在存活分析中，競爭風險是需要考量的重要因素，其中子分佈函數指存在競爭風險下一段時間內發生感興趣事件的機率，而如何更有效估算此函數的共變量效應，一直以來是學者們關心的議題，也是本篇論文所研究的重點。

Fine和Gray (1999) 針對子分佈提出比例危險模型的假設方法，已成為最常見處理競爭風險的迴歸分析，不過其估計子分佈比例危險模型時需對設限分佈建模及估計。另外，Mao和Lin (2017) 提出利用無母數最大概似估計法作為估計子分佈比例危險模型的方式，其方法不需建模和估計設限分佈，但需對所有競爭風險事件進行建模，在此Mao和Lin 假設所有競爭風險事件都服從子分佈比例危險模型，然而我們注意到此假設並不能套用到真實資料，這個問題也反映在Mao和Lin的程式計算上產生收斂問題的現象中

在本文中，我們提出在子分佈比例危險模型下一種新的最大概似方法，我們只對部分而不是所有的競爭風險事件建模子分佈比例危險模型，其餘的競爭風險事件時間則利用給定這些事件發生下的條件機率分佈進行建模。我們認為這種方法對於處理競爭風險是很自然的，並且在真實世界中此模型假設是可以成立的。模擬結果也顯示，我們提出的方法在計算上總是收斂，且利用最大概似估計法估計子分佈比例危險模型的參數結果具有良好的統計性質；特別是即使在剩餘競爭風險事件的條件機率分佈被假設錯誤下，估計出來的偏差和有效性與Fine和Gary的方法相當。

In the survival analysis, the competing risks are important issues to be addressed. The subdistribution function refers to the probability of an event occurring by some time point in the presence of competing risks. Valid and efficient estimation of this function, as well as the covariate effects on this function, has been a topic of concern in literature and is the focus of this thesis.

Fine and Gray (1999) proposed the proportional hazards (PH) model for the subdistribution, which has become a popular practice for regression analysis of competing risks. The Fine and Gray method for estimating the subdistribution PH model requires modeling and estimating the censoring distribution. Alternatively, Mao and Lin (2017) proposed the non-parametric maximum likelihood approach to the subdistribution PH model (or the Fine-Gray model), which does not require modeling and estimating the censoring distribution, but requires modeling all the competing risks events. Mao and Lin (2017) assume all the competing risks events follow the Fine-Gray model. However, it is noticed that such an assumption cannot hold in actual data. This issue is also reflected in the phenomenon that the Mao and Lin’s procedure suffers from the convergence problem in computation.

In this thesis, we propose a new maximum likelihood approach to the Fine-Gray model. Instead of assuming that all the competing risks follow the Fine-Gray model, we propose only modeling parts of the competing risks events by the Fine-Gray subdistribution model, while modeling all the remaining competing risks events through the conditional event-time distribution given the probability of occurring these remaining events. We believe that this formulation is natural to deal with competing risks events, and the modeling assumption can be satisfied in real data. Simulation results show that, the proposed method always converges in computation, and the resulting maximum likelihood estimator for the parameters in the subdistribution PH model has nice statistical properties. In particular, the bias and the efficiency of the proposed estimator are comparable to those of the Fine –Gray method even when the model for the conditional distribution of the remaining competing risks events is misspecified.

Cox, D.R. (1972), “Regression Models and Life Tables” (with discussion), J.R. Statist. Soc., Ser. B, 34, 187-220.

Chen, Y.H. (2010), “Semiparametric Marginal Regression Analysis for Dependent Competing Risks under an Assumed Copula”, Journal of the Royal Statistical Society, 72, 235-251.

Chang, I.S., Hsiung, C.A., Wen, C.C., Wu, Y.J., Yang, C.C. (2007), “Non-parametric Maximum-Likelihood Estimation in a Semiparametric Mixture Model for Competing-Risks Data”, Scand. J. Statist., 34, 870-895.

Donoghoe, M.K., Gebski, V. (2017) “The importance of censoring in competing risks analysis of the subdistribution hazard” BMC Med. Res. Methodology.

Fine, J.P. and Gray, R.J. (1999) “A Proportional Hazards Model for the Subdistribution of a Competing Risk”, J. Am. Statist. Ass., 94, 496-509.

Fine, J.P. (1999) “Analysing competing risks data with transformation models”, J. R. Statist. Soc. B, 61, 817-830.

Hammer S.M., Katzenstein, D.A., Hughes M.D., Gundacker, H., Schooley, R.T., Haubrich, R.H., Henry, W.K., Lederman, M.M., Phair, J.P., Niu, M., Hirsch, M.S., and Merigan, T.C. (1996) “A trial comparing nucleoside monotherapy with combination therapy in HIV-infected adults with CD4 cell counts from 200 to 500 per cubic millimeter”, New Engl. J. Mad. 335, 1081-1089.

Huang, X. and Zhang, N. (2008) “Regression Survival Analysis with an Assumed Copula for Dependent Censoring: A Sensitivity Analysis Approach”, Biometrics 64, 1090-1099.

Mao, L. and Lin, D.Y. (2017) “Efficient Estimation of Semiparametric Transformation Models for the Cumulative Incidence of Competing Risks”, J.R. Statist. Soc. Ser. B, 79, 573-587.

Pfeiffer, R.M. and Gail, M.H.(2018) “Absolute Risk: Methods and Applications in Clinical Management and Public Health”

Prentice, R.L., Kalbfleish, J.D., Peterson, A.V., Flournoy, N., Farewell, V.T., and Breslow, N.E. (1978), “The Analysis of Failure Times in the Presence of Competing Risks”, Biometrics, 34, 541-554.