Cox Proportional Hazards Regression

The hazard ratio (HR) for a predictor is exp(β̂) where β̂ is the estimated regression coefficient. An HR of 1.5 means that at any point in time, subjects with one additional unit of the predictor have a 50% higher instantaneous event rate than the reference group. An HR below 1 indicates a protective association. Always report the HR with a 95% confidence interval; a p-value alone does not convey the direction or magnitude of the effect.

The standard approach is to compute Schoenfeld residuals for each covariate and test whether they show a trend over time using the Grambsch–Therneau test (cox.zph in R). A statistically significant test for a covariate (p < 0.05) indicates that the hazard ratio for that covariate is not constant over follow-up. A complementary visual check is the log-log survival plot: if the proportional hazards assumption holds, the curves for different groups should be roughly parallel on a log(−log) scale.

Several options exist depending on the nature of the violation. If the effect of a predictor changes smoothly over time, time-varying coefficients — also called a time-by-covariate interaction — can be added to the model. If a stratification variable violates the assumption but its effect is not of primary interest, the model can be stratified by that variable, allowing each stratum its own baseline hazard. Alternatively, an accelerated failure time (AFT) model or a flexible parametric model may be more appropriate.

Fine-Gray regression is appropriate when subjects can experience one of several distinct event types and you want to model the cumulative incidence of a specific event type in the presence of the others. In such a competing-risks setting, treating other-cause events as simple censoring in a Cox model can underestimate the cumulative incidence of the event of interest. The Fine-Gray model directly models the subdistribution hazard, which corresponds to the cumulative incidence function.