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題名 廣義線性模式下處理比較之最適設計
Optimal Designs for Treatment Comparisons under Generalized Linear Models
作者 何漢葳
Ho, Han Wei
貢獻者 丁兆平
Ting, Chao Ping
何漢葳
Ho, Han Wei
關鍵詞 廣義線性模式
集區設計
D-最適
A-最適
近似與正合設計
穩健性
Generalized linear models (GLMs)
block designs
D-optimality
A-optimality
approximate and exact designs
robustness
日期 2013
上傳時間 2-May-2016 13:49:13 (UTC+8)
摘要 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.
參考文獻 Agresti, A. (2012). Categorical Data Analysis, 3rd edition. Wiley, New York.
Atkinson, A. C., Donev, A. N. and Tobias, R. D. (2007). Optimum Experimental Designs, With SAS. Oxford University Press.
Atkinson, A. C., Fedorov, V. V., Herzberg, A. M. and Zhang, R. (2012). Elemental information matrices and optimal experimental design for generalized regression models. Journal of Statistical Planning and Inference, http://dx.doi.org/10.1016/j.jspi.2012.09.012
Balinski, M. L. and Young, H. P. (2001). Fair representation: meeting the ideal of one man, one vote, 2nd edition. Brookings Institution Press, Washington, D.C.
Brooks, B. P. (2006). The coefficients of the characteristic polynomial in terms of the eigenvalues and the elements of an nXn matrix. Applied Mathematics Letters 19, 511-515.
Chaloner, K., and Larntz, K. (1989). Optimal Bayesian design applied to logistic regression experiments. Journal of Statistical Planning and Inference 21, 191-208.
Cheng, C.-S., Majumdar D., Stufken, J. and Ture T. E. (1988). Optimal step-type designs for comparing test treatments with a control. Journal of the American Statistical Association 83, 477-482.
Dror, H. A. and Steinberg, D. M. (2006). Robust experimental design for multivariate generalized linear models. Technometrics 48, 520-529.
Ford, I., Torsney, B. and Wu, C. F. J. (1992). The use of a canonical form in the construction of locally optimal designs for non-linear problems. Journal of the Royal Statistical Society. Series B 54, 569-583.
Hedayat, A. S. and Majumdar, D. (1984). A-optimal incomplete block designs for control-test treatment comparisons. Technometrics 26, 363-370.
Hedayat, A. S. and Majumdar, D. (1985). Families of A-optimal block designs for comparing test treatments with a control. Annals of Statistics, 13, 757-767.
Khuri, A.I., Mukherjee, B., Sinha, B. K. and Ghosh, M.(2006). Design issues for generalized linear models: a review. Statistical Science 3, 376-399.
Kiefer, J. C. (1971). The role of symmetric and approximation in exact design optimality. In Statistical Decision Theory and Related Topics, Proceedings of a Symposium, Purdue University 1971, Ed. S. S. Gupta and J. Yackel. pp. 109-118. Academic Press, New York.
Mathew, T., and Sinha, B. K. (2001). Optimal designs for binary data under logistic regression. Journal of Statistical Planning and Inference 93, 295-307.
McCulloch, C. E., Searle, S. R. and Neuhaus J. M. (2008). Generalized, Linear, and Mixed Models, 2nd edition. Wiley, New York.
McGree, J. M., and Ecclestona, J. A. (2012). Robust Designs for Poisson Regression Models. Technometrics 54, 64-72.
Nelder, J. A., and Wedderburn, R. W. M. (1972). Generalized linear models. Journal of the Royal Statistical Society: Series A 135, 370-384.
Pukelsheim, F. and Rieder, S. (1992). Efficient rounding of approximate designs. Biometrika 79, 763-770.
Pukelsheim, F. (2006). Optimal Design of Experiments. Society for Industrial and Applied Mathematics, Philadelphia.
Rodriguez-Torreblanca, C. and Rodriguez-Diaz, J. M. (2007). Locally D- and c-optimal designs for Poisson and negative binomial regression models. Metrika 66, 161-172.
Russell, K. G., Wood, D. C., Eccleston, J. A. (2009). D-optimal designs for Poisson regression models. Statistica Sinica 19, 721-730.
Sitter, R. R. and Torsney, B. (1995). Optimal designs for binary response experiments with two design variables. Statistica Sinica 5, 405-419.
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.
Wang, Y., Smith, E. P. and Ye, K. (2006). Sequential designs for a Poisson regression model. Journal of Statistical Planning and Inference 136, 3187-3202.
Waterhouse, T. H., Woods, D. C., Eccleston, J. A. and Lewis, S. M. (2007). Design selection criteria for discrimination/estimation for nested models and a binomial response. Journal of Statistical Planning and Inference 138, 132-144.
Woods, D. C., Lewis, S. M., Eccleston, J. A., and Russell, K. G. (2006). Designs for generalized linear models with several variables and model uncertainty. Technometrics 48, 284-292.
Woods, D.C. and Lewis, S.M. (2011). Continuous optimal designs for generalized linear models under model uncertainty. Journal of Statistical Theory and Practice 5, 137-145.
Yang, J., Mandal, A., and Majumdar, D. (2012). Optimal designs for two-level factorial experiments with binary response. Statistica Sinica 22, 885-907.
Yang, M. (2008). A-optimal designs for generalized linear model with two parameters. Journal of Statistical Planning and Inference, 138, 624-641.
Yang, M. and Stufken J. (2009). Support points of locally optimal designs for nonlinear models with two parameters. The Annals of Statistics 37, 518-541.
Yang, M., Zhang, B. and Huang, S. (2011). Optimal designs for binary response experiments with multiple variables. Statistica Sinica 21, 1415-1430.
描述 博士
國立政治大學
統計學系
93354503
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0933545032
資料類型 thesis
dc.contributor.advisor 丁兆平zh_TW
dc.contributor.advisor Ting, Chao Pingen_US
dc.contributor.author (Authors) 何漢葳zh_TW
dc.contributor.author (Authors) Ho, Han Weien_US
dc.creator (作者) 何漢葳zh_TW
dc.creator (作者) Ho, Han Weien_US
dc.date (日期) 2013en_US
dc.date.accessioned 2-May-2016 13:49:13 (UTC+8)-
dc.date.available 2-May-2016 13:49:13 (UTC+8)-
dc.date.issued (上傳時間) 2-May-2016 13:49:13 (UTC+8)-
dc.identifier (Other Identifiers) G0933545032en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/89053-
dc.description (描述) 博士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 統計學系zh_TW
dc.description (描述) 93354503zh_TW
dc.description.abstract (摘要) 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。		zh_TW
dc.description.abstract (摘要) 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.		en_US
dc.description.tableofcontents 1 Introduction 1
2 Information Matrices of GLMs 10
3 Completely Randomized Designs 15
3.1 D-optimal Designs 15
3.2 D-optimal Designs for c1=···=cm and cm+1=···=cv, v≥3 22
3.3 D-optimal Designs for v=2 40
3.4 D-optimal Designs for c1=···= cm, cm+1=···=cm+k, cm+k+1=···=cv 41
3.5 Robustness of D-optimal Designs 43
3.6 A-optimal Designs 45
4 Randomized Block Designs 47
4.1 D-optimal Designs for v=2 49
4.2 D-optimal Designs for v=3 51
4.3 A-optimal Designs 53
5 Conclusion and Future Research 54		zh_TW
dc.format.extent 889655 bytes-
dc.format.mimetype application/pdf-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0933545032en_US
dc.subject (關鍵詞) 廣義線性模式zh_TW
dc.subject (關鍵詞) 集區設計zh_TW
dc.subject (關鍵詞) D-最適zh_TW
dc.subject (關鍵詞) A-最適zh_TW
dc.subject (關鍵詞) 近似與正合設計zh_TW
dc.subject (關鍵詞) 穩健性zh_TW
dc.subject (關鍵詞) Generalized linear models (GLMs)en_US
dc.subject (關鍵詞) block designsen_US
dc.subject (關鍵詞) D-optimalityen_US
dc.subject (關鍵詞) A-optimalityen_US
dc.subject (關鍵詞) approximate and exact designsen_US
dc.subject (關鍵詞) robustnessen_US
dc.title (題名) 廣義線性模式下處理比較之最適設計zh_TW
dc.title (題名) Optimal Designs for Treatment Comparisons under Generalized Linear Modelsen_US
dc.type (資料類型) thesisen_US
dc.relation.reference (參考文獻) Agresti, A. (2012). Categorical Data Analysis, 3rd edition. Wiley, New York.
Atkinson, A. C., Donev, A. N. and Tobias, R. D. (2007). Optimum Experimental Designs, With SAS. Oxford University Press.
Atkinson, A. C., Fedorov, V. V., Herzberg, A. M. and Zhang, R. (2012). Elemental information matrices and optimal experimental design for generalized regression models. Journal of Statistical Planning and Inference, http://dx.doi.org/10.1016/j.jspi.2012.09.012
Balinski, M. L. and Young, H. P. (2001). Fair representation: meeting the ideal of one man, one vote, 2nd edition. Brookings Institution Press, Washington, D.C.
Brooks, B. P. (2006). The coefficients of the characteristic polynomial in terms of the eigenvalues and the elements of an nXn matrix. Applied Mathematics Letters 19, 511-515.
Chaloner, K., and Larntz, K. (1989). Optimal Bayesian design applied to logistic regression experiments. Journal of Statistical Planning and Inference 21, 191-208.
Cheng, C.-S., Majumdar D., Stufken, J. and Ture T. E. (1988). Optimal step-type designs for comparing test treatments with a control. Journal of the American Statistical Association 83, 477-482.
Dror, H. A. and Steinberg, D. M. (2006). Robust experimental design for multivariate generalized linear models. Technometrics 48, 520-529.
Ford, I., Torsney, B. and Wu, C. F. J. (1992). The use of a canonical form in the construction of locally optimal designs for non-linear problems. Journal of the Royal Statistical Society. Series B 54, 569-583.
Hedayat, A. S. and Majumdar, D. (1984). A-optimal incomplete block designs for control-test treatment comparisons. Technometrics 26, 363-370.
Hedayat, A. S. and Majumdar, D. (1985). Families of A-optimal block designs for comparing test treatments with a control. Annals of Statistics, 13, 757-767.
Khuri, A.I., Mukherjee, B., Sinha, B. K. and Ghosh, M.(2006). Design issues for generalized linear models: a review. Statistical Science 3, 376-399.
Kiefer, J. C. (1971). The role of symmetric and approximation in exact design optimality. In Statistical Decision Theory and Related Topics, Proceedings of a Symposium, Purdue University 1971, Ed. S. S. Gupta and J. Yackel. pp. 109-118. Academic Press, New York.
Mathew, T., and Sinha, B. K. (2001). Optimal designs for binary data under logistic regression. Journal of Statistical Planning and Inference 93, 295-307.
McCulloch, C. E., Searle, S. R. and Neuhaus J. M. (2008). Generalized, Linear, and Mixed Models, 2nd edition. Wiley, New York.
McGree, J. M., and Ecclestona, J. A. (2012). Robust Designs for Poisson Regression Models. Technometrics 54, 64-72.
Nelder, J. A., and Wedderburn, R. W. M. (1972). Generalized linear models. Journal of the Royal Statistical Society: Series A 135, 370-384.
Pukelsheim, F. and Rieder, S. (1992). Efficient rounding of approximate designs. Biometrika 79, 763-770.
Pukelsheim, F. (2006). Optimal Design of Experiments. Society for Industrial and Applied Mathematics, Philadelphia.
Rodriguez-Torreblanca, C. and Rodriguez-Diaz, J. M. (2007). Locally D- and c-optimal designs for Poisson and negative binomial regression models. Metrika 66, 161-172.
Russell, K. G., Wood, D. C., Eccleston, J. A. (2009). D-optimal designs for Poisson regression models. Statistica Sinica 19, 721-730.
Sitter, R. R. and Torsney, B. (1995). Optimal designs for binary response experiments with two design variables. Statistica Sinica 5, 405-419.
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.
Wang, Y., Smith, E. P. and Ye, K. (2006). Sequential designs for a Poisson regression model. Journal of Statistical Planning and Inference 136, 3187-3202.
Waterhouse, T. H., Woods, D. C., Eccleston, J. A. and Lewis, S. M. (2007). Design selection criteria for discrimination/estimation for nested models and a binomial response. Journal of Statistical Planning and Inference 138, 132-144.
Woods, D. C., Lewis, S. M., Eccleston, J. A., and Russell, K. G. (2006). Designs for generalized linear models with several variables and model uncertainty. Technometrics 48, 284-292.
Woods, D.C. and Lewis, S.M. (2011). Continuous optimal designs for generalized linear models under model uncertainty. Journal of Statistical Theory and Practice 5, 137-145.
Yang, J., Mandal, A., and Majumdar, D. (2012). Optimal designs for two-level factorial experiments with binary response. Statistica Sinica 22, 885-907.
Yang, M. (2008). A-optimal designs for generalized linear model with two parameters. Journal of Statistical Planning and Inference, 138, 624-641.
Yang, M. and Stufken J. (2009). Support points of locally optimal designs for nonlinear models with two parameters. The Annals of Statistics 37, 518-541.
Yang, M., Zhang, B. and Huang, S. (2011). Optimal designs for binary response experiments with multiple variables. Statistica Sinica 21, 1415-1430.		zh_TW