A bounded hazard ratio Cox model for the effect of time to treatment on mortality

In resource-limited or time-sensitive care settings, there is interest in assessing the impact of time to treatment (TTT) on mortality. Traditional Cox proportional hazards models, which specify the effect of TTT as an unrestricted term in the log hazard ratio, can produce counterintuitive results where the survival probability may not decrease monotonically with longer delays. Moreover, hazard ratios from such models quantify the effect of TTT conditional on surviving until treatment, rather than the effect of delayed treatment at baseline. We propose a class of bounded hazard ratio (BHR) Cox models that constrain the hazard ratio for TTT to attenuate towards the null with increasing treatment delay, such that hazard for death after treatment cannot exceed the hazard without treatment. Estimation can be performed using direct optimization of the partial log-likelihood or with an iterative linearized estimation procedure for large sample sizes. From BHR models, the estimated hazard ratio curve describes how treatment benefit diminishes with delay. Additionally, we propose a survival probability difference that provides an absolute measure for comparing survival under different treatment delays. We evaluate model performance in simulations and apply the method to examine treatment delays for colon cancer with data from the National Cancer Database.