Hazard Ratio

Hazard Ratio: Interpretation, Proportional Hazards and Use in Survival Models

In health economics the hazard ratio (HR) is the usual summary of a treatment effect on overall survival, progression or any other time-to-event outcome, and it is often the single number carried from a trial report into a survival or state-transition model. It is a ratio of two hazard rates, so it is a relative measure and says nothing about absolute benefit until it is applied to a baseline survival curve. This page covers the formula and its link to the Cox model, an illustrative two-year example that compares the HR with the relative risk, odds ratio, median and restricted mean survival time, how the proportional hazards assumption is checked, and how an HR is used to extrapolate survival and derive transition probabilities in economic evaluation.

Compare event rates within the risk sets

The hazard $h(t)$ describes an instantaneous event rate at time $t$ among individuals who have not yet experienced that event and remain under observation. Comparing treatment and comparator hazards at the same time gives $\operatorname{HR}(t)=h_T(t)/h_C(t)$, provided the comparator hazard is positive. The event, groups, time origin and analysis population all need to be specified, because these choices determine what the ratio refers to.

An HR below one indicates a lower treatment-group hazard of the named event at the time being compared; an HR above one indicates a higher hazard. For a harmful event such as death, lower can be favourable, while for a desirable event such as recovery the direction reverses. A hazard ratio of 0.70 is a 30% lower hazard under a constant-ratio interpretation, not automatically a 30% lower two-year event probability or a 30% longer survival time.

Link proportional hazards to a baseline curve

In a proportional-hazards model, the treatment-to-comparator ratio is assumed constant over the period modelled. A Cox regression with a binary treatment covariate commonly writes the treatment HR as $\exp(\beta)$, with the model conditioning on any included covariates. The baseline hazard can still change over time; a constant ratio is not the same as a constant hazard within either arm.

$$ h_T(t)=r,h_C(t),\qquad r=\exp(\beta), \qquad S_T(t)=\bigl[S_C(t)\bigr]^r, $$

where $h_T(t)$ and $h_C(t)$ are the treatment and comparator hazards at time $t$, $\beta$ is the Cox regression coefficient for treatment, $r$ is the hazard ratio, and $S_T(t)$ and $S_C(t)$ are the probabilities of remaining free of the event to time $t$ in each arm. The survival-curve relation follows under proportional hazards from the same baseline population and time origin. It also assumes the survival function represents the specified event process appropriately, without silently treating a competing event as ordinary independent censoring. A ratio by itself cannot determine an absolute survival probability; the comparator survival curve is needed.

Work through a two-year calculation

For illustration, suppose the comparator has a constant event hazard of 0.10 per year and treatment has a hazard of 0.07 per year. Their ratio is $\mathrm{HR}=0.07/0.10=0.70$, and under these illustrative exponential assumptions the two-year survivals are $S_C(2)=\exp(-0.20)\approx0.8187$ and $S_T(2)=\exp(-0.14)\approx0.8694$. The same treatment curve follows from the proportional-hazards relation, because $S_T(2)=S_C(2)^{0.70}=\exp(-0.20\times0.70)=\exp(-0.14)$. The corresponding two-year event probabilities are about 0.1813 and 0.1306.

The ratio of cumulative event probabilities, the relative risk, is about $\mathrm{RR}=0.1306/0.1813\approx0.72$, and the odds ratio is about $\mathrm{OR}=0.1503/0.2214\approx0.68$, so the three relative measures differ even in this simple case. Their absolute event-probability difference is about $\Delta=0.1813-0.1306\approx0.051$, or about 5.1 percentage points. The restricted mean survival time to two years, the area under each survival curve, is $(1-S(2))/h$ for a constant hazard, giving about 1.8127 and 1.8663 years, a gain of about 0.054 years or roughly 20 days within the two-year window. Because both hazards are constant here, the median survival times are $\ln 2/0.10\approx6.93$ and $\ln 2/0.07\approx9.90$ years, a ratio of $\mathrm{HR}^{-1}=1/0.70\approx1.43$; that link between the HR and the ratio of medians holds only for constant hazards. These results depend on both the baseline hazard and the two-year horizon; the numbers are not estimates for a real treatment.

Examine whether one ratio fits over time

Hazards can cross, treatment effects can wane, or delayed effects can arise, making $\operatorname{HR}(t)$ vary with time. A single estimated Cox HR can then be a time-dependent, study-specific summary rather than a constant multiplier suitable for every year of extrapolation. Assessment of the proportional hazards assumption draws on survival and hazard displays, formal diagnostics, follow-up duration and biological plausibility.

Two diagnostics are standard. Under proportional hazards, plots of $\log{-\log S(t)}$, the log-cumulative hazard, against log time are roughly parallel for the two arms, with a vertical gap equal to the log HR. Tests based on scaled Schoenfeld residuals, developed by Grambsch and Therneau, check whether the estimated log HR drifts with time. The NICE manual PMG36 (section 4.6.22) states that the assumption should always be assessed, preferably using log-cumulative hazard plots as advised in NICE DSU Technical Support Document 14, visual inspection of hazard plots or HRs over time, and interpretation of tests reported in the original trial publications.

Non-proportional hazards are a frequent concern in immuno-oncology appraisals. NICE DSU Technical Support Document 21 notes that delayed responses to treatment and the existence of long-term survivors with immunotherapies have been hypothesised to produce complex hazard functions, and it describes flexible parametric (spline-based), cure and mixture models that allow the treatment effect to vary over time. When hazards are not proportional, the restricted mean survival time offers a summary of the difference between arms that does not rely on the assumption, as Royston and Parmar proposed for randomised trials.

The assumption observed during a trial does not automatically remain credible after its end. The NICE manual PMG36 (sections 4.6.22 to 4.6.25) notes that treatment HRs may be constant or change over time, that alternative methods described in Technical Support Document 21 should be considered when proportional hazards do not hold, and that the clinical plausibility of extrapolated hazard functions should routinely be assessed. Alternative time-varying or separate survival models should be considered when warranted by evidence.

Keep competing risks and estimands clear

A cause-specific hazard ratio compares instantaneous rates of a named event among people still free of the relevant event history; a subdistribution hazard ratio from the Fine and Gray model uses a different risk-set construction linked to cumulative incidence. These ratios are not interchangeable, and neither alone supplies every competing-event probability needed for a model. The estimand needs to be specified before a published number is transferred into a state-transition model.

Censoring assumptions, treatment switching, covariate adjustment, and patient selection can also alter the estimate or its interpretation. In a randomised trial, randomisation supports a treatment comparison, but the HR remains a time-to-event summary with its own assumptions: the people still at risk at later times have been selected by earlier events, so period-specific HRs carry what Hernán described as a built-in selection bias. An observational HR requires additional attention to confounding and selection before any causal claim.

Carry uncertainty into health-economic modelling

Cost-effectiveness models often convert comparative survival into life-years, QALYs, and associated costs over a horizon longer than trial follow-up. A relative hazard should be applied only to an appropriate baseline hazard or survival model, after which absolute outcomes are calculated and the extrapolation tested. Parameter uncertainty in the HR, baseline survival uncertainty, structural uncertainty, and differences in the target population can all matter.

In a cohort state-transition model the HR acts on rates, not on probabilities. With a constant hazard $h$ and cycle length $u$, the per-cycle transition probability and its treatment-arm counterpart are

$$ p_C=1-\exp(-h_C,u),\qquad p_T=1-\exp(-r,h_C,u)=1-(1-p_C)^{r}, $$

where $p_C$ and $p_T$ are the comparator and treatment probabilities of the event within one cycle, $h_C$ is the comparator hazard, $u$ is the cycle length and $r$ is the hazard ratio. In the illustrative example with annual cycles, $p_C=1-\exp(-0.10)\approx0.0952$ and $p_T=1-\exp(-0.07)\approx0.0676$. Multiplying the probability directly by the HR would give about 0.0666 instead; the error is small here but grows as probabilities rise or cycles lengthen.

1. Report the event definition, reference group, time origin, follow-up, HR estimate, and confidence interval.
2. State whether the ratio is adjusted and which covariates or stratification were used.
3. Check proportional hazards over the observed period and the plausibility of extending it.
4. Use the relevant baseline curve to derive absolute survival and event probabilities.
5. Treat cause-specific and subdistribution hazard ratios as distinct inputs when competing events exist.
6. Compare long-term modelled survival with appropriate external evidence and clinical judgement.

Sources

• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford: Oxford University Press; 2006.
• Cox DR. Regression models and life-tables. Journal of the Royal Statistical Society Series B (Methodological). 1972;34(2):187-202. https://doi.org/10.1111/j.2517-6161.1972.tb00899.x
• Fine JP, Gray RJ. A proportional hazards model for the subdistribution of a competing risk. Journal of the American Statistical Association. 1999;94(446):496-509. https://doi.org/10.1080/01621459.1999.10474144
• Fleurence RL, Hollenbeak CS. Rates and probabilities in economic modelling: transformation, translation and appropriate application. PharmacoEconomics. 2007;25(1):3-6. https://doi.org/10.2165/00019053-200725010-00002
• Grambsch PM, Therneau TM. Proportional hazards tests and diagnostics based on weighted residuals. Biometrika. 1994;81(3):515-526. https://doi.org/10.1093/biomet/81.3.515
• Hernán MA. The hazards of hazard ratios. Epidemiology. 2010;21(1):13-15. https://doi.org/10.1097/EDE.0b013e3181c1ea43
• Latimer NR. NICE DSU Technical Support Document 14: Survival analysis for economic evaluations alongside clinical trials, extrapolation with patient-level data. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; 2011 (last updated March 2013). https://www.sheffield.ac.uk/nice-dsu/tsds/survival-analysis
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36), section 4.6 Modelling methods. London: NICE; 2022 (last updated 31 March 2026). https://www.nice.org.uk/process/pmg36/chapter/economic-evaluation-2
• Royston P, Parmar MKB. Restricted mean survival time: an alternative to the hazard ratio for the design and analysis of randomized trials with a time-to-event outcome. BMC Medical Research Methodology. 2013;13:152. https://doi.org/10.1186/1471-2288-13-152
• Rutherford MJ, Lambert PC, Sweeting MJ, Pennington B, Crowther MJ, Abrams KR, Latimer NR. NICE DSU Technical Support Document 21: Flexible methods for survival analysis. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; 2020 (updated version March 2022). https://www.sheffield.ac.uk/nice-dsu/tsds/flexible-methods-survival-analysis