Regression Splines in the Cox Model with Application to Covariate Effects in Liver Disease

Abstract

The Cox proportional hazards model restricts the log hazard ratio to be linear in the covariates. A smooth nonlinear covariate effect may go undetected in this model but can be well approximated by a spline function. A survival model based on data from a clinical trial of primary biliary cirrhosis is developed using regression splines, and the resulting log hazard ratio estimates are compared with those from nonparametric methods. We remove the linear restriction on the log hazard ratio by transforming a continuous covariate into a vector of fixed knot basis splines (B-splines). B-splines are known to produce better-conditioned systems of equations than the truncated power basis when used as interpolants, and show similar behavior when fitting proportional hazards models. We describe the procedures for, and the issues arising in, the estimation and the testing of the B-spline coefficients. Although inference is not well developed for some nonparametric methods that estimate covariate effects, the asymptotic theory for Cox model B-spline estimates is relatively straightforward. An S function for fitting B-splines in Cox regression models is available.

Key Words:

• Proportional hazards
• Relative risk
• Survival analysis
• B-spline
• Spline approximation