23 Survival analysis
23.1 Definitions
23.2 Hazard, rate, incidence
They are the same word, using in different fields, hazard = rate = incidence:
- Rate: number of new events that occur during specified time period.
- Incidence: when the event is a disease, it is called incidence.
- Hazard: in survival analysis, it is called hazard.
Let consider a discrete
23.3 Hazard function
Hazard function \(h(t)\) gives the probability that someone die at time \(t\), given that they have survived up to that time \(t\) (survived during the period \([0;t]\)).
\[h(t) = \frac{f(t)}{\mathbb{P}(T > t)} = \frac{f(t)}{S(t)}\]
Where \(f(t)\) is the time to first
23.4 Cumulative hazard function
Cumulative hazard function \(F(t)\) gives the probability that someone die by time \(t\) (die anytime during \([0;t]\)).
\[F(t) = \mathbb{P}(T \leq t) = \sum_{t = 0}^t h(t)\]
23.5 Survival function
The survival function \(S(t)\) gives the probability that someone will survive beyond time \(t\) (they have survived \([0;t]\) and will continue to survive).
\[S(t) = \mathbb{P}(T > t) = 1 - \mathbb{P}(T \leq t) = 1 - F(t)\]
23.6 IDM and survival analysis
Much of infectious disease modeling is derived from, or at least shares, key concepts with survival analysis (Hay et al., n.d.).
| Survival analysis | Infectious disease modelling |
|---|---|
| Hazard function | Infection rate, force of infection |
| Survival function | Probability of not infected |