ABSTRACT

Neural Network .............................................................................277 8.6 Conclusion ................................................................................................... 279 References ............................................................................................................ 279

Survival analysis is concerned with the time that elapses till the occurrence of some event of interest in the study (e.g., death of a patient who has been followed up since diagnosis of a particular illness). Usually a number of cases

are monitored or followed up. Some cases may drop out of the follow-up regime for various reasons. The result is that for these cases, the time till an event of interest is unknown and the time when they are lost from scrutiny is referred to as a “censoring time.” Times of events are “event times.” The probability distribution of time-to-event is usually skewed. The existence of both censoring times and skewness make for diffi culties in the estimation of the underlying probability distribution of time-to-event. Basic equations of survival analysis are as follows:

Probability density function of time-to-event, f(t) is defi ned as

f(t)dt = probability of an event occurring in time interval (t, t + dt)

Survival function, S(t), is defi ned as

S(t) = probability of no event in time interval (0, t)

Hazard function, h(t) is defi ned as

h(t)dt = conditional probability of an event occurring in the time interval (t, t + dt), given that no event has occurred in (0, t)

h(t)dt = f(t)dt

_____ S(t)

from which we get

S(t) = exp{–H(t)} (8.1)

where H(t) is the cumulative hazard function given by

h(x)dx

In most situations of interest, the probability distributions are conditioned on some covariates or factors such as patient characteristics, clinical variables or factors, and the type of treatment given.