A novel approach to determine two optimal cut-points of a continuous predictor with a U-shaped relationship to hazard ratio in survival data: simulation and application

Background: In clinical and epidemiological researches, continuous predictors are often discretized into categorical variables for classification of patients. When the relationship between a continuous predictor and log relative hazards is U-shaped in survival data, there is a lack of a satisfying solution to find optimal cut-points to discretize the continuous predictor. In this study, we propose a novel approach named optimal equal-HR method to discretize a continuous variable that has a U-shaped relationship with log relative hazards in survival data.

Methods: The main idea of the optimal equal-HR method is to find two optimal cut-points that have equal log relative hazard values and result in Cox models with minimum AIC value. An R package 'CutpointsOEHR' has been developed for easy implementation of the optimal equal-HR method. A Monte Carlo simulation study was carried out to investigate the performance of the optimal equal-HR method. In the simulation process, different censoring proportions, baseline hazard functions and asymmetry levels of U-shaped relationships were chosen. To compare the optimal equal-HR method with other common approaches, the predictive performance of Cox models with variables discretized by different cut-points was assessed.

Results: Simulation results showed that in asymmetric U-shape scenarios the optimal equal-HR method had better performance than the median split method, the upper and lower quantiles method, and the minimum p-value method regarding discrimination ability and overall performance of Cox models. The optimal equal-HR method was applied to a real dataset of small cell lung cancer. The real data example demonstrated that the optimal equal-HR method could provide clinical meaningful cut-points and had good predictive performance in Cox models.

Conclusions: In general, the optimal equal-HR method is recommended to discretize a continuous predictor with right-censored outcomes if the predictor has an asymmetric U-shaped relationship with log relative hazards based on Cox regression models.