The COM-Poisson Cure Rate Model for Survival Data-Computational Aspects

康威－馬克士威－泊松治癒率模型對於存活資料之計算觀點

何致晟(Zhisheng He) ； 江村剛志(Takeshi Emura)

《中國統計學報》 57卷1期 (2019/03) Pp. 1-42

摘要

The Conway-Maxwell-Poisson (COM-Poisson) distribution is useful to account for a cure proportion in survival data. With this model, two computational approaches for calculating maximum likelihood estimates have been developed in the literature: one based on the method in the gamlss R package that employs the first-order derivatives of the log-likelihood, and the other based on the EM algorithm that employs the complete-data likelihood. In this paper, we propose a robust version of the Newton-Raphson (NR) algorithm, where the robustness is introduced by random perturbations to the initial values and by log-transformations to positive parameters. We provide the expressions of the derivatives of the log-likelihood under the Bernoulli cure model and computer codes for implementation. Since the NR algorithm employs the first- and second-derivatives of the log-likelihood, it converges more quickly than the method of the gamlss R package. We also review the EM algorithms and compare the computational performance between the NR and EM algorithms via simulations. We also include a novel data to be fitted to the COM-Poisson cure model, and discuss the consequence of performing the two algorithms.

關鍵字

EM algorithm ； Generalized gamma distribution ； Newton-Raphson algorithm ； Survival analysis ； Weibull distribution

並列摘要

康威－馬克士威－泊松分配（Conway－Maxwell－Poisson）可以用於描述存活資料中之治癒比例，基於此模型，在文獻中已有兩種最大概似估計量之計算方法被提出，一種方法爲R gamlss套件所使用的方法，其利用了對數概似函數之一階導數，另一種方法則是使用了完整資料概似函數之最大期望演算法。在本篇文章中，我們提出了一個穩健的牛頓－拉弗森演算法，其穩健性是來自於對起始值之隨機擾動，以及對恆正參數之對數轉換，在伯努力治癒模型下，我們提供了概似函數之導數表示法與電腦程式碼給讀者使用。由於牛頓－拉弗森演算法使用了概似函數之一、二階導數，故其收斂速度較R gamlss套件快，同時我們也回顧了最大期望演算法，並利用模擬分析將其表現與牛頓－拉弗森演算法做比較，除此之外，我們還提出了一筆新的資料來配適康威－馬克士威－泊松治癒模型，並且討論使用兩種演算法之結果。

並列關鍵字

最大期望演算法 ； 廣義伽瑪分配 ； 牛頓－拉弗森演算法 ； 存活分析 ； 韋伯分配

參考文獻

Emura T, Pan CH (2017). Parametric maximum likelihood inference and goodnessof-fit tests for dependently left-truncated data, a copula-based approach, Stat Pap, doi: 10.1007/s00362-017-0947-z.

Balakrishnan N, Pal S (2015). An EM algorithm for the estimation of parameters of a flexible cure rate model with generalized gamma lifetime and model discrimination using likelihood- and information-based methods. Computational Statistics, 30: 151-189.

Balakrishnan N, Pal S (2016). Expectation maximization-based likelihood inference for flexible cure rate models with Weibull lifetimes. Stat Methods Med Res, 25(4): 1535–1563.

Boag JW (1949). Maximum likelihood estimates of the proportion of patients cured by cancer therapy. Journal of the Royal Statistical Society. Series B (Methodological), 11(1): 15-53.

Conway RW, Maxwell WL (1962). A queuing model with state dependent services rates. Journal of Industrial Engineering, 12: 132-136.