多反應迴歸模型下之最適設計簡介

An Introduction to Optimal Designs for Multiresponse Regression Models

許湘伶(Hsiang-Ling Hsu) ； 羅夢娜(Mong-Na Lo Huang)

《中國統計學報》 42卷4期 (2004/12) Pp. 365-381

https://doi.org/10.29973/JCSA.200412.0003

摘要

在本文中，我們介紹有關多反應變數模型下之一些最適設計的研究結果。此類模型在許多實際問題中，常被用來描述實驗當中一些相關變數間之變化趨勢，例如在一化學反應過程當中，不同階段下所觀測到的反應變數，可以視為一具有多反應變數之實驗。Draper和Hunter (1996)討論在一化學反應過程中，若對應之多反應變數觀利值，有一明確的非線性模型可描述其變化的情形下，與其相關之最適設計問題。而Uciński和Bogacka (2004)則探討當化學反應過程之模型，僅能由某些微分方程之解所描述時，如何區別模型之最適設計問題。本文將特別介紹此二種不同型式下之相關最適設計結果。另外也將介紹Krafft和Schaefer (1992)和Bischoff (1993,1995)所給出關於多反應變數線性模型下之一些重要的分解資訊矩陣行列式的定理。此類定理說明瞭在特殊模型結構下，相對應之最適設計，並不會因模型之共變異矩陣之改變而有所不同。同時亦給出在多反應變數下，多項式模型之正合最適設計的一些結果。最後，我們介紹在二反應變數之迥歸模型下的 D-和D（下標 s）-最適設計，以及生物分析下有關估計相對效力或位移參數之最適設計的相關研究。

關鍵字

生物分析 ； 二反應變數模型 ； 分解定理 ； 線性或非線性模型 ； 多反應變數 ； 最適設計 ； 相對效力

並列摘要

In this paper, some results of the research for optimal designs in multiresponse models are introduced. This type of models arises naturally in many applications. For example, in the process of a chemical reaction, the response variables observed under different stages can be regarded as multiresponse variables in the experiment. Draper and Hunter (1966) discussed the corresponding optimal designs with explicit non-linear model on multiresponse variables for a chemical reaction process. Lately Uciński and Bogacka (2004) investigated the optimal designs for discriminating two possible non-linear models in a chemical reaction process, where the reaction process can only be described from the solution of some given differential equations. The dif ferences between the two approaches will be illustrated with examples. Moreover, we introduce some important factorization theorems given by Krafft and Schaefer (1992) and Bischoff (1993, 1995) in linear models with multiresponse variables. These factorization theorems show that the corresponding optimal designs at special models or design structures remain to be optimal, regardless of what the covariance matrix may be. Furthermore, they also presented some results about the exact optimal designs in multiresponse polynomial models. Finally, we introduce the D-and D(subscript s)-optimal designs and the locally c optimal designs for the estimation of relative potency or equivalently the location shift parameter in the parallel linear regression models with dual responses.

並列關鍵字

Bioassay ； dual response model ； factorization theorem ； linear or non-linear models ； multiresponse ； optimal design ； relative potency

參考文獻

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