Survival Analysis — An Introduction


April 30, 2021

Suppose we place an individual understudy for a particular viral infection. Upon testing the patient, it is observed that the patient has already been infected by the virus, now since we don’t know the exact time at which the patient was infected with the virus, we have to truncate the survival time from the left. This is called left censoring.


Hazard function, h(t), is the probability that an event occuring at a particular time t. To put it in another way it is the instantaneous rate of change of probability of an event at time t. Hazard function gives the risk of having an event at any instant, given the individual has survived till that instant.

let us define f(t) as,

From Eqn. 4,


From Equation 3,

Now Equation 6 becomes,

Taking integral on both sides,

This concludes the proof that establishes the relationship between survival and hazard function.