# hazard rate and cumulative incidence function

Assume an event of interest $E$ and the competing risk of death $D$.

Let $h(t)$ the hazard rate $h(t)=\frac{Pr\left\{(t < T \leq t + \Delta t)| (T > t)\right\}}{dt}\\$ and $S(t)$ the survival function

If we calculate the cumulative incidence function $CIF$ for $E$, as

$CIF=Pr(T<t)=\int_{0}^{t}hSdu$

What is the relationship between the $h(t)$ and the $CIF´=\frac{CIF}{dt}\\$ ? Is then $CIF´={Pr(t<T<T+dt)}\\$?

• What is $S$? $h(t)$ is a function whose integral over the positive real line diverges and what you call the CIF has a different relation to $h(t)$. In particular, $$P\{T \leq t\}=1 - \exp\left(-\int_0^t h(u)\,\mathrm du\right),$$ where the integrand is the area under $h(\cdot)$ between $0$ and $t$. As a special case, constant hazard rate $h(t)=\lambda$ makes $T$ work out to be an exponential random variable with mean $\lambda^{-1}$. – Dilip Sarwate May 3 '13 at 2:22
• $S(t)$ is the survival function – nostock May 3 '13 at 8:15
• If $S(t)$ is the survival function, then your calculation of CIF $= P\{T \leq t\}$ or $P\{T < t\}$ is incorrect. Note that $$P\{T \leq t\} = 1 - P\{T > t\} = 1 - S(t),$$ so that no integration, and especially not an integration of $h(\cdot)S(\cdot)$, is needed for the calculation of the CIF from the survival function. – Dilip Sarwate May 3 '13 at 13:31