Hazard ratio: meaning, interpretation and examples

A hazard ratio compares how fast an event happens in two groups. The hazard is the rate at which the event occurs at a given moment among the people who have not yet had it, and the hazard ratio is one group's hazard divided by the other's. A hazard ratio of 1 means the same rate in both groups. When the hazards are proportional, a hazard ratio of 0.7 means that at every moment the rate in the first group is 70% of the rate in the second: 30% lower.

It is the usual effect measure for survival data. It comes out of the Cox model (Cox, 1972) and, approximately, from the counts of the log-rank test. It is also easy to misread. It is not a ratio of risks, it is not a ratio of survival times, and it does not say how many events a treatment prevents.

The hazard, in plain terms

The hazard at time t is the instantaneous rate of the event at t among those still at risk, in events per person per unit of time. A hazard of 0.2 per year means, if it stays constant, 20 events per 100 person-years at risk. It is a rate, not a probability: it can be larger than 1, its size depends on the time unit, and it can change as follow-up goes on. Survival and cumulative risk are built from it by accumulating the hazard over time.

How a hazard ratio is estimated

The log-rank test compares the number of events observed in each group with the number expected if the groups had the same hazard (Bland and Altman, 2004). An approximate hazard ratio can be taken from the same counts, as the ratio of observed to expected events in one group divided by the same ratio in the other (Berry et al., 1991), though that is reliable only when the hazard ratio is not large. More often it comes from a regression model. In a Cox model the hazard for a person with covariate x is h₀(t) exp(βx), where h₀(t) is a baseline hazard that is left unspecified, and the hazard ratio for a one-unit difference in x is exp(β). Parametric models such as the exponential and the Weibull also give hazard ratios. Spruance et al. (2004) give an introduction to hazard ratios in clinical trials.

The estimate and its standard error are on the log scale, so the 95% confidence interval is exp(β ± 1.96 × SE), where SE is the standard error of β, and it is not symmetric around the hazard ratio. The log scale is also why a hazard ratio of 2 and one of 0.5 are equally strong in opposite directions, and why forest plots of hazard ratios use a logarithmic axis.

A worked example

Suppose the hazard of death in a control group is a constant 0.2 per year and a treatment multiplies it by 0.7. The hazard ratio is 0.7 at every time, and the treated hazard is 0.14 per year. The survival curves follow from the hazards: S(t) = exp(−0.2t) for the controls and exp(−0.14t) for the treated. What the same hazard ratio means for the number of deaths then depends on how long you look:

After 1 year: 18.1% of the controls have died, against 13.1% of the treated. The risk ratio is 0.72 and the risk difference 5.1 percentage points. The risk ratio is close to the hazard ratio because the cumulative risk is still low.
After 5 years: 63.2% against 50.3%. The risk ratio is 0.80 and the risk difference 12.9 percentage points.
After 10 years: 86.5% against 75.3%. The risk ratio is 0.87, and the risk difference has fallen back to 11.1 percentage points, because most of the controls have now died too.
Median survival: 3.5 years for the controls and 5.0 for the treated, since the median is ln 2 divided by the hazard. The ratio of the medians is 1.43, which is 1/0.7. That holds when the hazard is constant (the exponential model) and not in general.

The baseline matters as much as the hazard ratio. Applied to a baseline hazard of 0.02 per year instead of 0.2, the same 0.7 gives 5-year risks of 9.5% and 6.8%, a difference of under 3 percentage points.

Common misreadings

“A hazard ratio of 0.7 means 30% fewer deaths.” If the hazards are proportional, it means a 30% lower rate at each moment among those still at risk. The reduction in the number of deaths depends on the follow-up and on the baseline risk, as the example shows. Sashegyi and Ferry (2017) make the point that one minus the hazard ratio is not a risk reduction in the usual sense.
“It is a relative risk.” A risk ratio compares the proportions who have had the event by a given time. A hazard ratio compares rates. The two are close when the cumulative risk is low in both groups, with a rare event or short follow-up, and move apart as it grows.
“Survival is 1/0.7 times as long.” Only under a constant hazard or, more generally, a Weibull model with the same shape k in both groups, where the ratio of survival times, for the median or any percentile, is the hazard ratio to the power −1/k. A constant hazard is k = 1 and gives 1/0.7.
“It describes the whole follow-up.” Only if the hazards are proportional. Otherwise a single hazard ratio is an average over time that depends on how long people were followed (Hernán, 2010).

When a hazard ratio can mislead

The hazards are not proportional. A delayed benefit, a waning effect or curves that cross make one number a poor summary. What is the proportional hazards assumption? covers how to check, and What a hazard ratio means when hazards are not proportional what the number then estimates.
It is not a clean causal contrast. Even in a randomised trial, people who are most susceptible have their events early and leave the risk set, and if treatment changes how quickly that happens, the groups left at later times are no longer comparable and may differ in more than treatment. Hernán (2010) called this the built-in selection bias of the hazard ratio.
Adjusting changes it. The hazard ratio is non-collapsible, so the hazard ratio adjusted for a prognostic covariate generally differs from the unadjusted one, even when the covariate is not a confounder, because the two answer different questions, a conditional and a marginal one. See What is non-collapsibility?

What to report beside a hazard ratio

Which group is the reference, and, for a continuous covariate, the unit it is per (per year of age, per standard deviation).
The 95% confidence interval and not only the p-value, and whether the model is adjusted and for what.
How proportionality was checked, or how the hazard ratio changes over time.
An absolute measure beside it: survival or risk at stated times together with the baseline, or the difference in restricted mean survival time (RMST or a hazard ratio?).

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What is a hazard ratio?