A Stochastic EM Algorithm for Cure Rate Model with Negative Binomial Competing Risks and Non-homogeneous Lifetime
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In this paper, we consider a long-term survival model under a competing risks scenario. Since the number of competing risks is unobserved, we assume it to follow a negative binomial distribution that can capture both over- and under-dispersion we usually encounter when modeling count data. The distribution of the progression time, corresponding to each competing risk, is associated with a set of risk factors that allow us to capture the non-homogeneous patient population. We also provide flexibility in modeling the cure or the long-term survival rate, which is considered as a function of risk factors. Considering the latent competing risks as missing data, we develop a variation of the well-known expectation maximization (EM) algorithm, called the stochastic EM algorithm (SEM), which is the main contribution of this paper. We show that the SEM algorithm avoids calculation of complicated expectations, which is a major advantage of the SEM algorithm over the EM algorithm. Our proposed procedure allows the objective function to be maximized 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 function, with lower parameter dimension, can be maximized independently. Through an extensive Monte Carlo simulation study we show the performance of the proposed SEM algorithm through calculated bias, root mean square error, and coverage probability of the asymptotic confidence interval. We also show that the SEM algorithm is not sensitive to the choice of the initial values. Finally, for illustration, we analyze a breast cancer survival data.
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