二元反應實驗很常出現在在各個領域的研究中，比如說在毒性實驗中反應變數為受測個體的存活或死亡，在火工件實驗中反應變數為火工件是否爆炸等。通常研究者會使用廣義線性模型來配適二元反應變數及解釋變數之相關資料。本論文的第一部分為研究二元反應變數下之廣義線性模型的穩建設計問題。我們將會提出一個將加權機率誤差之極大值最小化的穩健設計準則，稱為WB-最適準則，並在有兩種或多種可能的模型假設下研究其相關的設計問題。
此外，基於科學上的原因或是其特定的反應機率結構，研究者會傾向於使用一般非線性模型來配適一些特定類型的二元反應實驗資料，比如說用來估計某種特性盛行率的群組試驗實驗。本論文的第二部分是研究在試驗的敏感度和特異度都非100%的情況下，群組試驗模型的最適設計問題。當敏感度和特異度都給定或都未知的情形下，我們分別給出了關於全體參數估計的D-最適設計準則以及只著重於盛行率估計的Ds-最適設計準則之下的唯一最適設計結果。

Binary response experiments are widely used in a lot of scientific studies, such as responses with death or survival in a toxicity study, explosion or no reaction in a pyrotechnics experiment. Typically, a generalized linear model is adopted to fit the response probability curve under binary response experiments. In the first part of this dissertation, generalized linear models for binary response data are considered. The main purpose is to investigate robust design problems for estimation of the response probability curve with model uncertainty consideration. A minimax type of model robust design criterion, called WB- optimum in short is proposed, based on minimization of the maximum of the weighted squared probability bias function under two rival models. The corresponding design issues are investigated and results under the above design criterion for given rival models with several commonly seen symmetric links are presented.
On the other hand, general nonlinear models for binary response experiments are adopted because of some scientific reasons or their inherent structures, for example, estimating a rare prevalence through a group-testing procedure. In the second part of this dissertation, optimal design problems for group-testing models are investigated, while the sensitivity and specificity of the test of a certain trial are not 100%. The D- and Ds-optimal design problems are studied, under two situations that the sensitivity and specificity are specified or need to be estimated.

摘要 i
Abstract ii
1 Introduction 1
2 Preliminaries 4
2.1 Generalized linear models . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Group-testing model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Model robust designs for estimation of the response probability 8
3.1 The WB-optimal criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 WB-optimal models and designs . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3 The choices of compromise weight functions . . . . . . . . . . . . . . . . . 18
4 Design comparisons 20
4.1 Performance comparisons of WB- and D-optimal designs: logit-probit case 21
4.2 Performance comparisons of WBD- and D-optimal designs: logit-double exponential case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.3 Performance comparisons of WB- and TLR-optimal designs: logit-probit case 24
4.4 Sensitivity analysis for misspecification of the parameter vector . . . . . . . 26
4.5 More than two candidate links . . . . . . . . . . . . . . . . . . . . . . . . . 27
5 Optimal group-testing designs with testing errors 29
5.1 The D-optimal designs for the prevalence with specified testing error rates 29
5.2 The D-optimal designs for the prevalence and testing error rates . . . . . . 32
5.3 The Ds-optimal designs for the prevalence with unknown testing error rates 35
6 Discussions and future works 38
Bibliography 42
A Proofs of Theorems 45
A.1 Proof of Theorem 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
A.2 Proof of Theorem 3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
A.3 Proof of Theorem 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
A.4 Proof of Theorem 5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
A.5 Proof of Theorem 5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
A.6 Proof of Theorem 5.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Abdelbasit, K. M. and Plackett, R. L. (1983). Experimental design for binary data.
Journal of the American Statistical Association, 78:90-98.
Atkinson, A. C. (2008). DT-optimum designs for model discrimination and parameter
estimation. Journal of Statistical Planning and Inference, 138:56-64.
Atkinson, A. C. and Bogacka, B. (1997). Compound D- and Ds-optimum designs for
determining the order of a chemical reaction. Technometrics, 39:347-356.
Atkinson, A. C., Donev, A. N., and Tobias, R. D. (2007). Optimum experimental designs,
with SAS. Oxford University Press, Oxford.
Atkinson, A. C. and Fedorov, V. V. (1975). Optimal design: Experiments for discriminating
between several models. Biometrika, 62:289-304.
Biedermann, S., Dette, H., and Pepelyshev, A. (2006). Some robust design strategies
for percentile estimation in binary response models. Canadian Journal of Statistics,
34:603-622.
Biedermann, S., Dette, H., and Zhu, W. (2007). Compound optimal designs for percentile
estimation in dose-response models with restricted design intervals. Journal of
Statistical Planning and Inference, 137:3838-3847.
Chambers, E. A. and Cox, D. R. (1967). Discrimination between alternative binary
response models. Biometrika, 54:573-578.
Cook, R. D. and Wong, W. K. (1994). On the equivalence of constrained and compound
optimal designs. Journal of the American Statistical Association, 89:687-692.
Czado, C. and Santner, T. J. (1992). The e ect of link misspeci cation on binary regression
inference. Journal of Statistical Planning and Inference, 33:213-231.
Dette, H. (1990). A generalization of D- and D1-optimal designs in polynomial regression.
The Annals of Statistics, 18:1784-1804.
Dette, H. and Sahm, M. (1998). Minimax optimal designs in nonlinear regression models.
Statistica Sinica, 8:1249-1264.
Dette, H. and Studden, W. J. (1994). Optimal designs with respect to Elfving's partial
minimax criterion in polynomial regression. Annals of the Institute of Statistical
Mathematics, 46:389-403.
Dorfman, R. (1943). The detection of defective members of large populations. The Annals
of Mathematical Statistics, 14:436-440.
Fedorov, V. V. (1972). Theory of Optimal Experiments. Academic Press, New York.
Fuh, C. D., Lee, J. S., and Liaw, C. M. (2003). The design aspect of the bruceton test
for pyrotechnics sensitivity analysis. Journal of Data Science, 1:83-101.
Huang, M.-N. L. and Studden, W. J. (1988). Model robust extrapolation designs. Journal
of Statistical Planning and Inference, 18:1-24.
Hughes-Oliver, J. M. (2006). Pooling experiments for blood screening and drug discovery.
In Dean, A. and Lewis, S., editors, Screening: Methods for Experimentation in Industry,
Drug Discovery, and Genetics, pages 48-68. Springer, New York.
Hughes-Oliver, J. M. and Rosenberger, W. F. (2000). E cient estimation of the prevalence
of multiple traits. Biometrika, 87:315-327.
Hughes-Oliver, J. M. and Swallow, W. H. (1994). A two-stage adaptive group-testing
procedure for estimating small proportions. Journal of the American Statistical Asso-
ciation, 89:982-993.
Karlin, S. and Studden, W. J. (1966). Tchebyche Systems with Applications in Analysis
and Statistics. Wiley, New York.
Liu, A., Liu, C., Zhang, Z., and Albert, P. S. (2012). Optimality of group testing in the
presence of misclassi cation. Biometrika, 99:245-251.
Minkin, S. (1987). Optimal designs for binary data. Journal of the American Statistical
Association, 82:1098-1103.
Muller, W. G. and Ponce de Leon, A. C. M. (1996). Discrimination between two binary
data models: Sequentially designed experiments. Journal of Statistical Computation
and Simulation, 55:87-100.
Sitter, R. R. and Fainaru, I. (1997). Optimal designs for the logit and probit models for
binary data. The Canadian Journal of Statistics, 25:175-190.
Sitter, R. R. and Wu, C. F. J. (1993). Optimal designs for binary response experiments:
Fieller, d, and a criteria. Scandinavian Journal of Statistics, 20:329-341.
Tu, X. M., Litvak, E., and Pagano, M. (1995). On the informativeness and accuracy of
pooled testing in estimating prevalence of a rare disease: Application to HIV screening.
Biometrika, 82:287-297.
Waterhouse, T. H., Woods, D. C., Eccleston, J. A., and Lewis, S. M. (2008). Design selection
criteria for discrimination/estimation for nested models and a binomial response.
Journal of Statistical Planning and Inference, 138:132-144.
White, H. (1982). Maximum likelihood estimation of misspeci ed models. Econometrica,
50:1-25.
Wu, C. F. J. (1988). Optimal design for percentile estimation of a quantal response curve.
In Dodge, J., Fedorov, V. V., and Wynn, H. P., editors, Optimal Design and Analysis
of Experiments, pages 213-223. North Holland, Amsterdam.
Yanagisawa, Y. (1988). Designs for discrimination between binary response models. Jour-
nal of Statistical Planning and Inference, 19:31-41.
Yanagisawa, Y. (1990). Designs for discrimination between bivariate binary response
models. Biometrical Journal, 32:25-34.