Hazard Ratios

Definition

The hazard ratio is a measure commonly used in survival analysis to compare the risk of occurrence of an event of interest (e.g., death) in two groups (e.g., treatment group vs. control group) at a given time. For example, the hazards ratio can be used to describe the outcome of therapeutic trials where the question is to what extent treatment can delay the apparition of a disease.

The hazard ratio can be calculated using the hazard rate which are defined as

$$ {{\lambda (}}t{)} = {\text{li}}{{\text{m}}_{{h \to {0 + }}}}\frac{{\Pr (t \leq T \leq t + h|T \geq t)}}{h} $$

(1)

The hazard rate specifies the instantaneous rate at which failures occur for items that are surviving at time t and gives the risk of failure per unit time.

The hazard ratio is simply equal to ${\text{HR}} = \frac{{{{\lambda }}(t|\text{group}\;1)}}{{{{\lambda }}(t|{\text{group}}\;2)}}$. If HR = 1, the risk of occurrence of the event of interest is the same in the two groups of patients. If HR > 1 (HR...