Hazard Ratio Calculator

An HR of 0.75 means the treatment group has a 25% lower instantaneous rate of the event (death, recurrence, etc.) compared to the control group at any given time during follow-up. It does not mean 25% fewer total events — the actual difference depends on follow-up duration and baseline event rates.

Relative risk (RR) compares cumulative event probabilities at a fixed time point. Hazard ratio compares instantaneous event rates across the entire study period, accounting for censoring (participants lost to follow-up). HR is preferred for time-to-event data; RR is simpler for fixed-time binary outcomes.

A hazard ratio is statistically significant at the 0.05 level when its 95% confidence interval does not include 1.0. HR 0.72 (CI: 0.55-0.94) is significant because the CI is entirely below 1.0. HR 0.72 (CI: 0.48-1.08) is not significant because the CI crosses 1.0.

Cox regression is the standard statistical model for estimating hazard ratios while adjusting for covariates (age, sex, disease stage, etc.). It models the hazard function as a function of predictor variables, assuming the ratio of hazards between any two groups remains constant over time (proportional hazards assumption).

Yes. HR > 1 means the treatment group has a higher event rate. HR 1.50 indicates a 50% higher hazard in the treatment group — the intervention increases rather than decreases the risk. This can indicate a harmful treatment or simply that the reference group is the treatment group.

The assumption that the hazard ratio remains constant over time. If a treatment works well initially but its benefit fades (or vice versa), the assumption is violated. Violations can be detected by Schoenfeld residual tests or by examining Kaplan-Meier curves that cross or diverge non-proportionally.

Subtract the HR from 1 and multiply by 100. HR 0.72 = (1 - 0.72) × 100 = 28% risk reduction. HR 1.30 = (1.30 - 1) × 100 = 30% risk increase. This gives the percentage change in the instantaneous event rate.

Odds ratios (OR) compare the odds of an event between groups at a single time point and are used in case-control studies and logistic regression. Hazard ratios compare event rates over time and are used in survival analysis. For rare events (< 10%), OR approximates RR, but HR accounts for event timing and censoring.

A Kaplan-Meier curve is a step-function graph showing the probability of surviving (or remaining event-free) over time for each group. The visual separation between curves reflects the hazard ratio. Curves that separate early suggest an immediate treatment effect; delayed separation suggests a latent benefit.

In oncology, HR typically compares overall survival (OS) or progression-free survival (PFS). HR 0.70 for OS means a 30% reduction in instantaneous death rate. HR 0.50 for PFS means a 50% reduction in progression/death rate. Both should be evaluated alongside median survival improvements and absolute risk reduction at landmark time points.

Sample size depends on the expected HR, desired statistical power (typically 80-90%), significance level (0.05), and expected event rate. Detecting HR 0.75 with 80% power requires approximately 380 total events. Smaller HRs (0.90) require far more events (1,600+). Use dedicated sample size software for precise calculations.