA 84, 8180-8184.