Firth's logistic regression with rare events: accurate effect estimates and predictions?
Firth's logistic regression with rare events: accurate effect estimates and predictions?
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DOI：
10.1002/sim.7273
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发表时间：
2017-06-30
影响因子：
2
通讯作者：
Geroldinger, Angelika
中科院分区：
文献类型：
--
作者：
Puhr, Rainer;Heinze, Georg;Geroldinger, Angelika
Firth的逻辑回归已经成为小样本二元结果分析的标准方法。虽然它减少了系数的最大似然估计中的偏差，但在预测概率中引入了对二分之一的偏差。结果的不平衡性越强，预测概率的偏差就越严重。我们提出了两种对Firth逻辑回归的简单修改，从而得到无偏预测概率。第一种方法通过对截距进行事后调整来校正预测的概率。另一种是基于Firth惩罚的替代公式作为迭代数据增强过程。我们建议的修改包括引入一个指标变量来区分增强数据中的原始观测值和伪观测值。在一项全面的模拟研究中，将这些方法与基于Firth惩罚的其他改进预测的尝试以及用于常规使用的其他已发表的惩罚策略进行比较。例如，我们考虑最近提出的最大似然和Firth逻辑回归之间的折衷。对模拟结果进行了详细的预测和效果估计。我们发现，与Firth惩罚相比，我们提出的两种方法不仅给出了无偏的预测概率，而且在解释变量的条件下提高了精度。虽然一种方法的结果与Firth惩罚法的结果相同，但另一种方法引入了一些偏差，但这可以通过均方误差的减少来补偿。最后，对微创心脏手术中动脉闭合装置的研究中所考虑的所有方法进行了说明和比较。版权所有：JohnWiley & Sons， Ltd。
Firth's logistic regression has become a standard approach for the analysis of binary outcomes with small samples. Whereas it reduces the bias in maximum likelihood estimates of coefficients, bias towards one-half is introduced in the predicted probabilities. The stronger the imbalance of the outcome, the more severe is the bias in the predicted probabilities. We propose two simple modifications of Firth's logistic regression resulting in unbiased predicted probabilities. The first corrects the predicted probabilities by a post hoc adjustment of the intercept. The other is based on an alternative formulation of Firth's penalization as an iterative data augmentation procedure. Our suggested modification consists in introducing an indicator variable that distinguishes between original and pseudo-observations in the augmented data. In a comprehensive simulation study, these approaches are compared with other attempts to improve predictions based on Firth's penalization and to other published penalization strategies intended for routine use. For instance, we consider a recently suggested compromise between maximum likelihood and Firth's logistic regression. Simulation results are scrutinized with regard to prediction and effect estimation. We find that both our suggested methods do not only give unbiased predicted probabilities but also improve the accuracy conditional on explanatory variables compared with Firth's penalization. While one method results in effect estimates identical to those of Firth's penalization, the other introduces some bias, but this is compensated by a decrease in the mean squared error. Finally, all methods considered are illustrated and compared for a study on arterial closure devices in minimally invasive cardiac surgery. Copyright (C) 2017 JohnWiley & Sons, Ltd.