結果依賴抽樣設計 (outcome-dependent sampling design, ODS) 是一種已被證實可以提高有效性且降低研究成本的抽樣方法。其利用簡單隨機抽樣以及機率依賴抽樣來取得研究樣本。而現今 ODS 已被發展且調整至多維度資料當中。在最近的文獻中，ODS 已被延伸至抽取擁有多個觀測值的樣本，譬如來自同一病人的多重疾病或是多個家族成員中的某種疾病。然而到目前為止，只有每個集群包含兩個觀測值的情況被考慮過。
在此研究中，我們對於在加速失效模型 (accelerated failure time, AFT) 下將 ODS抽樣方法推廣至更高維度的存活資料中且考慮觀測值之間的相關性結構感到興趣。我們通過進行廣泛的模擬實驗來對比多種抽樣設計以及不同的相關性結構，其中包含當補充樣本數較少的處境。此外，我們還建立最佳化的設計和配置來充分提高有效性。最後我們套用所提出的方法於真實牙齒資料中進行分析。

The outcome-dependent sampling design (ODS) is a sampling design that has been shown to improve efficiency and reduce study cost. The ODS sample consists of not only a simple random sample (SRS) but also some supplemental samples through a probability sampling process which depends on the outcome level. Today, it has been developed to adjust for multivariate data under ODS. In the recent literatures, the multivariate ODS design has been extended to select the samples with multiple observations such as multiple disease outcomes from one patient or any certain disease outcome from family members. However, only the case where each cluster has two observations was discussed so far.
In this thesis, we are interested in extending multivariate outcome-dependent sampling design to higher dimensional failure time data under an accelerated failure time (AFT) model, and consider the correlation structures between the observations. We compare several sampling designs along with different correlation structures by conducting extensive simulation studies, including the situation particularly when the sizes of the supplemental samples are small. In addition, we establish optimal designs and allocations, which can substantially gain more efficiency. We apply our proposed methods to the dental data for analysis in the end.

[1] Buckley, J., & James, I. (1979). Linear regression with censored data. Biometrika, 66, 429-436.
[2] Caplan, D. J., Li, Y., Wang, W., Kang, S., Marchini, L., Cowen, H. J., & Yan, J. (2019). Dental Restoration Longevity among Geriatric and Special Needs Patients. JDR Clinical and Translational Research, 4, 41-48.
[3] Chiou, S. H., Kang, S., Kim, J. & Yan, J. (2014a). Marginal semiparametric multivariate accelerated failure time model with generalized estimating equations. Lifetime Data Analysis, 20, 599-618.
[4] Chiou, S. H., Kang, S., & Yan, J. (2014b). Fitting accelerated failure time model in routine survival analysis with R package aftgee. Journal of Statistical Software, 61, 1-23.
[5] Ding, J., Lu, T. S., Cai, J., & Zhou, H. (2017). Recent progresses in outcome-dependent sampling with failure time data. Lifetime Data Analysis, 23, 57-82.
[6] Higgins, M., Province, M., Heiss, G., Eckfeldt, J., Ellison, R. C., Folsom, A. R., Rao, D. C., Sprafka, J. M., & Williams, R. (1996). NHLBI Family Heart Study: objectives and design. American Journal of Epidemiology, 143, 1219-1228.
[7] Jin, Z., Lin, D. Y., & Ying, Z. (2006). On least-squares regression with censored data. Biometrika, 93, 147-161.
[8] Johnson, L. M., & Strawderman, R. L. (2009). Induced smoothing for the semiparametric accelerated failure time model: asymptotics and extensions to clustered data. Biometrika, 96, 577-590.
[9] Liang, K. Y., & Zeger, S. L. (1986). Longitudinal data analysis using generalized linear models. Biometrika, 73, 13-22.
[10] Lu, T. S., Kang, S., & Zhou, H. (2020). Semiparametric accelerated failure time modeling for multivariate failure times under multivariate outcome-dependent sampling designs. Statistics and Its Interface, 13, 373-383.
[11] Lu, T. S., Longnecker, M. P., & Zhou, H. (2018). Statistical inferences for data from studies conducted with an aggregated multivariate outcome-dependent sample design. Statistics in Medicine, 36, 985-997.
[12] Zhou, H., Weaver, M. A., Qin, J., Longnecker, M. P., & 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.