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Firth adjusted score function for monotone likelihood in the mixture cure fraction model

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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.

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Data availibility statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

The authors would like to thank two anonymous referees for their constructive comments and suggestions leading to an improved version of this paper. The first author thanks Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES, Brazil) for supporting his research. The second author acknowledge the support from Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq, Brazil).

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  1. Departamento de Estatística, ICEx, Universidade Federal de Minas Gerais, Av. Antônio Carlos, 6627, Belo Horizonte, MG, 31270-901, Brazil

    Frederico Machado Almeida, Enrico Antônio Colosimo & Vinícius Diniz Mayrink

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  1. Frederico Machado Almeida
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  2. Enrico Antônio Colosimo
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  3. Vinícius Diniz Mayrink
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Correspondence to Vinícius Diniz Mayrink.

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Almeida, F.M., Colosimo, E.A. & Mayrink, V.D. Firth adjusted score function for monotone likelihood in the mixture cure fraction model. Lifetime Data Anal 27, 131–155 (2021). https://doi.org/10.1007/s10985-020-09510-4

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