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.

在生物医学研究中广泛研究了某些个体是长期幸存者、对感兴趣的事件免疫或不敏感的情况的模型。在涉及高审查率的小样本量的情况下，拟合回归可能会出现问题，因为某些系数的最大似然估计可能是无穷大。这种现象称为单调似然，它发生在存在许多分类协变量的情况下，特别是当一个协变量水平与任何失败无关时（在生存分析中）或当分类协变量完美地预测二元响应时（在逻辑回归中） . 一个众所周知的解决方案是对 Firth 方法的改编，该方法最初是为了减少估计偏差而创建的。该方法通过惩罚似然函数来提供有限估计。混合物固化模型中的偏差校正是文献中很少讨论的主题，它构成了这项工作的核心贡献。为了在这种情况下处理这一点，我们建议基于 Firth 方法导出调整后的得分函数。广泛的蒙特卡罗模拟研究表明惩罚最大似然估计具有良好的推理性能。该分析通过涉及在巴西医院 das Clínicas/UFMG 接受的黑色素瘤患者的实际应用来说明。这是一个相对新颖的数据集，受单调似然问题影响，包含治愈个体。我们建议基于 Firth 方法导出调整后的评分函数。广泛的蒙特卡罗模拟研究表明惩罚最大似然估计具有良好的推理性能。该分析通过涉及在巴西医院 das Clínicas/UFMG 接受的黑色素瘤患者的实际应用来说明。这是一个相对新颖的数据集，受单调似然问题影响，包含治愈个体。我们建议基于 Firth 方法导出调整后的评分函数。广泛的蒙特卡罗模拟研究表明惩罚最大似然估计具有良好的推理性能。该分析通过涉及在巴西医院 das Clínicas/UFMG 接受的黑色素瘤患者的实际应用来说明。这是一个相对新颖的数据集，受单调似然问题影响，包含治愈个体。