貝氏存活分析之右設限資料之研究

Bayesian Survival Analysis For Right Censor Data

王聖文(Sheng-Wen Wang)

指導教授 ： 吳裕振

中原大學/理學院/應用數學研究所/碩士(2007年)

https://doi.org/10.6840/cycu200700322

摘要

這篇論文主要研究無母數貝氏右設限資料(right censor data)之存活分析。我們使用伯氏多項式(Bernstein polynomial)當作事前分配的隨機過程來建構累積分配函數。我們知道累積分配函數是一個遞增函數，所以我們可用伯氏多項式來描述其累積分配函數。
本篇論文的架構如下。第一節是介紹右設限資料之背景。第二節在說明伯氏多項式的圖形和係數之間的關係並且使用Bernstein-Weierstrass定理:對於任意緊緻子集的連續函數皆可以從伯氏基底(Bernstein bases)所成的多項式來逼近(見Altomare, F. & Campiti, M. (1994))。
第三節利用伯氏多項式來建構分配函數的模型及事前分配的支助(support of prior)。第四節寫下右設限資料的概似函數且對於參數給事前分配並利用貝氏的方法，我們可得到事後分配。
第五節主要是估計參數，而此參數為伯氏多項式之維度和它的係數，所以其事後分配的計算相當複雜，因此我們用馬可夫鍊蒙地卡羅法(MCMC；Markov chain Monte Carlo)之Metropolis-Hastings-Green (見Green, P. G. (1995))的演算法來計算事後分配之參數。
除了考慮Metropolis-Hastings-Green的演算法之外，我們也用了古典獨立的Metropolis-Hastings的演算法並與Metropolis-Hastings-Green的演算法作比較。第六節是模擬計算。我們的貝氏估計與在頻率方面所獲得右設限資料下之存活函數的無母數最大概似估計量(Non-parametric maximal likelihood estimator:Kaplan-Meier Estimator)作比較。
第七節是討論，在此篇論文之後，未來研究方向如Cox-model之研究如研究共變數(例如,男生和女生,抽菸和不抽菸等)(見Chang, I. S., Wen, C. C. and Wu, Y. J. (2007))之間的關係。

關鍵字

Metropolis-Hastings-Green algorithm ； 右設限 ； 貝氏存活 ； 伯氏多項式

並列摘要

This paper studies the non-parametric Bayesian survival analysis for right censor data mainly. We use the Bernstein polynomial as the random process of prior to construct the cumulative distribution function. We know the cumulative distribution function is a increasing function, so we can use the Bernstein polynomial to describe the cumulative distribution function.
The structure of this paper as follows. The first section is a introduction of background of the right censor data. The second section describes the relation of the shape of the graph of the Bernstein polynomial and the coefficients, and use the Bernstein-Weierstrass theorem:For the continuous function of any compact subset can be approximated by the polynomial of Bernstein bases.(cf. Altomare, F. & Campiti, M. (1994))
In the third section we will use the Bernstein polynomial to construct the model of the distribution function and the support of prior. The fourth section writes the likelihood function of right censor data and gives the prior distribution for parameter. We can get the posterior distribution by using the Bayesian method.
The fifth section estimates the parameters mainly, and the parameters are the dimension of the Bernstein polynomial and its coefficients. So the computation of the posterior distribution is quite complicated. Thus weuse the Metropolis-Hastings-Green algorithm of MCMC method (Markov chain Monte Carlo)to compute the posterior distribution. Besides considering the Metropolis-Hastings-Green algorithm(cf. Green, P. G. (1995)),
we also use the classic and independent Metropolis-Hastings algorithm and compare them with each other.
The sixth section is simulating computation. In frequency:the NPMLE(Kaplan-Meier Estimator) of the cumulative distribution function of the right censor data compared with our Bayesian estimator. The
seventh section is a discussion. After this paper, the direction of future research, like the research of the Cox-model. For instance,the research of the covariance.(e.g. men and women, smoking and non-smoking, etc.)(cf. Chang, I. S., Wen, C. C. and Wu, Y. J. (2007)).

並列關鍵字

Bernstein polynomial ； Bayesian Survival ； right censor ； Metropolis-Hastings-Green algorithm

參考文獻

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5.Green, P. G. (1995). Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82, 711-732.

7.Robert, C. P. & Casella, G. (1999). Monte Carlo statistical methods. Springer-Verlag, New York.

8.Tierney, L. (1994). Markov chains for exploring posterior distributions. Ann. Statist. 22, 1701-1728 (discussion : 1728-1762).