11.1 Survival and Censoring Times

For each individual, we suppose that there is a true survival time , T , as well survival time as a true censoring time , C . (The survival time is also known as the failure censoring time or the event time .) The survival time represents the time at which the time event of interest occurs: for instance, the time at which the patient dies, failure time or the customer cancels his or her subscription. By contrast, the censoring event time time is the time at which censoring occurs: for example, the time at which the patient drops out of the study or the study ends.

We observe either the survival time T or else the censoring time C . Specifically, we observe the random variable

\[Y = \min(T, C) \quad (11.1)\]

In other words, if the event occurs before censoring (i.e. T < C ) then we observe the true survival time T ; however, if censoring occurs before the event ( T > C ) then we observe the censoring time. We also observe a status indicator,

\[\delta = \begin{cases} 1 & \text{if } T \le C \\ 0 & \text{if } T > C. \end{cases}\]

Thus, δ = 1 if we observe the true survival time, and δ = 0 if we instead observe the censoring time.

Now, suppose we observe n ( Y, δ ) pairs, which we denote as ( y 1 , δ 1) , . . . , ( yn, δn ). Figure 11.1 displays an example from a (fictitious) medical study in which we observe n = 4 patients for a 365-day follow-up period. For patients 1 and 3, we observe the time to event (such as death or disease relapse) T = ti . Patient 2 was alive when the study ended, and patient 4 dropped out of the study, or was “lost to follow-up”; for these patients we observe C = ci . Therefore, y 1 = t 1, y 3 = t 3, y 2 = c 2, y 4 = c 4, δ 1 = δ 3 = 1, and δ 2 = δ 4 = 0.