Hazard Ratio

The HR is a multiplicative measure centered at 1.0, not an additive risk difference.

• HR = 1.0: No difference in hazard between groups.
• HR < 1.0: Reduced hazard (protective effect). An HR of 0.70 indicates a 30% reduction in the instantaneous risk of the event.
• HR > 1.0: Increased hazard. An HR of 2.0 indicates a two-fold increase in instantaneous risk.

Critical nuance: The HR is a ratio of rates, not a ratio of median survival times. A 30% reduction in hazard does not translate to a 30% increase in median survival.

The standard Cox model assumes the HR is constant over time. This is the proportional hazards (PH) assumption.

• The hazard curves for two groups must not cross.
• The effect of a covariate is assumed to be time-invariant.
• Violation example: A surgical intervention may have high initial risk (HR > 1) but long-term benefit (HR < 1). This crossing of hazards invalidates the single-number HR summary.
• Diagnostics: Tested using Schoenfeld residuals and the Grambsch-Therneau test. A significant p-value indicates non-proportionality.
• Remedies: Use time-varying coefficients, stratified Cox models, or alternative estimands like Restricted Mean Survival Time (RMST).

The interpretation of an HR depends critically on the model structure.

• Conditional HR: From a multivariable Cox model. It represents the effect of a treatment holding all other covariates constant. This is a subject-specific effect.
• Marginal HR: From an unadjusted model or a randomized trial. It represents the population-averaged effect.

Key distinction: Due to the non-collapsibility of the hazard ratio, conditional and marginal HRs can differ even in randomized trials when prognostic covariates are included. The conditional HR is typically further from 1.0 than the marginal HR. Always specify which type you are reporting.

The 95% confidence interval (CI) quantifies the precision of the estimated HR and is driven by the number of events, not the total sample size.

• Event-driven: A study with 1,000 patients but only 50 events has wide CIs. A study with 200 patients and 150 events has narrow CIs.
• Crossing 1.0: If the 95% CI includes 1.0 (e.g., 0.85–1.15), the result is not statistically significant at the α=0.05 level.
• Asymmetry on log scale: CIs are computed on the log(HR) scale and exponentiated, making them asymmetric around the point estimate on the HR scale.
• Reporting standard: Always report HR (95% CI, p-value). Example: HR 0.72 (95% CI 0.58–0.89, p=0.002).

The HR is frequently misinterpreted in clinical literature. Avoid these errors:

• It is NOT a risk ratio: The HR conditions on survival to time t, while a risk ratio compares cumulative incidence at a fixed time. They diverge as event rates increase.
• It is NOT a median ratio: An HR of 0.5 does not mean patients live twice as long. The relationship between HR and median survival depends on the shape of the baseline hazard.
• It does not imply constant benefit: Even under PH, a constant HR means a constant relative effect, but the absolute benefit varies with baseline risk.
• Censoring assumptions matter: The HR estimate assumes non-informative censoring. If patients drop out for reasons related to their prognosis, the HR is biased.

When the PH assumption fails, the single HR is misleading. Alternative approaches provide a more nuanced picture.

• Time-varying coefficient: Model the HR as a function of time, e.g., HR(t) = exp(β₀ + β₁ × log(t)). This captures waning or increasing effects.
• Landmark analysis: Estimate HRs conditional on survival to a specific landmark time, providing piecewise constant estimates.
• Reporting pattern: Describe how the effect changes. Example: "The treatment effect was strongest in the first 6 months (HR 0.55) and attenuated thereafter (HR 0.85)."
• Clinical relevance: A time-varying HR often reflects the biological mechanism—acute effects vs. long-term disease modification.