Using fractional polynomials and restricted cubic splines to model non-proportional hazards or time-varying covariate effects in the Cox regression model

The Cox proportional hazards model is used extensively in clinical and epidemiological research. A key assumption of this model is that of proportional hazards. A variable satisfies the proportional hazards assumption if the effect of that variable on the hazard function is constant over time. When the proportional hazards assumption is violated for a given variable, a common approach is to modify the model so that the regression coefficient associated with the given variable is assumed to be a linear function of time (or of log-time), rather than being constant or fixed. However, this is an unnecessarily restrictive assumption. We describe two different methods to allow a regression coefficient, and thus the hazard ratio, in a Cox model to vary as a flexible function of time. These methods use either fractional polynomials or restricted cubic splines to model the log-hazard ratio as a function of time. We illustrate the utility of these methods using data on 12 705 patients who presented to a hospital emergency department with a primary diagnosis of heart failure. We used a Cox model to assess the association between elevated cardiac troponin at presentation and the hazard of death after adjustment for an extensive set of covariates. SAS code for implementing the restricted cubic spline approach is provided, while an existing Stata function allows for the use of fractional polynomials.

Automated summary

The Cox proportional hazards model, commonly used in clinical and epidemiological research, assumes proportional hazards for variables. When this assumption is violated, there are two methods to allow regression coefficients to vary as a flexible function of time. This flexibility improves the modeling of data and enhances the accuracy of the model.