Hazard Ratio

Definition

A hazard ratio is the ratio of the hazard rates — the instantaneous risk of an event occurring — between two groups. In a Cox proportional-hazards model, the hazard ratio quantifies how much faster or slower one group experiences the event compared with another, assuming the ratio remains constant over time.

Why It Matters

In survival analysis, the hazard ratio is the standard measure of treatment effect. Unlike the odds ratio, which is a static comparison of proportions, the hazard ratio accounts for the timing of events and the presence of censored observations. It is the primary outcome reported in oncology trials, cardiovascular studies, and any research where the speed of an event matters as much as its occurrence.

Example

A cancer trial compares a new immunotherapy against standard chemotherapy. The Cox regression yields a hazard ratio of 0.65 for the immunotherapy group (95% CI [0.50, 0.85], p = 0.002). This means that, at any given time after randomisation, patients receiving immunotherapy have 35% lower hazard of disease progression than those on chemotherapy. Median progression-free survival is 14 months versus 9 months, a difference driven by the consistently lower hazard over time.

Related Terms

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

Software Notes

• SPSS: Analyze > Survival > Cox Regression. Enter the time variable, status indicator, and covariates. SPSS reports hazard ratios as exp(B) with confidence intervals in the Variables in the Equation table.
• R: survival::coxph(Surv(time, status) ~ group, data = df). summary(model) returns hazard ratios, standard errors, z-scores, and p-values. ggforest() from survminer draws forest plots of hazard ratios.
• Stata: stcox group x1 x2 after stset time, failure(status). Stata reports hazard ratios by default. estat phtest checks the proportional-hazards assumption.