臺灣博碩士論文加值系統

對設限存活資料(censored survival data)分析，Rodrigues等(2009)提出用Conway-Maxwell-Poisson (COM-Poisson)分佈為治愈模型(cure rate model)。對COM-Poisson治愈模型之特例——伯努利治愈模型(Bernoulli cure rate model)，考慮使用不同之運算演算法，以最大概似估計法(maximum likelihood estimation)得參數之估計值。據Balakrishnan與Pal於2016以韋伯分佈(Weibull distribution)及於2015以廣義伽瑪分佈(generalized gamma distribution)，假設為其壽命分佈(lifetime distribution)。進而導出之評分函數(score function)與黑塞矩陣(Hessian matrix)，用以牛頓-拉弗森演算法(Newton-Raphson algorithm)及最大期望演算法(EM algorithm)。模擬為分析比較此二種演算法之表現。末了，實際資料分析作詳加闡明此方法模型。

Rodrigues et al. (2009) proposed the Conway-Maxwell-Poisson (COM-Poisson) distribution as a model for a cure rate in censored survival data. We consider computational algorithms for maximum likelihood estimation under the Bernoulli cure rate model, a special case of the COM-Poisson cure rate model. The Weibull distribution (Balakrishnan and Pal 2016) and the generalized gamma distribution (Balakrishnan and Pal 2015) are considered as lifetime distributions. We obtain all the expressions of the score function and Hessian matrix to perform the Newton-Raphson and EM algorithms. Simulations are conducted to compare the performance between the EM algorithm and Newton-Raphson algorithms. Finally, a real data is analyzed to illustrate the methods.

Contents
Chapter 1 Introduction ………………………………………...…………………………….1
Chapter 2 Background ………………………………………………………………………3
2.1 The Conway-Maxwell-Poisson (COM-Poisson) distribution …………………………..3
Example: The Bernoulli distribution …………………………………………….……….4
2.2 Lifetime distributions ………………………………………………………….…..……5
Example 1: The Weibull distribution …………………………………...............………..6
Example 2: The generalized gamma distribution …………………………………...……6
2.3 Long-term survival function ……………………………………………………………8
Chapter 3 Maximum likelihood estimation ………………………………………...………9
3.1 Right-censored data with cure …………………………………………………….…….9
3.2 Log-Likelihood under the Bernoulli cure model …………………………………...….11
Example 1: Weibull lifetime with Bernoulli cure ………………………………..……..12
Example 2: Generalized gamma lifetime with Bernoulli cure ………………………….12
3.3 EM-algorithm …………………………………………………………………....…….13
3.3.1 The complete data likelihood function ……………………………………………13
3.3.2 Cure rate model with Weibull lifetime ……………………………………..……..15
Example: Bernoulli cure rate model with Weibull lifetime ………………………...…..16
3.3.3 Cure rate model with generalized gamma lifetime …………………………….....18
Example: Bernoulli cure rate model with generalized gamma lifetime ………………...19
Chapter 4 Computational algorithms ……………………………………………………..21
4.1 Newton-Raphson algorithm ……………………………………………………….…..21
Example 1: Randomized Newton-Raphson algorithm with Weibull lifetime ………..…21
Example 2: Algorithm with generalized gamma lifetime …………………………..…..23
4.2 EM-algorithm ………………………………………………………………………….24
Example 1: M-step by randomized Newton-Raphson with Weibull lifetime …………..24
Example 2: M-step algorithm with generalized gamma lifetime ……………………….26
4.3 Interval estimation ………………………………………………………………….….27
Chapter 5 Simulation ……………………………………………………………….………30
5.1 simulation design …………………………………………………………………..….30
5.2 Simulation results ……………………………………………………………………..33
Chapter 6 Data analysis …………………………………………………...………………..39
6.1 Tumor metastasis data …………………………………………………………………39
6.2 Model fitting with Bernoulli cure model ……………………………….……………..40
6.2.1 Weibull lifetime ……………………………………………………………….…..40
6.3 Discuss under generalized gamma lifetime ……………………………………….…..45
6.3.1 Covariate with gender ………………………………………………………….....45
6.3.2 Covariate with tumor position …………………………………………………….48
Chapter 7 Conclusion and discussion ……………………………………………..……….52
Reference……………………………………………………………………………….….54
Appendix A …………………………………...……………………………………………..55
Appendix B ………………………………………………………………………………….56
Appendix C …………………………………………………………………………….……62
Appendix D…………………………………………………………………………………….65

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