In survival analysis, the hazard ratio (HR) is the ratio of the hazard rates corresponding to the conditions described by two levels of an explanatory variable. For example, in a drug study, if the treated population dies at twice the rate as the control population, the hazard ratio is 2, indicating a higher hazard of death from the treatment.
The definition of the Cox proportional hazard model is:
h(t) = h0(t)exp(β1X1 + … + β n X n )
The definition of HR is:
HR = h1(t) / h2(t) =
h0(t)exp(β1X1 + … + β n X n ) / h (t) = h0(t)exp(β1X'1 + … + β n X' n ) =
exp(β1(X1 - X'1) + … + β n (X n - X' n ))
The natural logarithm of HR is:
ln(HR) = β1(X1 - X'1) + … + β n (X n - X' n )
For two groups that differ only in treatment condition, the ratio of the hazard functions is given by e β, where β is the estimated treatment effect derived from the regression model. This hazard ratio (the ratio of the predicted hazard for a member of one group to the predicted hazard for a member of the other group) is given by holding everything else constant (that is, assuming proportionality of the hazard functions).
For a continuous explanatory variable, the same interpretation applies to a unit difference.
Researchers consider probabilities lower than .05 to be significant and provide a 95% confidence interval for the hazard ratio. Statistically significant hazard ratios cannot include unity (one) in their confidence intervals.
Suppose that you have the following Cox proportional hazard model:
h(t) = h0(t)exp(β1X AGE + β2X GENDER + β1X AGE*GENDER + β2X WEIGHT )
You can use the preceding model to calculate hazard ratios such as:
- The hazard ratio when AGE increases 1 unit
- The hazard ratio among AGE=20, 40, 60 at the group in which GENDER is female
- The hazard ratio when WEIGHT increases 1 unit at the group in which GENDER is male and AGE = (20, 40)
- The hazard ratio between the groups (GENDER=1, AGE=20, WEIGHT=80) and (GENDER=0, AGE=60, WEIGHT=70)
- The hazard ratio when AGE increases 1 unit and WEIGHT increases 10 units