結果依賴採樣設計(Outcome-dependent sampling, ODS) 在很多流行病學或大型資料庫的研究中已經被證明是具有成本效益的抽樣方式且廣為採用，而多維度依賴採樣設計(multivariate outcome-dependent sampling design, MODS) 則是將ODS 延伸到群聚或長期追蹤資料，尤其在一位實驗對象有多於兩個觀測值時，存在著無法忽略的相關性。過去的文獻中，模擬設定的取樣大多來自母體樣本數很大，較少討論到若母體資料稀少的情況該如何抽樣，因此我們想更進一步討論此設計在這樣的情況是否能有效的估計。本研究的目標是針對在小樣本的情況下，我們進行大量的模擬，討論此設計在不同參數設定下目標參數的表現以及在有限的母體下，找出最佳配置的選取。最後，我們也將用此模型分析來自Busselton Health Survey 的實際資料。

In most epidemiological or large database studies, the outcome-dependent sampling(ODS) design has been shown to enhance study efficiency and reduce the cost. The multivariate ODS (MODS) design is a further generalization for clustered or longitudinal data sampled under the ODS design. When the case is that one has at least two responses or there is more than one observation in the same cluster, the correlation between the responses can not be ignored. In the previous studies, most random samples were selected from large underlying cohorts. Therefore, we aim to consider relatively smaller underlying populations and see if the efficiency of the estimators still holds. Through conducting extensive simulation studies, we discuss the optimal allocation and design of the MODS sample under various settings. We also apply our proposed approach to
analyze a real data set from the Busselton Health Survey Study.

Shan, G., Wilding, G. E., Hutson, A. D., & Gerstenberger, S. (2016). Optimal
adaptive two-stage designs for early phase II clinical trials. Statistics in medicine, 35(8), 1257–1266.
Yu, J., Liu, Y., Sandler, D. P., & Zhou, H. (2015). Statistical inference for the additive hazards model under outcome-dependent sampling. The Canadian journal of statistics = Revue canadienne de statistique, 43(3), 436–453.
Longnecker, M. P., Hoffman, H. J., Klebanoff, M. A., Brock, J. W., Zhou, H.,Needham, L., Adera, T., Guo, X., & Gray, K. A. (2004). In utero exposure to polychlorinated biphenyls and sensorineural hearing loss in 8-year-old children. Neurotoxicology and teratology, 26(5), 629–637.
Knuiman, M. W., Divitini, M. L., Olynyk, J. K., Cullen, D. J., & Bartholomew, H.C. (2003). Serum ferritin and cardiovascular disease: a 17-year follow-up study in Busselton, Western Australia. American journal of epidemiology, 158(2), 144–149.
Owen, A. B. (1990). Empirical likelihood Ratio Confidence Regions. Annals of Statistics 18, 90-120.
Prentice, R. (1986). A Case-Cohort Design for Epidemiologic Cohort Studies and Disease Prevention Trials. Biometrika, 73(1), 1-11. doi:10.2307/2336266
Qin, J. and Lawless, J. F. (1994). Empirical likelihood and general estimating equations. Annals of Statistics 22, 300-325.
Simon R. (1989). Optimal two-stage designs for phase II clinical trials. Controlled clinical trials, 10(1), 1–10.
Kang, S., & Cai, J. (2009). Marginal hazards model for case-cohort studies with multiple disease outcomes. Biometrika, 96(4), 887–901.
Lu, T. S., Longnecker, M. P., & Zhou, H. (2017). Statistical inferences for data from studies conducted with an aggregated multivariate outcome-dependent sample design. Statistics in medicine, 36(6), 985–997.
Zhou, H., Weaver, M. A., Qin, J., Longnecker, M. P., and Wang, M. C. (2002). A semi- parametric empirical likelihood method for data from an outcome-dependent sampling scheme with a continuous outcome. Biometrics 58, 413-421.
張雪愉（2011）。連續型資料的結果依賴採樣設計之配置選擇。國立臺灣師範大學數學系碩士論文。