現今的實驗無論是工業、化學乃至於生理學等方面，常同時包含了定性及定量的可控因子，而針對這樣的實驗其最適設計也比只含有定量因子的實驗更為複雜。對於定量因子，在反應曲面方法論（response surface methodology）中，其取值的範圍（稱為設計空間）往往決定於實驗者所感興趣及關心的範圍，有別於常用的立方形的設計空間，當實驗者對於反應值的推估在空間中的不同方向上的因子設計，都有相同程度的重視時，採用球形的設計空間是相當適合的。對於模型的考量，二階反應曲面模型常用來做為近似曲面，因此對於定量因子，本論文考慮其完整的二次效應：包含線性效應、二因子交互作用及二次曲線效應，並進一步探討當定量因子效應與定性因子效應有交互作用的情形時之Ｄ型最適設計，其中包含了近似最適設計（approximate optimal designs）及正合最適設計（exact optimal designs）。
論文中首先探討在模型中的定性與定量因子無交互作用之假設下，且定量因子取值在圓形設計空間中的正合Ｄ型最適設計。針對只有定量因子的二階反應曲面模型及含有一個兩水準的定性因子之反應曲面模型，文中證明了正多邊形結構的設計為正合Ｄ型最適設計。當定性因子具有多個水準時，在本研究中提出了一個架構方法，能在以正多邊形為基礎的設計中得到正合Ｄ型最適設計。當總實驗個數增加時，必然會有更多的實驗個數在正合Ｄ型最適設計的正多邊形結構上，因此在論文裡也給出具有最少支撐點（minimal support points）的最適設計，結果顯示，不論總實驗個數為多少，正合Ｄ型最適設計在每個定性因子的水準中的支撐點不會超過13個。
當考慮模型中的定量因子與定性因子有交互作用時，特別是以下的定量因子效應在不同的定性因子水準中有不同的表現時：線性效應、線性效應與二因子交互作用、線性效應與二次效應，近似Ｄ型最適設計可以由中央合成設計（central composite design）的支撐點架構而成。在不同的模型考慮之下，論文中也比較了最適設計間的穏健性。
由於正合設計的Ｄ型最適性當因子個數增加時，即使設計的結構可以被找
出，仍往往無法給出理論證明，因此有許多的文獻提出用演算法在有限個候選設
計中找出表現最佳的，或是根據近似Ｄ型最適設計的權重將實驗單位分配到對應
的支撐點上而得到正合設計。在此，由於球面上連續型均勻設計（continuous
uniform design）再加上適當比例的球體中心點所構成的設計可證明為一連續型的
Ｄ型最適設計，根據此特殊的結構，將Pukelsheim 和Rieder（1992）所提出的高
效進位法（Efficient rounding method）在最適設計上的應用做適當的調整，並針
對二維的圓形設計空間進行研究，結果顯示用此修正方法得出的正合設計都有很
好的Ｄ型最適性。
以上所討論的定性因子均在固定效應假設之下，但有許多的情形，定性因子
效應必須考慮為具備隨機性，在隨機區集效應的假設之下，且考慮區集大小為2
時，本論文也探討了二階反應曲面模型在圓形設計空間裡的最適設計，我們發現
近似Ｄ型最適設計具有一些有趣的結果，特別是其結構也與正多邊形有密切的關
聯。

Experiments with both quantitative and qualitative factors always complicate the selections of experimental settings and the statistical analysis for data. Response surface methodology (RSM) provides the systematic procedures such as the steepest ascent method to develop and improve the response models through the optimal settings of quantitative factors. However the sequential method lacks of exploring the direction of the maximum increase in the response among the qualitative levels. In this dissertation the optimal designs for experiments with both qualitative and quantitative factors are investigated. Focused on the second-order response surface model for quantitative factors, which is widely used in RSM as a good approximation for the true response surface, the approximate and exact D-optimal designs are proposed for the model containing the qualitative effects. On spherical design regions, the D-optimal designs have particular structures for considering the qualitative effects to be fixed or random.
In this study, the exact D-optimal designs for a second-order response surface model on a circular design region with qualitative factors are proposed. For this model, the interactions between the quantitative and qualitative factors are assumed to be negligible. Based on this design region, an exact D-optimal design with regular polygon structure is made up according to the remainder terms of the numbers of experimental trials at each qualitative levels divided by 6. The complete proofs of exact D-optimality for models including two quantitative factors and one 2-level qualitative factor are presented as well as those for a model with only quantitative factors. When the qualitative factor has more than 2 levels, a method is proposed for constructing exact designs based on the polygonal structure with high efficiency. Furthermore, a procedure for minimizing the number of support points for the quantitative factors of exact D-optimal designs is also proposed for practical consideration. There are no more than 13 support points for the quantitative factors at an individual qualitative level.
When the effects between the quantitative and qualitative factors are taken into consideration, approximate D-optimal designs are investigated for models in which the qualitative effects interact with, respectively, the linear quantitative effects, or the linear effects and 2-factor interactions of the quantitative factors or quadratic effects of the quantitative factors. It is shown that, at each qualitative level, the corresponding D-optimal design consists of three portions as a central composite design but with different weights on the cube portion, star portion and center points. Central composite design (CCD) is widely applied in many fields to construct a second-order response surface model with quantitative factors to help to increase the precision of the estimated model. A chemical study is illustrated to show that the effects of the qualitative factor interacts with 2-factor interactions of the quantitative factors are important but absent in a second-order model including a qualitative factor treated as a coded variable.
The verification of the D-optimality for exact designs has become more and more intricate when the qualitative levels or the number of quantitative factors increase, even when the patterns of the exact optimal designs have been speculated. The efficient rounding method proposed by Pukelsheim and Rieder (1992) is a model-free approach and it generates an exact design by apportioning the number of trials on the same support points of a given design. For constructing the exact designs with high efficiencies, a modified efficient rounding method is proposed and is based on the polygonal structure of the approximate D-optimal design on a circular design region. This modification is still based on the same rounding approach by apportioning the number of trials to the concentric circles where the support points of the given design are standing on. Then a regular polygon design will be assigned on the circles by the apportionments. For illustration, the exact designs for a third-order response surface model with qualitative factors are presented as well as those for the second-order model. The results show that nearly D-optimal designs are obtained by the modified procedure and the improvement in D-efficiency is very significant.
When the factors with the levels selected randomly from a population, they are treated as with random effects. Especially for the qualitative effects caused by the experimental units that the experimenter is not interested in, one should consider the model with random block effects. In this model, the observations on the same unit are assumed to be correlated and they are uncorrelated between different units. Then the mean response surface is still considered as second-order for quantitative factors but the covariance matrix of the observations is different from the identity matrix. In the fourth part of this dissertation, the locally D-optimal designs on a circular design region are proposed for given the value of the correlations. These optimal designs with the structures based on the regular polygons are similar to the D-optimal designs for the uncorrelated model.

1 Introduction 1
1.1 Experiments with qualitative and quantitative factors 2
1.1.1 Example 1: solubility of fluticasone propionate 2
1.1.2 Example 2: removal of sulfur dioxide 2
1.2 Model descriptions and literature reviews 3
1.2.1 Fixed effect models 4
1.2.2 Random block-effects model 8
1.3 Preliminaries 9
1.3.1 Information matrix for fixed effect models 9
1.3.2 Polygonal designs on unit circle 11
1.3.3 Central composite design on B_k(k) 13
2 Model without interactions between the quantitative and qualitative factors 17
2.1 Exact D-optimal designs for the second-order response surface model 18
2.1.1 Exact D-optimal designs 18
2.1.2 Minimal numbers of support points 19
2.2 Designs for models with both quantitative and qualitative factors 24
2.2.1 Exact D-optimal designs for J = 2 25
2.2.2 Construction method and results for J ≥ 3 29
2.3 Discussion 31
3 Model with interactions between the quantitative and qualitative factors 35
3.1 Approximate D-optimal designs 39
3.2 Model robust design 43
3.3 Discussion 44
4 Efficient designs from rounding method 47
4.1 Efficient rounding designs on X_J × B_2(1) 48
4.1.1 Modified efficient rounding procedure 48
4.1.2 Exact D-optimal designs for the third-order response surface model on B_2(1) 50
4.1.3 Efficient rounding designs for the 3rd-order response surface model 56
4.2 Discussion 60
5 D-optimal designs in the presence of random block effects 63
5.1 Information matrix 64
5.2 Optimal designs for block size two on B_2(2) 66
5.3 Discussion 74
Appendix A 75
A.1 Fluticasone propionate solubility 75
A.2 Sulfur dioxide (SO2) removal 77
Appendix B 79
B.1 Proofs for Chapter 2 79
B.1.1 Proposition 16 79
B.1.2 Proof of Lemma 3 81
B.1.3 Proof of Lemma 7 82
B.1.4 Proof of Theorem 8 85
B.1.5 Proof of Theorem 9 87
B.2 Proofs for Chapter 3 88
B.2.1 Proof of Lemma 12 88
B.2.2 Proof of Theorem 13 89
References 91

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