Cox Regression

Definition

Cox proportional-hazards regression is a semi-parametric survival model that relates the hazard — the instantaneous rate of experiencing an event — to one or more explanatory variables through a log-linear function. It assumes proportional hazards, meaning the ratio of hazard rates between groups remains constant over time. This assumption can be checked using Schoenfeld residuals.

Why It Matters

The Cox model is the most widely used regression method in survival analysis because it makes no assumptions about the shape of the underlying baseline hazard. This flexibility allows researchers to focus on understanding how covariates affect the relative risk of an event, rather than modelling the complex temporal dynamics of the baseline risk. It is the standard tool in oncology, cardiology, and any field studying time-to-event outcomes.

Example

A cardiovascular study examines whether statin use reduces the risk of heart attack, adjusting for age, sex, and smoking status. A Cox regression yields a hazard ratio of 0.70 for statin users (95% CI [0.55, 0.89], p = 0.004). This means that, after adjustment, statin users have approximately 30% lower hazard of heart attack at any given time compared with non-users, assuming proportional hazards hold.

Related Terms

• Survival Analysis
• Hazard Ratio
• Kaplan-Meier Curve
• Regression Analysis
• P-Value
• Confidence Interval

Software Notes

• SPSS: Analyze > Survival > Cox Regression. Enter time variable, status indicator, and covariates. Click Options to request hazard ratios (exp(B)) and confidence intervals. Check Plots for survival functions.
• R: survival::coxph(Surv(time, status) ~ x1 + x2, data = df). summary(model) returns hazard ratios, standard errors, z-scores, and p-values. cox.zph(model) tests the proportional-hazards assumption.
• Stata: stcox x1 x2 after stset time, failure(status). Stata reports hazard ratios by default. estat phtest checks proportional hazards. stcurve, hazard plots estimated hazard functions.