Modified score function for monotone likelihood in the semiparametric mixture cure model

Abstract

The cure fraction models are intended to analyze lifetime data from populations where some individuals are immune to the event under study, and allow a joint estimation of the distribution related to the cured and susceptible subjects, as opposed to the usual approach ignoring the cure rate. In situations involving small sample sizes with many censored times, the detection of nonfinite coefficients may arise via maximum likelihood. This phenomenon is commonly known as monotone likelihood (ML), occurring in the Cox and logistic regression models when many categorical and unbalanced covariates are present. An existing solution to prevent the issue is based on the Firth correction, originally developed to reduce the estimation bias. The method ensures finite estimates by penalizing the likelihood function. In the context of mixture cure models, the ML issue is rarely discussed in the literature; therefore, this topic can be seen as the first contribution of our paper. The second major contribution, not well addressed elsewhere, is the study of the ML issue in cure mixture modeling under the flexibility of a semiparametric framework to handle the baseline hazard. We derive the modified score function based on the Firth approach and explore finite sample size properties of the estimators via a Monte Carlo scheme. The simulation results indicate that the performance of coefficients related to the binary covariates are strongly affected to the imbalance degree. A real illustration, in the melanoma dataset, is discussed using a relatively novel data set collected in a Brazilian university hospital.

Bayesian solution to the monotone likelihood in the standard mixture cure model

“…In other words, there are few values $$$ {x}_{1i}=1 $$$ compared to $$$ {x}_{1i}=0 $$$. As discussed in Almeida et al (2021b), depending on the sample size, the following percentages of $$$ {x}_{1i}=1 $$$ are obtained: $$$ 10\% $$$ ($$$ n=50 $$$), $$$ 2.5\% $$$ ($$$ n=200 $$$), $$$ 0.83\% $$$ ($$$ n=600 $$$), and $$$ 0.50\% $$$ ($$$ n=1,000 $$$). Note that the proportion of 1's in the binary covariate decreases as the sample size increases.…”

“…The first proposal in this paper follows the steps in Almeida et al (2021a) to model h 0 (t|⋅) by assuming the Weibull distribution for the survival times, with scale 𝜆 * = 𝜆 exp(x ⊤ 𝜷) and shape 𝛼; consider 𝜆 = exp{𝛽 0 }. The second proposal is to handle h 0 (t|⋅) with a semi-parametric specification, which is established under the profiled approach in Almeida et al (2021b). In this semi-parametric case, assume that the m distinct failure times are ordered as follows t (1) < • • • < t (m) .…”

“…The first proposal in this paper follows the steps in Almeida et al (2021a) to model $$$ {h}_0\left(t|\cdotp \right) $$$ by assuming the Weibull distribution for the survival times, with scale $$$ {\lambda}^{\ast }=\lambda \exp \left({\mathbf{x}}^{\top}\boldsymbol{\beta} \right) $$$ and shape $$$ \alpha $$$; consider $$$ \lambda =\exp \left\{{\beta}_0\right\} $$$. The second proposal is to handle $$$ {h}_0\left(t|\cdotp \right) $$$ with a semi‐parametric specification, which is established under the profiled approach in Almeida et al (2021b). In this semi‐parametric case, assume that the $$$ m $$$ distinct failure times are ordered as follows $$$ {t}_{(1)}<\cdots <{t}_{(m)} $$$.…”

“…However, despite this similarity between the penalized $$$ \log $$$‐likelihood function $$$ {\tilde{\ell}}_c^{\ast}\left(\boldsymbol{\psi} \right) $$$ and the logarithm of the posterior distribution, all inferences are based on the frequentist approach. Additionally, the third‐order cumulants $$$ {\nu}_{r,u,k} $$$ for both latency and incidence parts, under the parametric and semi‐parametric specifications, can be found in Almeida et al (2021a, 2021b). The use of Firth correction to handle the ML issue is a topic easily found in the literature; some examples are Kolassa (1997), Heinze and Schemper (2001), Zorn (2005), Rainey (2016), Lima and Cribari‐Neto (2016), Kolassa (2016), and Kenne Pagui and Colosimo (2020).…”

“…Motivated by the frequentist studies in (Almeida, Colosimo, & Mayrink, 2021a, 2021b), related to the ML issue in the cure fraction models, the main contributions of the present paper are the following: (i) handle the ML issue by evaluating both Firth correction and the Bayesian view; (ii) investigate other flexible penalty functions (non‐Jeffreys prior distributions) and compare the result with those from the Firth method; (iii) conduct a sensitivity analysis to explore different levels of prior information; (iv) develop a comprehensive MC simulation study assuming different specifications for the latency distribution (parametric and semiparametric cases) to evaluate the posterior uncertainty; and (v) analyze the melanoma skin cancer data set assuming the Firth correction and other penalties in the mixture cure model.…”