本篇論文中，我們關注的是二元(bivariate)半競爭風險資料(semi-competing risks)的統合分析(meta-analysis)，其中一個事件時間是另一個事件時間的競爭風險。統合分析是收集來自不同獨立研究的資料並總結研究結果的一種統計方法。Rondeau等人(2015)提出了半競爭風險資料統合分析的joint frailty model，其中frailty應用於不同研究之間的異質性(heterogeneity)。他們提出的模型是Cox比例風險模型的延伸，其中加了兩個事件時間之間的條件獨立假設。Emura等人(2017a)利用Copula函數來放寬joint frailty model的條件獨立性假設，並提出了joint frailty-copula model。在joint frailty-copula model中的基線風險函數(baseline hazard function)是透過nonparametric method估計的。在本論文中，我們提出基線風險函數的Weibull分佈和joint frailty-copula model中frailty的Gamma分佈。我們證明Weibull model構成了Gamma frailty分佈的共軛模型，Weibull model給出了邊際動差，生存函數，分位數和平均剩餘壽命的明確表達式。這些數學特性不可能在spline models下推導出來。在點估計部分，使用最大概似估計法來估計模型中的未知參數，並開發其計算程式。在區間估計的部分，提出了兩種不同的構建標準誤和信賴區間的方法，並比較了它們的表現。我們進行模擬研究來驗證所提出方法的準確性。最後，我們使用卵巢癌患者資料來說明所提出的方法。

In this thesis, we are concerned about the meta-analysis of bivariate semi-competing risks data, where one event time is a competing risk for another event time. Meta-analysis is a statistical method that collects data from different independent studies and concludes research results. Rondeau et al. (2015) proposed the joint frailty model for the meta-analysis of semi-competing risks data, where frailty is applied for heterogeneity between different studies. The model they proposed is an extension of the Cox proportional hazard model, in which the assumption of conditional independence between two event times given frailty is imposed. Emura et al. (2017a) used the copula function to relax the conditional independence assumption of the joint frailty model, and proposed the joint frailty-copula model. The baseline hazard functions in the joint frailty-copula model are estimated non-parametrically by splines. In this thesis, we propose the Weibull distribution for baseline hazard functions and the gamma distribution for frailty in the joint frailty-copula model. We show that the Weibull model constitutes a conjugate model for the gamma frailty distribution, and that the Weibull models give explicit expressions for the marginal moments, survival functions, quantiles, and mean residual lifetimes. These mathematical properties are not possible to derive under the spline models. In the point estimation part, the maximum likelihood estimation method is used to estimate the unknown parameters in the model, and its computer programs are developed. In the part of the interval estimation, two different methods for constructing the standard error and confidence intervals are proposed and their performances are compared. We conduct simulation studies to examine the accuracy of the proposed methods. Finally, we use an ovarian cancer patient data to illustrate the proposed method.

Contents
摘要 ...........................................................................................................................................i
Abstract ......................................................................................................................................ii
誌謝辭 .....................................................................................................................................iii
Contents ....................................................................................................................................iv
Chapter 1 Introduction ................................................................................................................1
Chapter 2 Review .......................................................................................................................3
2.1 Univariate Weibull distribution ..................................................................................3
2.2 Review of copulas .......................................................................................................6
Chapter 3 Joint models for meta-analysis ...................................................................................8
3.1 Meta-analytic data .......................................................................................................8
3.2 The joint frailty-copula model .....................................................................................8
3.3 The Weibull joint frailty-copula model .....................................................................10
3.4 Properties ..................................................................................................................10
Chapter 4 Estimation procedure ...............................................................................................19
4.1 Semi-competing risks data ........................................................................................19
4.2 Maximum likelihood inference .................................................................................21
4.3 Interval estimation ....................................................................................................23
Chapter 5 Computation .............................................................................................................26
Chapter 6 Simulation ................................................................................................................28
6.1 Simulation design .....................................................................................................28
6.2 Simulation results .....................................................................................................29
Chapter 7 Data analysis ............................................................................................................33
7.1 The ovarian cancer data .............................................................................................33
7.2 Variable selection .....................................................................................................38
7.3 Fitted results ..............................................................................................................41
7.4 Prognostic prediction.................................................................................................42
Chapter 8 Conclusion and discussion .......................................................................................48
Appendix A ..............................................................................................................................50
Appendix B ..............................................................................................................................63
Appendix C ..............................................................................................................................79
References ................................................................................................................................83

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