Solutions to the monotone likelihood in the standard mixture Cure fraction model

Abstract

Survival models for situations where some individuals are long-term survivors, immune or non-susceptible to the event of interest are extensively studied in biomedical research. Fitting a regression can be problematic in situations involving small sample sizes with many censored times, since the maximum likelihood estimates of some coefficients may be infinity. This phenomenon is commonly known as Monotone Likelihood (ML), occurring in the presence of many categorical and unbalanced covariates. A well-known solution is an adaptation of the Firth's method, originally created to reduce the maximum likelihood estimation bias. The method ensures finite estimates by penalizing the likelihood function, where the penalty term might be interpreted as the Jeffreys invariant prior, largely used in the Bayesian framework. The ML issue in the context involving mixture cure models is a topic rarely discussed in the literature, and it configures a central contribution of this work. In order to handle this point in such context, we propose to derive the adjusted score function based on the Firth method. The second major contribution is to investigate other flexible penalty functions (prior distributions), in which all inference procedures will be based on the posterior samples. An extensive Monte Carlo simulation study indicates good inference performance for the penalized estimates, especially in the Bayesian framework. The analysis is illustrated through a real application involving patients with melanoma assisted at the Hospital das Clínicas/UFMG. This is a relatively novel data set affected by the monotone likelihood issue and containing cured individuals.