Hazard Ratio Explained: How to Interpret HRs in Survival Analysis

A hazard ratio compares how quickly an event, such as death or disease progression, occurs in two groups. It is the usual effect measure for time-to-event endpoints such as overall survival, and it feeds indirect comparisons and health economic models. It is also easy to misread.

Risk is the probability of an event over a period, such as five years; hazard is a rate at a moment in time, which is why the Cochrane Handbook calls it instantaneous risk.

Survival methods also handle censoring: patients without the event when the study closes, or who leave early, contribute follow-up time but no event. This works if censoring is non-informative, meaning censored patients are as likely to have the event later as those still followed.

Hazard ratio formula: how HRs are calculated

In words, the hazard ratio formula is the event rate in one group divided by the event rate in the other. In published trials it is usually estimated with a Cox proportional hazards regression model.

The Cox proportional hazards model

The Cox model, introduced by David Cox in 1972, writes a patient’s hazard as a baseline hazard multiplied by the exponential of their covariates: h(t) = h₀(t) × exp(b₁x₁ + … + b_px_p). The baseline hazard is left unspecified, so the model is called semi-parametric. For a treatment indicator with coefficient b, the hazard ratio is exp(b), and the software reports the log hazard ratio and its standard error directly.

Other ways to calculate a hazard ratio

1. Observed and expected events. The log-rank test counts the events expected in each group if there were no difference. Each group’s observed-to-expected ratio (O/E) is its relative hazard, and their ratio approximates the HR. In a lung cancer trial used as the worked example in a British Journal of Cancer tutorial, 54 relapses were observed against 70.6 expected with radiotherapy plus chemotherapy and 70 against 53.4 with radiotherapy alone: 0.765 ÷ 1.311 = 0.58. The tutorial’s authors still prefer Cox regression.
2. Log-rank statistics. The Cochrane Handbook gives the log HR as (O − E) ÷ V, with standard error 1 ÷ √V, where V is the log-rank variance.
3. A published HR and confidence interval. For meta-analysis, take natural logs: a 95% interval is 3.92 standard errors wide on the log scale, so SE = (ln upper − ln lower) ÷ 3.92.

How to interpret a hazard ratio and its 95% confidence interval

Hazard ratio interpretation starts from one rule: an HR is a relative change in the event rate. An HR of 0.70 means the hazard is 30% lower in the intervention arm, at any point in follow-up if hazards are proportional. The confidence interval shows how precisely it is estimated.

• HR below 1. The event occurs at a lower rate with the intervention. In the lung cancer trial above, an HR of 0.58 meant a 42% lower risk of relapse at any point in time.
• HR above 1. The event occurs at a higher rate. Whether that is good depends on the event: for time to healing, a higher hazard is a benefit.
• Confidence interval including 1. The data are compatible with no difference. A BMJ statistics note uses a screening example with HR 0.73 (95% CI 0.47 to 1.13) for death from colorectal cancer: a 27% lower hazard, but not statistically significant.

Spruance and colleagues add a probability reading: under proportional hazards, HR ÷ (1 + HR) is the chance that a patient in the intervention group has the event before a patient in the comparator group. An HR of 2 means about a 67% chance; an HR of 0.70 gives 0.70 ÷ 1.70 = 0.41.

Four common misreadings

Worked example: one hazard ratio, changing risks

The same HR produces different absolute effects at different times. This hypothetical example uses round numbers and constant hazards, so the HR stays at 0.70 throughout; it is not real trial data.

Suppose the comparator arm has 10 deaths per 100 patient-years (a hazard of 0.10 a year) and the new treatment 7 per 100 (0.07). The HR is 0.07 ÷ 0.10 = 0.70. With a constant hazard, the proportion still alive at time t is e^{−hazard × t}.

1. The risk ratio drifts towards 1. As deaths accumulate in both arms, the share who have died is never quite 30% lower; by 10 years it is 20% lower (50.3% against 63.2%).
2. The odds ratio drifts away from 1. It moves from 0.69 to 0.59 as events become common.
3. The absolute difference grows. It rises from about 3 percentage points at one year to about 13 at ten.

Median survival is 6.9 years with the comparator and 9.9 years with the new treatment. Their ratio equals the HR here only because the hazards are constant; real curves rarely behave like that, as the Spruance examples show. Restricted to 10 years, mean survival (the restricted mean survival time, explained below) is 6.32 against 7.19 years: a gain of 0.87 years.

Hazard ratio vs odds ratio vs relative risk

A relative risk (risk ratio) compares the proportion of people with the event by a fixed time; an odds ratio compares the odds of the event by that time; a hazard ratio compares event rates throughout follow-up and uses censored data.

Hazard ratio vs odds ratio

The Cochrane Handbook notes that odds and risks differ little when events are rare but a lot when they are common, and that misreading an odds ratio as a risk ratio exaggerates the effect, especially when risks exceed about 20%. The worked example shows the same pattern against the HR: by ten years the odds ratio is 0.59 while the HR is 0.70.

Hazard ratio vs relative risk

Time-to-event outcomes can be analysed as a yes/no outcome at a fixed time point, giving a risk ratio, but that ignores when events happen. Cochrane also warns against treating time to event as a continuous outcome, because censored participants would have to be excluded, which almost certainly introduces bias. The HR uses all follow-up; the price is the proportional hazards assumption.

The proportional hazards assumption and what happens when it fails

A single HR assumes that the ratio of hazards stays constant over follow-up, although each hazard may rise or fall. Under this assumption the hazard curves of the two groups cannot cross. When the ratio changes over time, one number no longer describes the treatment effect.

How to check it

• Log-cumulative hazard plots. Roughly parallel lines for the two arms support proportional hazards. NICE Decision Support Unit (DSU) Technical Support Document (TSD) 14 recommends them, together with external evidence and expert opinion.
• Formal tests. Schoenfeld residuals and tests of a treatment-by-time interaction.

Why a time-varying HR misleads

Hernán’s 2010 commentary in Epidemiology uses the Women’s Health Initiative trial of oestrogen plus progestin, which reported an HR of 1.24 for coronary heart disease. Year-by-year HRs ranged from 1.81 in year one to 0.70 from year six onwards, so the average HR would have been about 1.8 had the trial stopped after one year and about 1.2 after five. He also shows that later-period HRs carry a built-in selection bias, because the most susceptible patients have already had the event, and recommends reporting survival curves.

Immunotherapy: delayed and crossing curves

In an EMA-funded review of EMA assessment reports, posted as a preprint in 2024, 12 of 16 procedures with non-proportional hazards were in oncology, most involving immune checkpoint inhibitors. A delayed treatment effect was the most common pattern, with curves separating after roughly two to nine months; some crossed.

In KEYNOTE-189, pembrolizumab plus chemotherapy in non-squamous lung cancer, Tian and colleagues note that the overall survival curves did not separate for about the first three months. In their reanalysis of survival data reconstructed from the published curves, the unstratified HR was 0.52 (95% CI 0.40 to 0.68), but they also report 12-month survival of 68.3% against 48.8% and an 18-month restricted mean survival of 14.0 against 11.5 months, a 2.5-month gain.

Alternatives to a single HR

1. Restricted mean survival time (RMST). The area under the survival curve up to a chosen time: life expectancy over that window. Royston and Parmar argue the HR cannot be recommended as a general measure of treatment effect, while RMST is valid under any distribution of event times.
2. Survival at fixed time points. Milestone survival, such as 12-month overall survival, with differences in absolute terms.
3. Time-varying effects. Piecewise HRs for separate periods, or flexible parametric models that let the effect change.

Why HTA models do not plug in a hazard ratio directly

Cost-effectiveness models need mean survival over a horizon that usually runs far beyond the trial, while a trial gives an HR within limited follow-up. Analysts therefore fit and extrapolate survival curves, and use an HR only where its assumptions hold.

TSD 14 makes the main recommendations: estimate mean rather than median survival; use proportional hazards modelling only when clearly justified, otherwise fit separate parametric models of the same type to each arm; take any HR from the parametric model fitted to the data, not from a Cox model; and test how long the treatment effect lasts. TSD 21 covers flexible models for complex hazards, and TSD 16 and TSD 24 adjusting for control patients who switch to the new treatment. Extrapolation is explained in our guide to the partitioned survival model.

Practice varies between agencies. Reviewing the guidelines of 10 HTA agencies and HTA reports for 23 cancer medicines approved since 2014, Monnickendam and colleagues found that only NICE widely tested the proportional hazards assumption, and RMST appeared rarely.

From hazard ratio to QALYs

In a model, the HR scales a baseline rate; in a Markov model it applies to rates, not probabilities. Modelled survival becomes life years, utility weights turn them into QALYs, and QALYs drive the ICER (see what a QALY is). In the worked example, an invented utility of 0.75 turns the 0.87 life years gained within 10 years into about 0.65 QALYs; if both constant hazards continued indefinitely, mean survival would be 14.3 against 10 years, a gain of 4.3 life years from the same HR. Try the arithmetic in our QALY calculator.

The HR’s uncertainty matters too: its log standard error feeds probabilistic sensitivity analysis. Model structures are taught in Module 8: Economic Modelling.

Hazard ratios in indirect comparisons and EU joint clinical assessment

When trials do not compare the treatments a decision needs, analysts combine HRs from different trials on the log scale through a common comparator, as in the Bucher method or a network meta-analysis. Those results inherit the proportional hazards assumption of every trial in the network.

The EU HTA Coordination Group’s methodological guideline on direct and indirect comparisons, adopted on 8 March 2024, warns that the assumption is often implausible, most obviously when survival curves intersect, and that severe violations can bias the HR or make it uninterpretable. It points to network meta-analysis of parametric survival curves or fractional polynomials, where the effect is multidimensional, and requires every method’s assumptions to be specified and justified.

The Group’s outcomes guidance, adopted on 10 June 2024, lists the HR among ratio effect measures and asks for one measure of treatment effect to be pre-specified as the primary analysis. The mechanics are covered in our guide to network meta-analysis, and how assessors have treated indirect evidence in practice in our tracker of JCA reports. Evidence synthesis is taught in Module 4.

How to read a hazard ratio in an HTA submission: checklist

Before an HR enters a dossier, an indirect comparison or a model, a reviewer should be able to answer seven questions:

1. Which event and population? Overall or progression-free survival, or a composite, in which analysis set.
2. How was it estimated? Cox model, stratified or not, with which covariates.
3. How precise is it? The 95% CI, number of events and length of follow-up.
4. Are hazards proportional? Kaplan–Meier curves with numbers at risk, log-cumulative hazard plots and formal tests.
5. What is the absolute effect? Survival at fixed time points and RMST alongside the HR.
6. Was there treatment switching? If so, how survival was adjusted.
7. How does the model use it? From which fitted model, and for how long the effect is assumed to last.