群試迴歸模型中D-最適設計之理論極限;Theoretical Limits of $D$-Optimal Design in Group Testing Regression Models

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题名:	群試迴歸模型中D-最適設計之理論極限;Theoretical Limits of $D$-Optimal Design in Group Testing Regression Models
作者:	趙鴻儒;Chao, Hung-Ju
贡献者:	統計研究所
关键词:	互補雙對數鏈結函數;D-最適設計;群組試驗;群試迴歸模型;Complementary Log-Log Link Function;D-Optimal Design;Group Testing;Group Testing Regression Model
日期:	2025-08-25
上传时间:	2025-10-17 12:17:14 (UTC+8)
出版者:	國立中央大學
摘要:	在公共衛生及流行病學中，相較於傳統個別試驗，群組試驗透過合併樣本檢測，有助於針對低盛行率疾病之研究能降低成本與誤差風險。在群組試驗中，群組大小與群組中個體自變數的分佈，會顯著影響群試迴歸模型參數估計的有效性。在實務中，研究者僅能將個體樣本分組，無法完全控制自變數於組間及組內的分佈。鑑於此，本論文探討群試迴歸模型在「理想情況」下的最適設計問題，藉由指定自變數之分佈，以達到參數估計變異的理論下界。當群組大小固定且只有一個自變數時，本研究刻劃了使用互補雙對數 (complementary log-log, cloglog) 鏈結函數的群試迴歸模型之D-最適設計的理論特徵。對於多變數的情況，我們利用隨機交換演算法來尋找最適設計。最後，我們探討群組試驗結構對 cloglog 模型下最適設計的影響，並與傳統個別試驗情境進行比較。;In public health and epidemiology, compared to traditional individual testing, group testing offers a cost-effective and error-reducing approach for screening low-prevalence diseases. In group testing, the group size and the distribution of individual covariates both within and between groups can greatly influence the efficiency of parameter estimation in group-testing regression models. In practical applications, researchers are able to assign individual samples into groups but have limited control over the covariate distributions within or across these groups. In light of this, we explore the optimal design problem for group-testing regression models in this thesis, under an idealized framework where covariate distributions are specified to achieve the theoretical lower bound of estimator variance. When the group size is fixed and only a single covariate is involved, we derive the theoretical properties of the D-optimal design for a group-testing regression model using a complementary log-log (cloglog) link function. For models involving multiple covariates, the randomized-exchange algorithm is employed to obtain optimal designs. Finally, we analyze how various group structures impact the optimal designs under the cloglog model and compare these findings with those obtained under the conventional individual-testing setting.
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