¹Institut für Medizinische Biometrie, Informatik und Epidemiologie (IMBIE), Universitätsklinikum Bonn, Bonn, Germany
²Koblenz University of Applied Sciences, RheinAhrCampus Remagen, Remagen, Germany

Text

Introduction: The Cox proportional hazards model is a widely used method for analyzing clinical time-to-event data. The basic form of the model assumes the covariate effects on the hazard function to be constant over time. However, in many clinical settings, covariate effects may vary with time, and covariate interactions may significantly influence survival. Implementing a strategy for the selection of interactions and time-varying covariate effects within the framework of the Cox model often requires substantial manual effort, including pre-screening all covariates and their interactions, and conducting comprehensive model diagnostics. Often, this is only feasible when the number of covariates is small. Alternatively, stepwise selection procedures can be applied to filter out most important interactions and time-varying covariate effects, which, however, have been criticized for their instability.

Methods: We propose a linked-shrinkage adaptive lasso approach that enables the automatic selection of two-way interaction terms and time-varying covariate effects [1], [2]. The method applies an adaptive lasso framework using penalty weights estimated in a preceding main-effects-only ridge regression. Potential time-varying effects are modeled as piecewise constant functions and penalized more strongly to ensure selection only when supported by the data. We conducted a simulation study with varying numbers of main and two-way interaction effects (5-15 non-zero main effects plus 5-15 non-zero interaction effects), generating survival times for n ∈ {500, 1000} observations under four different scenarios: (i) no time-varying effects, (ii) time-varying continuous effects, (iii) time-varying dichotomous effects and (iv) time-varying continuous and dichotomous effects. We evaluated the proposed approach in terms of its ability to correctly identify informative main effects, two-way interaction terms, and time-varying covariate effects. In addition, we assessed the mean squared error (MSE) of predicted survival probabilities and benchmarked our proposed methods against several alternative modeling approaches, including stepwise Cox models, regularized Cox models with standard L1/L2 penalties and a hierarchical lasso [3]. Furthermore, we illustrate the proposed approach by analyzing real-world data from the National Cancer Institute Surveillance, Epidemiology, and End Results (SEER) program [4].

Results: In our experiments, the proposed linked-shrinkage adaptive lasso approach reliably identified true time-varying and two-way interaction effects. In case of no time-varying effects (scenario (i)), the selection proportion was < 2% for time-varying coefficients. Both continuous and dichotomous time-varying effects were identified correctly with selection proportions between 85%-90% and 92%-96%, respectively. The proposed method outperformed standard regularized Cox models with respect to the MSE, especially in the presence of time-varying effects.

Conclusion: By addressing the limitations of manual covariate selection and stepwise procedures, the proposed method extends penalized estimation techniques to Cox regression with time-varying coefficients. Further, it facilitates the simultaneous selection of relevant interaction terms and time-varying covariate effects.

The authors declare that they have no competing interests.

The authors declare that an ethics committee vote is not required.

Literatur

[1] van de Wiel MA, Amestoy M, Hoogland J. Linked shrinkage to improve estimation of interaction effects in regression models. Epidemiologic methods. 2024 Jul 9;13(1):20230039. DOI: 10.1515/em-2023-0039
[2] Zou H. The adaptive lasso and its oracle properties. Journal of the American Statistical Association. 2006;101(476):1418–1429. DOI: 10.1198/016214506000000735
[3] Bien J, Taylor J, Tibshirani R. A lasso for hierarchical interactions. Annals of statistics. 2013;41(3):1111–1141. DOI: 10.1214/13-AOS1096
[4] National Cancer Institute. Surveillance, Epidemiology, and End Results (SEER) Program. SEER*Stat Database. 2019. Available from: https://www.seer.cancer.gov