Restricted mean survival time approach versus time-varying coefficient Cox model for quantifying treatment effect when hazards are non-proportional

Abstract

Background: Although the Cox time-varying coefficient (TVC) model has been developed to address non-proportional hazard (PH), its use remains underexplored. Instead, the restricted mean survival time (RMST) has been widely used in non-PH settings to quantify treatment effects using life expectancy ratio (LER) and life expectancy difference (LED).

Methods: This study explores a novel extension of the Cox TVC model under non-PH to generate LER and LED to enable a direct comparison with RMST based on flexible parametric survival model (FPM). An intensive simulation study was conducted to compare the performance of FPM to the Cox TVC model under PH and non-PH assumptions. The survival time t was assumed to follow the Piecewise Exponential distribution with various censoring patterns generated from the Uniform distribution. Both methods were evaluated via a randomised clinical trial of nasopharyngeal cancer exhibiting increasing treatment benefit.

Results: Intensive simulations showed Cox TVC outperformed FPM under non-PH in terms of bias and coverage, with generally higher power observed in scenarios of crossing or diverging curves under low censoring. In real-world data, the FPM produced slightly larger LER and LED estimates than Cox TVC. Cox TVC has the advantage of assessing treatment effect at different milestones and detecting earlier difference when estimating using hazard ratio (HR).

Conclusion: Overall, Cox TVC is a viable option for summarising treatment effect using LER and LED under non-PH conditions. It would be beneficial to complement the reporting by providing estimates of HR at specific milestone to detect early differences.

Keywords: Event outcomes; Proportional hazards, Flexible parametric model, Time; Restricted mean survival time, Time; To; Varying coefficient Cox model, Non.

Fig. 1

Bias, 95% coverage probability, Power and Type I Error of FPM(3,1) and Cox TVC estimates of LER under PH assumption (Scenarios 1 to 3) for HR = 0.5, 1 and 1.5, with low, moderate and high censoring, n = 500 and varying t = 3 to 5

Fig. 2

Bias, 95% coverage probability and Power of FPM(3,1) and Cox TVC estimates of LER when survival curves diverge, assuming HR = 1.9 for t $\in [1,3)$, HR = 3 for t $\ge$ 3 (Scenario 4), and HR = 0.5 for t $\in [1,3)$, HR = 0.3 for t $\ge$ 3 (Scenario 5), with low, moderate and high censoring, n = 500

Fig. 3

Bias, 95% coverage probability and Power of FPM(3,1) and Cox TVC estimates of LER when survival curves cross, assuming HR = 1.5 for t $\in [1,3)$, HR = 3.5 for t $\ge$ 3 (Scenario 6), and HR = 0.5 for t $\in [1,3)$, HR = 0.29 for t $\ge$ 3 (Scenario 7), with low, moderate and high censoring, n = 500

Fig. 4

Overall survival curves of Kaplan–Meier, Cox TVC, Cox TVC_ln(t) and FPM(3,1) models (data from Wee, et al. [19])

Fig. 5

Estimation of the HR over time, with the corresponding 95% confidence interval and p-value (data from Wee, et al. [19])