Joint Models for longitudinal and time-to-event data are frequently used to model endogenous longitudinal covariates alongside a time-to-event outcome. However, the model class borrows some limitations of the survival submodels, including the necessity for non-separation for each category of categorical covariates. We therefore incorporate Firth's correction into the frequentist estimation procedure of joint models in order to make the model class applicable in settings with separation cases. We derive the needed quantities for the correction term and implement it in the Expectation-Maximization Algorithm for the parameter estimation in joint models. Our simulation study shows, that in data situations with separation issues, the Firth-corrected estimation procedure yields less biased estimates and the respective coefficients approach the estimated values observed in the non-separation cases. The application on a data set on satisfaction with and dropouts from vocational training demonstrates the advantages of the Firth-corrected joint model in a real world data set with separation. The results add to the literature on dropout from vocational training in Germany by explicitly modeling direct effects of socioeconomic and training-specific factors on the risk of dropout as well as their indirect contribution via satisfaction with the training.

翻译：纵向数据与时间事件数据的联合模型常用于在时间事件结局中建模内源性纵向协变量。然而，该模型类别继承了生存子模型的部分限制，其中包括分类协变量的每个类别必须满足非分离条件。为此，我们将Firth校正引入联合模型的频率学派估计流程，使该模型类别能够适用于存在分离问题的场景。我们推导了校正项所需的量，并将其嵌入联合模型参数估计的期望最大化算法中。模拟研究表明，在存在分离问题的数据情境下，Firth校正估计方法能产生偏差更小的估计值，且相应系数趋近于非分离情形下的估计值。基于一项关于职业培训满意度与退学问题的真实数据集应用，展示了Firth校正联合模型在处理存在分离现象的真实数据时的优势。研究结果通过明确建模社会经济因素和培训特定因素对退学风险的直接效应，以及这些因素通过培训满意度产生的间接贡献，丰富了关于德国职业培训退学问题的文献。