What is a hazard ratio?

A hazard ratio is the ratio of the instantaneous event rate in one group to the instantaneous event rate in another, calculated at each moment among the participants still alive and still event-free at that moment. A hazard ratio of 0.75 means the treated group experiences events at three-quarters of the control rate, averaged across follow-up. A hazard ratio is not a risk ratio.

Almost every misreading of a survival analysis traces to that last sentence. This page covers what a hazard is, how the Cox model produces the number, what a hazard ratio does and does not say about how long anyone lives, the proportional hazards assumption on which the single-number summary rests, why the number is not fully causal even in a randomised trial, why precision depends on events rather than participants, and how the mishandling shows up in a manuscript.

What a hazard is

The hazard at time t is the rate at which events occur among those who have survived to t. Formally it is the limit, as the interval shrinks to zero, of the probability of an event in the next instant given that the participant has reached t without one, divided by the length of that instant.

Two properties follow, and both are routinely forgotten.

A hazard is a rate, not a probability. Hazards have units of events per person-time and can exceed 1. A hazard of 1.4 per person-year is a perfectly ordinary quantity; a probability of 1.4 is not.

A hazard is conditional on survival to that point. The denominator of the hazard at 18 months contains only participants who reached 18 months. That conditioning is the source of most of the interpretive difficulty later on this page, because who reaches 18 months is itself affected by treatment.

The hazard ratio divides one group’s hazard function by another’s. When that quotient is constant over time, one number describes the whole comparison. When it is not, one number describes a weighted average whose weights nobody chose deliberately.

How the number is produced

A hazard ratio in a clinical paper almost always comes from a Cox proportional hazards regression, introduced by D. R. Cox in “Regression Models and Life-Tables” (Journal of the Royal Statistical Society, Series B, 1972). The Cox model specifies the hazard for an individual as an unspecified baseline hazard multiplied by the exponential of a linear predictor, so the baseline shape never has to be estimated.

Cox’s partial likelihood is what makes that possible. At each observed event time, the partial likelihood asks only which member of the risk set had the event, conditional on one occurring. The baseline hazard cancels from that conditional probability, and the coefficients are estimated from the ordering of events rather than from their spacing. This is why the Cox model is described as semi-parametric, and why a Cox hazard ratio carries no information about the time scale at all.

Two implementation details change published numbers and are worth naming. First, the log-rank test is the score test from the Cox partial likelihood with a single binary covariate, which is why the log-rank p-value and the Cox hazard ratio in the same paper agree so closely: they are two readings of one model. Second, tie handling differs by software — R’s survival::coxph defaults to the Efron approximation, SAS PROC PHREG defaults to Breslow — and with heavily tied event times, Breslow biases the estimate toward the null. Papers using coarse follow-up intervals, such as annual registry updates, generate exactly the tie structure where this matters.

Reading a hazard ratio correctly

A hazard ratio of 0.75 means a 25% lower event rate, not 25% fewer deaths, not 25% longer survival, and not a 25% chance of benefit for any individual patient. The correct sentence names the rate: “the treated group experienced events at 75% of the rate observed in the control group over the follow-up period.”

Three specific translations are wrong often enough to be worth stating separately.

A hazard ratio is not a risk ratio. A risk ratio compares cumulative probabilities at a stated time point: 12-month mortality of 18% versus 24% is a risk ratio of 0.75, an entirely different calculation. Hazard ratios and risk ratios coincide only when events are rare; as the cumulative incidence rises, the hazard ratio sits further from 1 than the corresponding risk ratio.

A hazard ratio contains no time units. Nothing in the number tells a reader whether the trial ran for six months or six years, or whether the two curves separated at week 3 or week 30. A hazard ratio of 0.60 from an eight-week study and one from a ten-year study are the same number and mean very different things clinically, which is why reviewers ask whether the effect size is meaningful rather than merely significant.

A hazard ratio maps to median survival only under an assumption. If both groups’ event times are exponentially distributed — constant hazard — then the ratio of medians is the reciprocal of the hazard ratio. A control median of 12 months with a hazard ratio of 0.75 implies a treated median of 16 months. When the hazard is not constant, which is the ordinary case, that arithmetic fails and the observed median difference can be much smaller or much larger than the hazard ratio suggests.

The proportional hazards assumption

The proportional hazards assumption states that the ratio of the two hazard functions is constant over the whole follow-up period. Proportional hazards does not require the hazards themselves to be constant — each may rise, fall or be lumpy — only that their ratio holds steady.

Three checks are standard, and a manuscript should report at least one.

Scaled Schoenfeld residuals. Grambsch and Therneau’s test (Biometrika, 1994), implemented as cox.zph in R, regresses the scaled Schoenfeld residuals on a function of time and tests whether the slope differs from zero. A non-zero slope means the log hazard ratio is drifting.

Complementary log-log plots. Plotting log(−log S(t)) against log t gives approximately parallel curves under proportional hazards and visibly converging or crossing curves when it fails.

An explicit treatment-by-time interaction. Fitting a time-varying coefficient and testing it is the most interpretable check, because a rejected test comes with an estimate of how the ratio moves.

One honest caveat belongs with all three: a global proportional hazards test has low power when the number of events is small, so “p = 0.42 for the Schoenfeld test” in a trial with 60 events is weak reassurance rather than evidence of proportionality. Report the plot alongside the p-value.

What a hazard ratio becomes when proportionality fails

When the hazard ratio varies over time, a Cox model still returns a single number, and that number is a weighted average of the time-varying log hazard ratio. The weights depend on the censoring and event distribution — roughly, on how many participants remain at risk at each time — which means the reported hazard ratio is partly a function of when the data were locked rather than of the treatment alone. Two analyses of the same trial with different follow-up durations can report materially different hazard ratios with no change in the underlying biology.

Three patterns produce this, and each has a recognisable curve shape. Delayed separation, common in immunotherapy trials where the curves overlap for months before diverging, dilutes the average toward the null. Diminishing effect, common where a procedural risk is front-loaded, does the opposite early on. And curves that actually cross make the single summary close to meaningless, because it averages a harm and a benefit into one figure that describes neither.

The standard remedy is restricted mean survival time. RMST is the area under the survival curve up to a pre-specified horizon τ, reported in months, and the difference in RMST between arms is a valid summary whether or not hazards are proportional — Uno and colleagues set out the argument for oncology in the Journal of Clinical Oncology in 2014. Two conditions apply: τ must be chosen before looking at the curves, and τ must lie within the follow-up of both arms. Landmark analyses and piecewise hazard ratios by interval are the other accepted options.

Non-collapsibility: why the adjusted and unadjusted numbers differ

The hazard ratio is non-collapsible, which means the adjusted and unadjusted values differ even when the covariate is not a confounder. This surprises people who expect a randomised trial to give the same answer either way.

The mechanism is subject heterogeneity. When a strong prognostic covariate is omitted, the high-risk participants in both arms experience events first, so the risk sets at later times are progressively enriched with lower-risk participants — and enriched unequally, because the arm with more events depletes its high-risk members faster. The marginal hazard ratio therefore drifts toward 1 relative to the covariate-adjusted one, and drifts further the longer follow-up runs.

Two practical consequences. A trial reporting both an unadjusted and a covariate-adjusted hazard ratio should not present the gap between them as evidence of imbalance or of confounding; some of that gap is arithmetic. And a hazard ratio is not transportable in the way a risk difference is: the adjusted hazard ratio answers a question about individuals with a given covariate profile, while the unadjusted one answers a question about the trial population, and the two are different estimands even when both are unbiased for their own target.

Why a hazard ratio is not simply a causal effect

A hazard ratio at any time after randomisation is conditional on survival to that time, and survival is a consequence of treatment. Miguel Hernán made the argument compactly in “The hazards of hazard ratios” (Epidemiology, 2010): at time zero the two arms are exchangeable by randomisation, but from the first event onward the comparison is made within risk sets that treatment itself has shaped. Conditioning on a variable that treatment affects is collider conditioning, and it can induce an association between treatment and unmeasured frailty inside the risk set.

The practical reading is a hedge worth stating plainly rather than a reason to abandon the method. The hazard ratio at time zero is causal in a randomised trial; the average hazard ratio over follow-up is a valid comparison of observed rates but has no clean interpretation as an individual-level causal contrast. This is one reason cumulative incidence differences and RMST have gained ground in trial reporting: both are collapsible, both retain the randomisation guarantee at every time point, and both are stated in units a clinician can use.

Observational hazard ratios carry the same issue plus every ordinary threat to validity. Time-zero misalignment is the most damaging: when follow-up is counted from a date before the exposure could be assigned, the resulting immortal time bias produces spuriously protective hazard ratios, and it appears repeatedly in registry and claims analyses of treatment effectiveness.

Precision comes from events, not participants

The precision of a hazard ratio is governed almost entirely by the number of events observed. For a two-arm trial with 1:1 allocation, the standard error of the log hazard ratio is approximately 2 divided by the square root of the total event count, with the number of randomised participants entering only through how many events they generate.

The arithmetic is worth doing once. With 380 events, the standard error of the log hazard ratio is about 0.103, so an observed hazard ratio of 0.75 carries a 95% confidence interval of roughly 0.61 to 0.92. With 100 events, the standard error is about 0.20, and the same point estimate of 0.75 gives roughly 0.51 to 1.11 — an interval that includes no effect.

Schoenfeld’s sample-size formula (Biometrics, 1983) inverts this: the required number of events is approximately 4 times the squared sum of the two normal quantiles, divided by the square of the log hazard ratio. At two-sided α = 0.05 and 80% power, detecting a hazard ratio of 0.75 needs about 380 events, a hazard ratio of 0.50 needs about 65, and a hazard ratio of 0.90 needs roughly 2,830. Because the log hazard ratio enters as a square, that escalation is steep as the effect approaches the null, which is why event-driven trial designs specify a target event count rather than a target enrolment, and why a paper reporting 4,000 participants and 41 events is underpowered regardless of how large the cohort sounds.

A related consequence for readers of manuscripts: a hazard ratio estimated with fewer than roughly 10 events per covariate is unstable, and a Cox model with 12 adjustment variables fitted to 55 events should be read as exploratory whatever the p-values say.

Competing risks change which hazard ratio you want

A competing risk is an event that prevents the event of interest from occurring — most commonly death from another cause. Two different hazard ratios exist in that setting and they answer different questions, so the choice needs stating rather than defaulting.

The cause-specific hazard ratio treats competing events as censored and describes the rate of the event of interest among those still event-free and still alive. It is the aetiological quantity: use it to ask whether an exposure influences the biological process.

The subdistribution hazard ratio, from Fine and Gray (Journal of the American Statistical Association, 1999), keeps participants who experienced a competing event in the risk set with declining weight, so the estimate maps directly onto cumulative incidence. It is the prognostic quantity: use it to ask what proportion of a population will experience the event by a given time.

Treating a competing event as ordinary censoring assumes it is independent of the event of interest, which is rarely defensible when the competing event is death. In an elderly cohort, censoring non-cardiac deaths and reporting a cause-specific hazard ratio for cardiac events as if it described absolute risk overstates the burden that treatment could remove. Competing-risks reporting is a standard statistical objection wherever the study population has substantial mortality from other causes, which is most of cardiology and much of oncology.

How the mishandling is detected in a real manuscript

Detection in a submitted manuscript is largely a matter of reading the hazard ratio against the figure, the methods and the abstract at once, because each mishandling leaves a tell in a different place.

The abstract sentence contradicts the estimand. “Treatment reduced mortality by 30% (HR 0.70)” is the defect to look for first. A 30% reduction in what — deaths, rate, risk at 12 months? Cross-check whether absolute event counts appear anywhere; frequently they do not.

The Kaplan–Meier figure and the single hazard ratio disagree. Curves that overlap for the first 20% of follow-up, converge late, or cross, reported alongside one unqualified hazard ratio and no mention of proportionality, indicate the assumption was never examined.

The methods section omits any proportional hazards check. Search the text for “Schoenfeld”, “proportional hazards”, “log-log” and “time-varying”. Absence of all four in a paper whose primary endpoint is a hazard ratio is a reportable gap.

Event counts are missing or small relative to the model. Count the adjustment covariates and divide the events by that number. Any ratio below about 10 warrants a comment.

The time origin is unstated or misaligned. In observational work, check that follow-up starts at the moment eligibility, exposure assignment and the start of follow-up coincide. Where they do not, immortal time is present.

Every covariate’s hazard ratio is interpreted. A model fitted to estimate one exposure effect is not simultaneously valid for the other coefficients in the table — that is the Table 2 fallacy, and adjusted hazard ratios for covariates are among its commonest carriers.

Subgroup hazard ratios appear without an interaction test. A forest plot of subgroup hazard ratios with confidence intervals but no formal test for interaction invites a claim the design cannot support.

What a peer reviewer says when it has been mishandled

Reviewer comments on hazard ratios are unusually formulaic, which makes them easy to anticipate before submission.

“The proportional hazards assumption is not addressed; please report a formal assessment.” “The survival curves appear to cross, so a single hazard ratio is not an adequate summary of the treatment effect; consider restricted mean survival time.” “The hazard ratio is interpreted throughout as a reduction in risk. Please report absolute event rates and the risk difference at clinically relevant time points.” “Competing risks appear to have been treated as censoring; please justify the independence assumption or report cumulative incidence.” “Please state the time origin and confirm that follow-up began at the point of treatment assignment.” “The number of events, rather than the number of participants, determines the precision of these estimates; please report the event count for each analysis.” “A hazard ratio is reported without a confidence interval.” “The adjusted and unadjusted hazard ratios differ substantially; the authors attribute this to confounding, but the hazard ratio is non-collapsible and some difference is expected.”

The third is the one worth pre-empting hardest. A reviewer who sees a rate ratio described as a risk reduction may conclude that the analysis was not conducted by someone who understood the model, and that conclusion contaminates their reading of everything else — a pattern visible across statistical objections generally.

How to report a hazard ratio

Report the hazard ratio with its 95% confidence interval and the total number of events, in that order, in both the abstract and the results.

State the estimand in words: which events, over what period, from what time origin, in whom. State how the proportional hazards assumption was assessed and what the assessment showed, including the plot. Where proportionality is doubtful, report RMST at a pre-specified horizon or piecewise hazard ratios rather than defending the single number.

Pair every hazard ratio with an absolute measure — event counts by arm, cumulative incidence at named time points, or an RMST difference in months. Of the changes on this page, this is the one that does most for a survival paper’s clinical readability, because it answers the question a clinician actually brought to the abstract. Name the tie-handling method and the software version. If a covariate-adjusted hazard ratio is the primary result, pre-specify the adjustment set and say so.

Limitations worth stating rather than hiding

The hazard ratio has genuine limitations that a careful paper acknowledges rather than works around silently.

A hazard ratio does not describe any individual’s experience, and no re-expression makes it do so. It is not collapsible, so it does not combine across strata the way a risk difference does. It has no clean causal interpretation after time zero. It depends on the censoring distribution when hazards are non-proportional, so it is partly an artefact of study duration. And it is silent about magnitude in units a patient recognises.

None of this makes the hazard ratio a bad estimate. It remains the most efficient available summary of a time-to-event comparison under proportional hazards, it handles censoring correctly, and the log-rank test built on the same model is the most powerful rank test against a proportional-hazards alternative. The failure mode is not using it; the failure mode is using it as though it answered the question a reader actually asked, which is usually about absolute benefit over a stated period. Papers that report both, and say why, give statistical review much less to object to.

Related

Collider bias · Confounding · Immortal time bias · Lead time bias · My survival curves cross · Table 2 fallacy

Checked before submission by causal language discipline, which flags a hazard ratio described as a risk reduction, and by the competing-risks check, which reports whether competing events were censored without justification.

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Frequently asked questions

What does a hazard ratio mean in plain terms?

A hazard ratio compares how fast events happen in two groups at each moment, counting only people still at risk. A hazard ratio of 0.70 means the treated group had events at 70% of the control group’s rate across follow-up. It describes a rate, not a number of people and not a length of time.

What does a hazard ratio of 0.75 mean?

A hazard ratio of 0.75 means a 25% lower event rate in the treated group, averaged over the follow-up period. It does not mean 25% fewer deaths, 25% longer survival, or a 25% chance of benefit for a given patient. Absolute event counts are needed to say what 0.75 is worth clinically.

Is a hazard ratio the same as a relative risk?

No. A relative risk compares cumulative probabilities at one stated time point; a hazard ratio compares instantaneous rates among those still at risk at every time point. The two agree closely only when events are rare. As cumulative incidence rises, the hazard ratio lies further from 1 than the corresponding relative risk.

What is a hazard ratio in a Cox regression?

A hazard ratio in a Cox proportional hazards model is the exponentiated regression coefficient, estimated from the ordering of event times by Cox’s partial likelihood. The baseline hazard cancels out of that likelihood, so the estimate carries no information about the underlying time scale — only about relative rates.

What does a hazard ratio greater than 1 mean?

A hazard ratio greater than 1 means the exposed or treated group experienced events at a higher rate than the comparison group. A hazard ratio of 1.4 indicates a 40% higher event rate. Whether that difference is real depends on the confidence interval, and whether it matters depends on the absolute event rates.

How do you interpret a hazard ratio in survival analysis?

Interpret a hazard ratio as a relative event rate over the whole follow-up period, then check three things: the confidence interval, the total event count, and whether the survival curves support a constant ratio. If the curves converge or cross, the single hazard ratio is an average of changing effects and should be replaced or supplemented.

Why is a hazard ratio not a percentage reduction in deaths?

A hazard ratio compares rates within shrinking risk sets, while a percentage reduction in deaths compares counts in fixed populations. The two coincide only under rare events and constant hazards. Reporting a hazard ratio of 0.70 as “30% fewer deaths” overstates absolute benefit whenever cumulative incidence is appreciable.