Institute of Statistical Research and Training, University of Dhaka, Dhaka, Bangladesh.
Department of Statistics, University of Dhaka, Dhaka, Bangladesh.
BMC Med Res Methodol. 2022 Jun 11;22(1):169. doi: 10.1186/s12874-022-01638-1.
Separation or monotone likelihood may exist in fitting process of the accelerated failure time (AFT) model using maximum likelihood approach when sample size is small and/or rate of censoring is high (rare event) or there is at least one strong covariate in the model, resulting in infinite estimates of at least one regression coefficient.
This paper investigated the properties of the maximum likelihood estimator (MLE) of the regression parameters of the AFT models for small sample and/or rare-event situation and addressed the problems by introducing a penalized likelihood approach. The penalized likelihood function and the corresponding score equation is derived by adding a penalty term to the existing likelihood function, which was originally proposed by Firth (Biometrika, 1993) for the exponential family models. Further, a post-hoc adjustment of intercept and scale parameters is discussed keeping them out of penalization to ensure accurate prediction of survival probability. The penalized method was illustrated for the widely used log-location-scale family models such as Weibull, Log-normal and Log-logistic distributions and compared the models and methods uisng an extensive simulation study.
The simulation study, performed separately for each of the log-location-scale models, showed that Firth's penalized likelihood succeeded to solve the problem of separation and achieve convergence, providing finite estimates of the regression coefficients, which are not often possible by the MLE. Furthermore, the proposed penalized method showed substantial improvement over MLE by providing smaller amount of bias, mean squared error (MSE), narrower confidence interval and reasonably accurate prediction of survival probabilities. The methods are illustrated using prostate cancer data with existence of separation, and results supported the simulation findings.
When sample size is small (≤ 50) or event is rare (i.e., censoring proportion is high) and/or there is any evidence of separation in the data, we recommend to use Firth's penalized likelihood method for fitting AFT model.
在使用最大似然法拟合加速失效时间 (AFT) 模型时，如果样本量较小且/或删失率较高（罕见事件），或者模型中至少有一个强协变量，则拟合过程中可能存在分离或单调似然，导致至少一个回归系数的估计值无穷大。
本文研究了小样本和/或罕见事件情况下 AFT 模型回归参数的最大似然估计量 (MLE) 的性质，并通过引入惩罚似然方法解决了这些问题。惩罚似然函数及其相应的得分方程是通过在原始似然函数中添加惩罚项而得出的，该函数最初由 Firth（Biometrika，1993 年）提出，用于指数族模型。此外，讨论了截距和比例参数的事后调整，将它们排除在惩罚之外，以确保生存概率的准确预测。惩罚方法适用于广泛使用的对数位置尺度族模型，如 Weibull、对数正态和对数逻辑分布，并使用广泛的模拟研究比较了模型和方法。
对于每个对数位置尺度模型分别进行的模拟研究表明，Firth 的惩罚似然成功地解决了分离问题并实现了收敛，提供了回归系数的有限估计值，这通常是 MLE 无法实现的。此外，所提出的惩罚方法通过提供更小的偏置、均方误差 (MSE)、更窄的置信区间和合理准确的生存概率预测，大大优于 MLE。该方法通过存在分离的前列腺癌数据进行说明，结果支持模拟结果。
当样本量较小（≤50）或事件罕见（即删失比例较高），并且/或者数据中存在任何分离的证据时，我们建议使用 Firth 的惩罚似然方法来拟合 AFT 模型。
