Graphical methods to illustrate the nature of the relation between a continuous variable and the outcome when using restricted cubic splines with a Cox proportional hazards model
Peter C. Austin
Abstract
Restricted cubic splines (RCS) allow analysts to model nonlinear relations between continuous covariates and the outcome in a regression model. When using RCS with the Cox proportional hazards model, there is no longer a single hazard ratio for the continuous variable. Instead, the hazard ratio depends on the values of the covariate for the two individuals being compared. Thus, using age as an example, when one assumes a linear relation between age and the log-hazard of the outcome there is a single hazard ratio comparing any two individuals whose age differs by 1 year. However, when allowing for a nonlinear relation between age and the log-hazard of the outcome, the hazard ratio comparing the hazard of the outcome between a 31- and a 30-year-old may differ from the hazard ratio comparing the hazard of the outcome between an 81- and an 80-year-old. We describe four methods to describe graphically the relation between a continuous variable and the outcome when using RCS with a Cox model. These graphical methods are based on plots of relative hazard ratios, cumulative incidence, hazards, and cumulative hazards against the continuous variable. Using a case study of patients presenting to hospital with heart failure and a series of mathematical derivations, we illustrate that the four methods will produce qualitatively similar conclusions about the nature of the relation between a continuous variable and the outcome. Use of these methods will allow for an intuitive communication of the nature of the relation between the variable and the outcome.