Firth adjusted score function for monotone likelihood in the mixture cure fraction model
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Firth adjusted score function for monotone likelihood in the mixture cure fraction model Frederico Machado Almeida1 · Enrico Antônio Colosimo1 · Vinícius Diniz Mayrink1 Received: 28 January 2020 / Accepted: 30 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract 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 high censoring rate, since the maximum likelihood estimates of some coefficients may be infinity. This phenomenon is called monotone likelihood, and it occurs in the presence of many categorical covariates, especially when one covariate level is not associated with any failure (in survival analysis) or when a categorical covariate perfectly predicts a binary response (in the logistic regression). A well known solution is an adaptation of the Firth method, originally created to reduce the estimation bias. The method provides a finite estimate by penalizing the likelihood function. Bias correction in the mixture cure model 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. An extensive Monte Carlo simulation study indicates good inference performance for the penalized maximum likelihood estimates. The analysis is illustrated through a real application involving patients with melanoma assisted at the Hospital das Clínicas/UFMG in Brazil. This is a relatively novel data set affected by the monotone likelihood issue and containing cured individuals.
Electronic supplementary material The online version of this article (https://doi.org/10.1007/s10985020-09510-4) contains supplementary material, which is available to authorized users.
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Vinícius Diniz Mayrink [email protected] Frederico Machado Almeida [email protected] Enrico Antônio Colosimo [email protected]
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Departamento de Estatística, ICEx, Universidade Federal de Minas Gerais, Av. Antônio Carlos, 6627, Belo Horizonte, MG 31270-901, Brazil
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F. M. Almeida et al.
Keywords Cure rate models · EM algorithm · Firth method · Logistic link function · Melanoma · Weibull model
1 Introduction A usual assumption for general survival models is that the individuals in the study are susceptible to experience the event of interest if they are followed long enough. Under this assumption, the estimated survival curve will eventually decrease to zero. However, in many real situations, including biomedical works, it is common to observe at the end of the study elements for which the event of interest does not occur, even after a very long period of follow-up. These event times are considered as infinity and the subjects are said to be cured/non-susceptible or long-term survivors; the remaining are called susceptible. Models that incorporate the propo
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