Stochastic EM Algorithm-based Likelihood Inference for Spatial Cure Rate Models Based on Some Flexible Distributions

Abstract

With geographic information systems (GIS) widely accessible, modeling spatially referenced data has become increasingly relevant. Spatial survival analysis addresses spatial dependence by analyzing location-linked time-to-event data. In this work, we adopt a competing risks framework, assuming that the number of causes leading to an event follows a discrete power series (PS) distribution. The PS cure rate model is highly flexible and can recover many standard cure models through specific choices of its power parameter and series function. We focus on three activation schemes: first, random, and last activation. Spatial effects are modeled via a Gaussian process (Gaussian random field) and incorporated into the cure rate models as spatial frailties, capturing the influence of geographic location on survival times for susceptible individuals and cure rate. We propose a flexible baseline hazard function based on the generalized extreme value (GEV) distribution, which unifies several continuous distributions. By adjusting scale and shape parameters, the GEV includes the Gumbel (Type I), Fréchet (Type II), and Weibull (Type III) distributions as special cases. Parameter estimation is performed using a Stochastic EM (SEM) algorithm, which avoids computing conditional expectations, is free from saddle point problems, insensitive to starting values, and performs well with small to moderate clinical sample sizes. Extensive simulations demonstrate robust convergence and strong model discrimination via information-based criteria. We apply the proposed models to a smoking cessation dataset, visualizing spatial effects on hazard using maps. Cure rates and survival probabilities are compared with and without spatial effects, and the necessity of including spatial frailties is confirmed through a likelihood ratio test. The results highlight the importance of accounting for geographic variation in survival analysis and demonstrate the flexibility and applicability of the proposed spatial PS cure rate framework.