In this paper, a long-term survival model under competing risks is considered. The unobserved number of competing risks is assumed to follow a negative binomial distribution that can capture both over- and under-dispersion. Considering the latent competing risks as missing data, a variation of the well-known expectation maximization (EM) algorithm, called the stochastic EM algorithm (SEM), is developed. It is shown that the SEM algorithm avoids calculation of complicated expectations, which is a major advantage of the SEM algorithm over the EM algorithm. The proposed procedure also allows the objective function to be split into two simpler functions, one corresponding to the parameters associated with the cure rate and the other corresponding to the parameters associated with the progression times. The advantage of this approach is that each simple function, with lower parameter dimension, can be maximized independently. An extensive Monte Carlo simulation study is carried out to compare the performances of the SEM and EM algorithms. Finally, a breast cancer survival data is analyzed and it is shown that the SEM algorithm performs better than the EM algorithm.
翻译:本文考虑了在相互竞争的风险下的长期生存模式。 未经观察的相竞风险数量假定会遵循一种负面的二进制分布, 能够捕捉出超分散和低分散。 考虑到潜在的相互竞争的风险, 缺少数据, 开发出众所周知的预期最大化算法的变异, 称为随机的EM 算法( SEM 算法 ) 。 显示SEM 算法避免计算复杂的预期, 这是SEM 算法的主要优势。 拟议的程序还允许将目标函数分成两个更简单的函数, 一个函数与治愈率相关的参数相对应,另一个函数则与递进时间相关的参数相对应。 这种方法的优点是, 每一个简单的函数, 低参数尺寸的, 都可以独立最大化。 一个广泛的蒙特卡洛模拟研究, 比较SEM 和EM 算法的性能。 最后, 对乳腺癌生存数据进行了分析, 并显示SEM 算法比EM 算法要好。