Hazard Ratio

Definition

The Hazard Ratio (HR) is the ratio of hazard function between two subjects. It measures the relative risk associated with a covariate change.

Categorical Predictor

Let $x_1$ be binary ($1$ = condition A, $0$ = otherwise), with other covariates held constant:

$h_1(t)=h_0(t)e^{{\beta }_1+{\beta }_2{x}_2+\cdots },h_2(t)=h_0(t)e^{0+{\beta }_2{x}_2+\cdots }$

The hazard ratio for condition A vs not-A:

$HR=\frac{h_1(t)}{h_2(t)}=\frac{h_0(t)e^{{\beta }_1+{\beta }_2{x}_2+\cdots }}{h_0(t)e^{{\beta }_2{x}_2+\cdots }}=e^{{\beta }_1}$

Numeric Predictor

Let $x_2$ be numeric. Compare subjects differing by 1 unit in $x_2$, all else equal:

$HR=\frac{h_0(t)e^{{\beta }_1{x}_1+{\beta }_2(a+1)+\cdots }}{h_0(t)e^{{\beta }_1{x}_1+{\beta }_2(a)+\cdots }}=e^{{\beta }_2}$

Interpretation of $\beta$

${\beta }_j$	$HR=e^{{\beta }_j}$	Meaning
${\beta }_j>0$	$HR>1$	Increasing $x_j$ by 1 unit raises hazard — higher risk
${\beta }_j=0$	$HR=1$	$x_j$ has no effect on hazard
${\beta }_j<0$	$HR<1$	Increasing $x_j$ by 1 unit lowers hazard — protective effect

Key Properties

1. Constant over time: $HR$ does not depend on $t$ (proportional hazards assumption)
2. Multiplicative: For a $\Delta$-unit change in $x_j$, $HR=e^{{\beta }_j\Delta }$
3. Independent of $h_0(t)$: HR depends only on $\beta$ and covariate differences

Interpretation

$HR=e^{{\beta }_j}$ means: for each 1-unit increase in $x_j$, the hazard is multiplied by $e^{{\beta }_j}$. An $HR$ of 2 means the event rate is twice as high; $HR=0.5$ means it’s halved.

Related

• Cox Proportional Hazards Model
• Hazard Function