Hazard ratio
In
For example, a scientific paper might use an HR to state something such as: "Adequate COVID-19 vaccination status was associated with significantly decreased risk for the composite of severe COVID-19 or mortality with a[n] HR of 0.20 (95% CI, 0.17–0.22)."[1] In essence, the hazard for the composite outcome was 80% lower among the vaccinated relative to those who were unvaccinated in the same study. So, for a hazardous outcome (e.g., severe disease or death), an HR below 1 indicates that the treatment (e.g., vaccination) is protective against the outcome of interest. In other cases, an HR greater than 1 indicates the treatment is favorable. For example, if the outcome is actually favorable (e.g., accepting a job offer to end a spell of unemployment), an HR greater than 1 indicates that seeking a job is favorable to not seeking one (if "treatment" is defined as seeking a job).[2]
Hazard ratios differ from relative risks (RRs) and odds ratios (ORs) in that RRs and ORs are cumulative over an entire study, using a defined endpoint, while HRs represent instantaneous risk over the study time period, or some subset thereof. Hazard ratios suffer somewhat less from selection bias with respect to the endpoints chosen and can indicate risks that happen before the endpoint.
Definition and derivation
Regression models are used to obtain hazard ratios and their
The instantaneous
where N(t) is the number at risk at the beginning of an interval. A hazard is the probability that a patient fails between and , given that they have survived up to time , divided by , as approaches zero.[4]
The hazard ratio is the effect on this hazard rate of a difference, such as group membership (for example, treatment or
Such models are generally classed
For two groups that differ only in treatment condition, the ratio of the hazard functions is given by , where is the estimate of treatment effect derived from the regression model. This hazard ratio, that is, the ratio between the predicted hazard for a member of one group and that for a member of the other group, is given by holding everything else constant, i.e. assuming proportionality of the hazard functions.[4]
For a continuous explanatory variable, the same interpretation applies to a unit difference. Other HR models have different formulations and the interpretation of the parameter estimates differs accordingly.
Interpretation
In its simplest form, the hazard ratio can be interpreted as the chance of an event occurring in the treatment arm divided by the chance of the event occurring in the control arm, or vice versa, of a study. The resolution of these endpoints are usually depicted using
Hazard ratios do not reflect a time unit of the study. The difference between hazard-based and time-based measures is akin to the difference between the odds of winning a race and the margin of victory.[3] When a study reports one hazard ratio per time period, it is assumed that difference between groups was proportional. Hazard ratios become meaningless when this assumption of proportionality is not met.[7][page needed]
If the proportional hazard assumption holds, a hazard ratio of one means equivalence in the
Conventionally, probabilities lower than 0.05 are considered
The proportional hazards assumption
The proportional hazards assumption for hazard ratio estimation is strong and often unreasonable.
If the hazard ratio between groups remain constant, this is not a problem for interpretation. However, interpretation of hazard ratios become impossible when selection bias exists between groups. For instance, a particularly risky surgery might result in the survival of a systematically more robust group who would have fared better under any of the competing treatment conditions, making it look as if the risky procedure was better. Follow-up time is also important. A cancer treatment associated with better remission rates might on follow-up be associated with higher relapse rates. The researchers' decision about when to follow up is arbitrary and may lead to very different reported hazard ratios.[12]
The hazard ratio and survival
Hazard ratios are often treated as a ratio of death probabilities.[4] For example, a hazard ratio of 2 is thought to mean that a group has twice the chance of dying than a comparison group. In the Cox-model, this can be shown to translate to the following relationship between group survival functions: (where r is the hazard ratio).[4] Therefore, with a hazard ratio of 2, if (20% survived at time t), (4% survived at t). The corresponding death probabilities are 0.8 and 0.96.[11] It should be clear that the hazard ratio is a relative measure of effect and tells us nothing about absolute risk.[13][page needed]
While hazard ratios allow for
- ; conversely, .
In the previous example, a hazard ratio of 2 corresponds to a 67% chance of an early death. The hazard ratio does not convey information about how soon the death will occur.[3]
The hazard ratio, treatment effect and time-based endpoints
Treatment effect depends on the underlying disease related to survival function, not just the hazard ratio. Since the hazard ratio does not give us direct time-to-event information, researchers have to report median endpoint times and calculate the median endpoint time ratio by dividing the control group median value by the treatment group median value.[citation needed]
While the median endpoint ratio is a relative speed measure, the hazard ratio is not.[3] The relationship between treatment effect and the hazard ratio is given as . A statistically important, but practically insignificant effect can produce a large hazard ratio, e.g. a treatment increasing the number of one-year survivors in a population from one in 10,000 to one in 1,000 has a hazard ratio of 10. It is unlikely that such a treatment would have had much impact on the median endpoint time ratio, which likely would have been close to unity, i.e. mortality was largely the same regardless of group membership and clinically insignificant.[citation needed]
By contrast, a treatment group in which 50% of infections are resolved after one week (versus 25% in the control) yields a hazard ratio of two. If it takes ten weeks for all cases in the treatment group and half of cases in the control group to resolve, the ten-week hazard ratio remains at two, but the median endpoint time ratio is ten, a clinically significant difference.
See also
- Survival analysis
- Failure rate and Hazard rate
- Proportional hazards models
- Relative risk
References
- PMID 35653428.
- S2CID 16100294– via Elsevier Science Direct.
- ^ PMID 15273082.)
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: CS1 maint: multiple names: authors list (link - ^ S2CID 46520247. Retrieved 7 December 2012.)
{{cite journal}}
: CS1 maint: multiple names: authors list (link - ^ Cox, D. R. (1972). "Regression-Models and Life-Tables" (PDF). Journal of the Royal Statistical Society. B (Methodological). 34 (2): 187–220. Archived from the original (PDF) on 20 June 2013. Retrieved 5 December 2012.
- PMID 21729314.)
{{cite journal}}
: CS1 maint: multiple names: authors list (link - ^ ISBN 9780123919137.
- ISBN 9780199730063.
- ^ ISBN 9781550093476. Retrieved 7 December 2012.
- ]
- ^ ISBN 9781590471357.
- PMID 20010207.
- ISBN 9780471461609.[page needed]