在工業、農業、或醫藥界的實驗中，經常必須拿數個不同的試驗處理

(test treatments)和一個已使用過的對照處理(control treatment)比較

。所謂的試驗處理可能是數組新的儀器、不同配方的新藥、或不同成份的

肥料等。以實驗新藥為例，研藥者想決定是否能以新藥取代原來所使用的

藥，故對v種新藥與原藥做比較，評估其藥效之差異。為了降低實驗中不

必要的誤差以增加其準確性，集區設計成為實驗者常用的設計方法之一；

又因A式最適設計是我們欲估計的對照處理效果(effect)與試驗處理效果

之差異之估計值最小的設計，基於此良好的統計特性，我們選擇A式最適

性為評判根據。古典的A式最適性並未將對照處理與試驗處理所具備的先

前資訊(prior information)加以考慮，以上例而言，我們不可能對原來

使用的藥一無所知，經由過去的實驗或臨床的反應，研藥者必已對其藥性

有某種程度的了解，直觀上，這種過去經驗的累積，影響到實驗配置上，

可能使對照處理的實驗次數減少，相對地可對試驗處理多做實驗，設計遂

更具意義。因而本文考慮在k>v的情形下之貝式最適集區設計，對先前分

配施以某種限制，依據準確設計理論(exact design theory)，推導單項

異種消除模型(one- way elimination of heterogeneity model)之下的

貝氏A式最適設計與Γ- minimax最適設計，使Majumdar(1992)的結果能適

用於完全集區設計。此種設計對先前分配具有強韌性，即當先前分配有所

偏誤，且其誤差在某一範圍內時，此設計仍為最適設計或仍可維持所謂的

高效度(high efficiency)。本文將列舉許多實例以說明此一特性。

We consider the problem of comparing a set of v test treatments

simultaneously with a control treatment when k>v. Following the

work of Majumdar(1992), we use exact design theory to derive

Bayes A-optimal designs and optimal Γ-minimax designs for the

one-way elimination of heterogeneity model. These designs have

the same properties as of Bayes A-optimal incomplete block

designs. We also provide several examples of robust optimal

designs and highly efficient designs.

Cheng.C.S. and Wu. C.F.(1980). Balanced Repeated Measurements Designs. The Annals of Statistics 8 1272-1283.
Cheng.C.S.. Majumdar. D.. Stufken. J.. and Türe. T.E.(1988). Optimal steptype designs for comparing test treatments with a control. Journal of the American Statistical Association 83 477-482.
Giovagnoli.A. and Verdinelli. I. (1983). Bayes D-optimal and E-optimal block desigus. Biometrika 70 695-706.
Giovagnoli. A.and Verdinelli.I.(1985).Optimal block designs under a hierarchical linear model. In Bayesian Statistics 2 (J.M. Bernardo. M.H. DeGroot. D. V. Lindly and A.F.M.Smith.eds.) 655-662. North –Holland. Amsterdam.
Hedayat. A.S.. Jacroux. M. and Majundar. D. (1988). Optimal designs for comparing test treatments with controls. Statistical Science 3 462-491.
Jacroux.M.and Majumdar. D.(1989). Optimal block designs for comparing test treatments with a control when k > r. Journal of Statistical Planning and Inference 23 381-396.
Majumdar. D. (1988). Optimal block designs for comparing new treatments with a standard treatment. In Optimal Design and Analysis of Experiments (Y. Dodge. V.V. Fedorov and H.P. Wynn. Eds.) 15-27.North –Holland. Amsterdam.
Majumdar. D. (1992). Optimal designs for comparing test treatments with a control utilizing prior information. The Annals of Statistics 20 216-237.
Majumdar. D. and Notz, W. I. (1983). Optimal incomplete block designs for comparing test treatments with a control. The Annals of Statistics 11 258-266.
Owen R.J. (1970). The optimum design of a two-factor experiment using prior information. The Annals of Mathematical Statistics 41 1971-1934.
Stufken. J.(1991). Bayesian optimal experimental design for treatment-control comparisous in the presence of two-way heterogeneity. Journal of Statistical Planning and Inference 27 51-63.