Signature
HR = h_T / h_C
| Inputs | Definition | Unit |
|---|---|---|
h_T | Instantaneous event rate among treated people still at risk at time t | events per person per unit of time |
h_C | Instantaneous event rate among comparator people still at risk at time t, greater than zero | events per person per unit of time |
HR | Ratio of the treatment hazard to the comparator hazard at the same time | ratio, no unit |
|---|
Function
Relative hazard function
Maps the hazards of an event in a treatment group and a comparator group at the same time t to their ratio. Under proportional hazards the ratio is constant over time, and it can then be applied to a baseline survival curve or a baseline transition probability to obtain absolute outcomes for the treatment group.
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Implementations
Excel
Hazard ratio from two named hazards
Excel divides the treatment hazard by the comparator hazard and returns #N/A when the comparator hazard is not positive.
=IF(ComparatorHazard>0,TreatmentHazard/ComparatorHazard,NA())
Assumptions
Hazards compared at the same time and on the same event
Both hazards refer to the same event definition, time origin and analysis population, and are measured at the same time since the origin.
Proportional hazards when one ratio summarises follow-up
Reporting a single HR for the whole follow-up assumes that the ratio is constant over time. The NICE manual states that the assumption should always be assessed, preferably with log-cumulative hazard plots.
Worked examples
Constant hazards of 0.07 and 0.10 per year
With illustrative constant hazards of 0.07 per year on treatment and 0.10 per year on the comparator, HR = 0.07 / 0.10 = 0.70, as in the article.
h_T = 0.07; h_C = 0.10; HR = 0.70
Common errors
Reading a hazard ratio as a relative risk
An HR of 0.70 is a 30% lower hazard, not a 30% lower probability of the event. In the article's two-year example the relative risk is about 0.72 and the odds ratio about 0.68, so the three measures differ even with constant hazards.
Giving period-specific hazard ratios a causal reading
People still at risk later in follow-up have been selected by earlier events, so an HR for a later period compares groups that may no longer be balanced, even in a randomised trial.
Sources
Hazard ratios under the proportional hazards model
Bradburn MJ, Clark TG, Love SB, Altman DG. Survival analysis part II: multivariate data analysis, an introduction to concepts and methods. British Journal of Cancer. 2003;89(3):431-436. Section on the Cox proportional hazards model (the hazard in any group is a constant multiple of the hazard in any other; exp(b) is the hazard ratio).
Built-in selection bias of period-specific hazard ratios
Hernán MA. The hazards of hazard ratios. Epidemiology. 2010;21(1):13-15.
Canonical Identity
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