The Impact of Violation of the Proportional Hazards Assumption on the Calibration of the Cox Proportional Hazards Model

Abstract

INTRODUCTION

The Cox proportional hazards regression model is frequently used to develop clinical prediction models for time-to-event outcomes, allowing clinicians to estimate an individual’s risk of experiencing the outcome within specified time horizons (e.g., estimate an individual’s 10-year risk of death) [1]. A key assumption of the Cox model is the proportional hazards assumption: the ratio of the hazard function for any two individuals is constant over time and the ratio is a function only of their covariates and the regression coefficients [2]. Although the impact of violation of proportional hazard assumption has been largely investigated to assess the magnitude of treatment effects especially in randomized clinical trials, less is known about the impact of violation of proportional hazard assumption in assessing reliable estimated predictions and their correspondence performances [3, 4, 5]. Calibration is an important aspect of the validation of clinical prediction models: it refers to the agreement between predicted and observed risk [6]. We evaluated the impact of the violation of the proportional hazards assumption on the calibration of clinical prediction models developed using the Cox model through Monte Carlo simulations.

METHODS

We conducted a set of Monte Carlo simulations to assess the impact of the magnitude of the violation of the proportional hazards assumption on the calibration of the Cox model. We compared the calibration of predictions obtained using a Cox regression model with those obtained using accelerated failure time (AFT) models, Royston and Parmar’s spline-based parametric survival models, and generalized linear models using pseudo-observations. Calibration performances were evaluated through different metrics such as the O-P ratio, Observed/Predicted ratio; ICI, Integrated Calibration Index; E50, E90 and calibration curves.

RESULTS

We found that violation of the proportional hazards assumption had generally a negligible impact on the calibration of predictions obtained using a Cox model using calibration measures.

CONCLUSIONS

The magnitude of the violation of the proportional hazards assumption had, at most, a minor impact on the calibration of the Cox regression model. The use of well-known alternative methods such as AFT parametric survival models did not result in improved calibration compared to the use of the Cox model. However, future research might provide more detailed insights about potential alternatives such as the use of generalized linear models through pseudo-observations and spline-based flexible parametric models.