The Hazard Ratio

Concept and Logic

The hazard function h(t) represents the instantaneous probability density of experiencing an event at time t, given survival up to that point. The hazard ratio is the ratio of two hazard functions: HR = h1(t) / h0(t). The Cox model assumes this ratio remains constant over time, meaning the cumulative hazard curves run in parallel. HR = 2 means that at any given moment, the event rate in the treatment group is twice that of the control group. It is a rate ratio, not a probability.

Computing and Reading the HR

The Cox proportional-hazards model produces log(HR) as a regression coefficient; the HR is obtained by exponentiating this coefficient: HR = e^beta. Results are reported with a 95 percent confidence interval and a p-value. If the confidence interval includes 1, the difference is not statistically significant. Kaplan-Meier curves do not directly yield a hazard ratio; they display survival probabilities. Each covariate in a Cox model has its own HR, interpreted with all other variables held constant.

Common Misconceptions

The most frequent mistake is interpreting the HR as an absolute risk or probability. HR = 0.5 does not mean the treatment cuts risk in half; it means the instantaneous event rate is halved. A second error is applying the Cox model without testing the proportional hazards assumption, which should be verified using Schoenfeld residuals or log-log plots. A third mistake is reporting a single HR across all time points when the assumption is violated. In such cases, restricted mean survival time (RMST) is a more appropriate summary measure.

Reporting and Why It Matters

The hazard ratio has become the universal language of survival analysis, used across clinical trials and epidemiological cohort studies. Correct reporting requires: the HR value, 95 percent confidence interval, p-value, follow-up duration, results of the proportional hazards assumption test, and a Kaplan-Meier curve. Reporting only the HR can mislead; complementary absolute measures such as median survival time or survival probability at a specific time point should also be provided. This multi-faceted presentation enables clinicians and policymakers to interpret findings accurately.

Sources