1.3 Some notation

We are now ready to introduce basic mathematical terminology and notation for survival analysis.

Let \(T\) the random variable that denotes the survival time, i.e., the time to an event. Since \(T\) denotes time, its possible values include all nonnegative numbers; that is, \(T\) can be any number equal to or greater than zero. Furthermore, \(t\) will be any specific value of interest for the random variable \(T\).

Additionally, when each subject has a random right censoring time \(C_i\) that is independent of their failure time \(T_i\), the data is represented by \((Y_i, \Delta_i)\) where \(Y_i = \min(T_i, C_i)\) and \(\Delta_i = I(T_i \le C_i)\), this \(\Delta\) define a \((0,1)\) random variable indicating either failure or censorship. That is, \(\Delta = 1\) for failure if the event occurs during the study period, or \(\Delta = 0\) if the survival time is censored by the end of the study period.