Units & Measures

Hazard Ratio (HR): Time-to-Event Risk Measure

Hazard ratios compare the instantaneous risk (hazard) of an event between two groups over time. An HR above 1 indicates higher instantaneous risk in the exposed group; an HR below 1 indicates lower risk. As a dimensionless quantity derived from rates, HRs are central to clinical trials, reliability studies, and public-health surveillance. This explainer covers definitions, history, modeling concepts, applications, and best practices.

Key facts

  • Quantity. HR = hazard in treatment group ÷ hazard in reference group; dimensionless.
  • Model. Commonly estimated via Cox proportional hazards regression using log-linear covariate effects.
  • Assumptions. Proportional hazards over time and independent censoring underpin standard interpretation.

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Calculators

Definition and estimation

The hazard function h(t) represents the instantaneous event rate at time t conditional on survival until t. The hazard ratio compares hazards between groups: HR(t) = h1(t) ÷ h0(t). When proportional hazards hold, HR is assumed constant over time and estimated by exponentiating coefficients from a Cox model. For parametric models (exponential, Weibull), HR derives from closed-form hazard functions. Always report whether HRs are time-dependent or adjusted for covariates.

Historical background

Sir David Cox introduced the proportional hazards model in 1972, providing a semi-parametric framework for estimating HRs without specifying baseline hazards. The approach revolutionized clinical trial analysis and reliability engineering by enabling covariate-adjusted time-to-event comparisons. Subsequent developments extended the model to stratified designs, time-varying covariates, and competing risks, ensuring HRs remain interpretable across diverse applications.

Concepts and diagnostics

Proportional hazards imply parallel log-cumulative hazard curves. Schoenfeld residuals, log–log plots, and time-by-covariate interactions test this assumption. When hazards cross, report time-specific HRs or adopt flexible models such as restricted mean survival time differences. Right-censoring should be independent of future event risk; informative censoring requires sensitivity analyses or joint models. Document handling of ties, stratification factors, and baseline hazard estimation methods.

Applications

  • Clinical trials. HRs summarize treatment efficacy for endpoints like progression-free survival or overall survival.
  • Reliability engineering. Component lifetimes under varying stress profiles are compared with HRs to guide maintenance schedules.
  • Environmental health. Exposure cohorts use HRs to link pollutant levels with time to hospitalization or mortality.
  • Cybersecurity and operations. Time-to-failure or time-to-incident analyses apply HRs to evaluate controls and patch strategies.

Importance and best practices

Report HRs with 95% confidence intervals derived from log-scale standard errors using the linked calculator. Provide median survival estimates or survival curves to complement HRs, especially when hazards are not proportional. Clearly state the time origin, censoring rules, covariates, and model diagnostics. Sensitivity analyses for competing risks, non-proportional hazards, and missing data strengthen conclusions and reproducibility.

Because HRs are dimensionless ratios of rates, careful documentation of event definitions, follow-up schedules, and data quality is essential. Align reporting with CONSORT or STROBE guidelines, and share code when possible to enable independent verification.

Calculators to support hazard-ratio work

Use these tools to build interval estimates and plan sample sizes that keep HR analyses transparent.

  • Confidence Interval Calculator

    Translate log hazard ratios and standard errors into interval estimates.

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  • Bayesian A/B Test Sample Size

    Approximate power targets by mapping desired HR detection thresholds to event counts.

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  • Mann–Whitney Sample Size

    Plan nonparametric time-to-event comparisons when proportional hazards are uncertain.

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