Hazard Ratio Calculator

Use the hazard ratio when comparing time-to-event outcomes between two groups—such as survival after treatment, time until relapse, or duration until equipment failure. The hazard ratio tells you how much more (or less) likely an event is at any given moment in one group compared to another.

What Is a Hazard Ratio?

The hazard ratio (HR) is the ratio of hazard rates between two groups:

$HR = \frac{h_1(t)}{h_0(t)}$

Where:

• $h_1(t)$ = hazard rate in the treatment/exposed group
• $h_0(t)$ = hazard rate in the control/unexposed group

Interpreting the Hazard Ratio

Example: HR = 0.70 for a new cancer treatment means patients on the new treatment have 30% lower risk of death at any given time compared to standard treatment.

Hazard vs. Risk

Risk is cumulative; hazard is instantaneous. Hazard ratio compares these instantaneous rates.

Calculating Hazard Ratio

From Cox Regression

Most hazard ratios come from Cox proportional hazards regression:

$h(t|X) = h_0(t) \exp(\beta X)$

The hazard ratio for a binary predictor X is:

$HR = e^\beta$

Where β is the regression coefficient.

From Summary Data

If you have event counts and person-time:

$HR = \frac{Events_1 / Person\text{-}time_1}{Events_0 / Person\text{-}time_0}$

Example:

• Treatment: 30 events in 500 person-years
• Control: 45 events in 480 person-years

$HR = \frac{30/500}{45/480} = \frac{0.060}{0.094} = 0.64$

Treatment reduces hazard by 36%.

Confidence Interval for HR

The log hazard ratio is approximately normal:

$\ln(HR) \pm z^* \times SE(\ln HR)$

Standard error from Cox regression output, or approximated:

$SE(\ln HR) \approx \sqrt{\frac{1}{E_1} + \frac{1}{E_0}}$

Where E₁ and E₀ are event counts in each group.

Example:

• HR = 0.64, ln(HR) = -0.446
• SE = √(1/30 + 1/45) = √(0.033 + 0.022) = 0.235
• 95% CI for ln(HR): -0.446 ± 1.96(0.235) = [-0.91, 0.02]
• 95% CI for HR: [exp(-0.91), exp(0.02)] = [0.40, 1.02]

Since the CI includes 1, the effect is not statistically significant at α = 0.05.

Proportional Hazards Assumption

The hazard ratio assumes proportional hazards: the ratio stays constant over time.

$\frac{h_1(t)}{h_0(t)} = HR \text{ (constant)}$

Checking the Assumption

1. Kaplan-Meier curves: Should not cross
2. Schoenfeld residuals: Should show no time trend
3. Log-log plot: Should be parallel lines

When It's Violated

If hazards aren't proportional:

• Report time-specific HRs
• Use restricted mean survival time (RMST)
• Stratify by time periods
• Use accelerated failure time models

HR vs. Other Measures

Clinical Interpretation

Reduction in Risk

$\text{Risk reduction} = 1 - HR$

HR = 0.70 → 30% reduction in risk

Increase in Risk

$\text{Risk increase} = HR - 1$

HR = 1.50 → 50% increase in risk

Number Needed to Treat

HR doesn't directly give NNT; you need absolute risks:

$NNT = \frac{1}{|Risk_1 - Risk_0|}$

Applications

Clinical Trials

• Survival after cancer treatment
• Time to disease progression
• Duration of treatment response

Epidemiology

• Mortality by exposure status
• Time to infection
• Age at disease onset

Reliability Engineering

• Time to failure by component type
• Mean time between failures comparison
• Warranty period analysis

Social Science

• Time to recidivism
• Duration of employment
• Time to marriage/divorce

Forest Plot Interpretation

Meta-analyses display HRs on forest plots:

• Vertical line at HR = 1 is null
• Points left of 1: treatment reduces hazard
• Points right of 1: treatment increases hazard
• CI not crossing 1: statistically significant

Adjusted vs. Unadjusted HR

Unadjusted HR: Compares groups without controlling for confounders

Adjusted HR: Controls for covariates in Cox regression

$h(t|X, Z) = h_0(t) \exp(\beta_X X + \beta_Z Z)$

The adjusted HR for X holds Z constant: $HR_X = e^{\beta_X}$

Stratified Analysis

When proportional hazards holds within strata but not overall:

$h(t|X) = h_{0,stratum}(t) \exp(\beta X)$

Each stratum has its own baseline hazard, but HR is assumed common.

Sample Size for HR

To detect a hazard ratio with adequate power:

$n = \frac{(z_\alpha + z_\beta)^2}{p_1 p_0 (\ln HR)^2}$

Where p₁ and p₀ are proportion of events in each group.

More events needed for HR closer to 1.

Common Mistakes

Confusing HR with RR

HR ≠ risk ratio. For long follow-up with many events, they can differ substantially.

Ignoring Non-Proportionality

If curves cross, a single HR is misleading. Report time-varying effects.

Over-interpreting Small HRs

HR = 0.98 is a 2% reduction but may not be clinically meaningful.

Forgetting Censoring

Hazard ratio analysis requires proper handling of censored observations.

Reporting Guidelines

APA/Medical format: "Patients receiving treatment A had a significantly lower mortality rate compared to treatment B (HR = 0.64, 95% CI [0.45, 0.91], p = 0.013)."

Include:

• Hazard ratio with CI
• P-value
• Number of events in each group
• Median survival times if appropriate
• Note if proportional hazards verified

Historical Context

David Cox developed the proportional hazards model in 1972, revolutionizing survival analysis. It allows covariate adjustment without specifying the baseline hazard function.

The hazard ratio became the standard effect measure for time-to-event data in clinical trials and epidemiology.

Related Calculations

• Kaplan-Meier Calculator — Survival curves
• Log-Rank Test — Compare survival distributions
• Relative Risk Calculator — Risk ratio
• NNT Calculator — Number needed to treat

Summary

The hazard ratio compares instantaneous event rates between groups. HR = 1 means no difference; HR < 1 means lower risk in the first group; HR > 1 means higher risk. Derived from Cox regression, HR assumes proportional hazards (constant ratio over time). Verify this assumption before interpreting. Report HR with 95% CI and verify proportionality. HR is the standard effect measure for survival analysis in clinical trials and epidemiological studies.