Signature
HR = exp(beta)
| Inputs | Definition | Unit |
|---|---|---|
beta | Estimated regression coefficient for treatment, coded 1 for treatment and 0 for the comparator | log hazard ratio, no unit |
HR | Hazard ratio for the treatment group relative to the reference group, adjusted for the other covariates in the model | 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.
Implementations
Excel
Hazard ratio and 95% confidence limits from a coefficient
With the coefficient in the named cell Coef and its standard error in SE, the three formulas return the hazard ratio and its lower and upper 95% confidence limits, computed on the log scale.
=EXP(Coef); =EXP(Coef-1.96*SE); =EXP(Coef+1.96*SE)
Assumptions
Binary treatment covariate and proportional hazards
Treatment is coded 0 and 1, and the model assumes that the hazards are proportional. A covariate coded in other units gives the hazard ratio per unit of that covariate.
Worked examples
Coefficient of minus 0.3567
A Cox coefficient of minus 0.3567 for treatment gives a hazard ratio of about 0.7000.
beta = -0.3567; HR = 0.7000
Common errors
Reporting the coefficient as the hazard ratio
Software output often lists the coefficient beside its exponent. Taking minus 0.3567 as the hazard ratio, or averaging hazard ratios rather than log hazard ratios in a synthesis, misstates the effect; pooling is done on the log scale.
Sources
Proportional hazards regression
Cox DR. Regression models and life-tables. Journal of the Royal Statistical Society Series B (Methodological). 1972;34(2):187-202.
Exponentiated Cox coefficients as hazard ratios
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.
Canonical Identity
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