4.6 Survival Analysis
Thi sis a test. Great.
The probability of survival until time \(t\) or the probability that the event will occurr after time \(t\) is defined as: \[\begin{equation} S(t) = P(T > t) = 1 - F(t) \end{equation}\]
The hazard (instantaneous risk) \[\begin{equation} h(t) = P(t < T \leq t + t\delta \mid T > t) = \frac{f(t)}{S(t)} \end{equation}\]
Notice that it is enough to know one of the terms \(f(t), S(t), h(t)\) to derive the remaining terms:
- \(f(t)\): \(S(t) = \int_{t}^{\infty}f(x)dx\); \(h(t) = \frac{f(t)}{\int_{t}^{\infty}f(x)dx}\)
- \(S(t)\): \(f(t) = \frac{dS(t)}{dt}\); \(h(t) = \frac{S'(t)}{S(t)} = -\frac{dlog(S(t))}{dt}\)
- \(h(t)\): \(S(t) = exp\left(\int_0^t h(u)du\right)\); \(f(t) = \frac{ d\left(1-exp\left(\int_0^t h(u)du\right)\right)}{dt}\)
This type of analysis assumes a hazard function that defines the probability of the occurrence of an event. En general it is called survival because of the interest on. I should not neet to update.
This is strange because I can not see the file.