本研究旨在建立廣義線性模式下之D-與A-最適設計(optimal designs)，並依不同處理結構(treatment structure)分成完全隨機設計(completely randomized design, CRD)與隨機集區設計(randomized block design, RBD)兩部分探討。
根據完全隨機設計所推導出之行列式的性質與理論結果，我們首先提出一個能快速大幅限縮尋找D-最適正合(exact)設計範圍的演算法。解析解的部分，則從將v個處理的變異數分為兩類出發，建立其D-最適近似(approximate)設計，並由此發現 (1) 各水準對應之樣本最適配置的上下界並非與水準間不同變異有關，而是與有多少處理之變異相同有關；(2) 即使是變異很大的處理，也必須分配觀察值，始能極大化行列式值。此意味著當v較大時，均分應不失為一有效率(efficient)的設計。至於正合設計，我們僅能得出某一處理特別大或特別小時的D-最適設計，並舉例說明求不出一般解的原因。
除此之外，我們亦求出當三個處理的變異數皆不同時之D-最適近似設計，以及v個處理皆不同時之A-最適近似設計。
至於最適隨機集區設計的建立，我們的重點放在v=2及v=3的情形，並假設集區樣本數(block size)為給定。當v=2時，各集區對應之行列式值不受其他集區的影響，故僅需依照完全隨機設計之所得，將各集區之行列式值分別最佳化，即可得出D-與A-最適設計。值得一提的是，若進一步假設各集區中兩處理變異的比例(>1)皆相同，且集區大小皆相同，則將各處理的「近似設計下最適總和」取最接近的整數，再均分給各集區，其結果未必為最適設計。當v=3時，即使只有2個集區，行列式也十分複雜，我們目前僅能證明當集區內各處理的變異相同時(不同集區之處理變異可不同)，均分給定之集區樣本數為D-最適設計。當集區內各處理的變異不全相同時，我們僅能先以2個集區為例，類比完全隨機設計的性質，舉例猜想當兩集區中處理之變異大小順序相同時，各處理最適樣本配置的多寡亦與變異大小呈反比。由於本研究對處理與集區兩者之效應假設為可加，因此可合理假設集區中處理之變異大小順序相同。 / The problem of finding D- and A-optimal designs for the zero- and one-way elimination of heterogeneity under generalized linear models is considered. Since GLM designs rely on the values of parameters to be estimated, our strategy is to employ the locally optimal designs. For the zero-way elimination model, a theorem-based algorithm is proposed to search for the D-optimal exact designs. A formula for the construction of D-optimal approximate design when values of unknown parameters are split into two, with respective sizes m and v-m, are derived. Analytic solutions provided to the exact counterpart, however, are restricted to the cases when m=1 and m=v-1. An example is given to explain the problem involved.
On the other hand, the upper bound and lower bound of the optimal number of replicates per treatment are proved dependent on m, rather than the unknown parameters. These bounds imply that designs having as equal number of replications for each treatment as possible are efficient in D-optimality.
In addition, a D-optimal approximate design when values of unknown parameters are divided into three groups is also obtained. A closed-form expression for an A-optimal approximate design for comparing arbitrary v treatments is given.
For the one-way elimination model, our focus is on studying the D-optimal designs for v=2 and v=3 with each block size given. The D- and A-optimality for v=2 can be achieved by assigning units proportional to square root of the ratio of two variances, which is larger than 1, to the treatment with smaller variance in each block separately. For v=3, the structure of determinant is much more complicated even for two blocks, and we can only show that, when treatment variances are the same within a block, design having equal number of replicates as possible in each block is a D-optimal block design. Some numerical evidences conjecture that a design satisfying the condition that the number of replicates are inversely proportional to the treatment variances per block is better in terms of D-optimality, as long as the ordering of treatment variances are the same across blocks, which is reasonable for an additive model as we assume.
| Identifer | oai:union.ndltd.org:CHENGCHI/G0933545032 |
| Creators | 何漢葳, Ho, Han Wei |
| Publisher | 國立政治大學 |
| Source Sets | National Chengchi University Libraries |
| Language | 中文 |
| Detected Language | English |
| Type | text |
| Rights | Copyright © nccu library on behalf of the copyright holders |
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