14.1 Basic Concepts

This section reviews the fundamentals of survival analysis, including the hazard probability density, and survival functions.

You can specify the survival distribution function either as a survival function or as a hazard function. Define \(F(t) = Pr(T \le t), \hspace{3mm} 0 < t < \infty\) as the cumulative risk function, the probability of dying on or before time \(t\). Then the survival function is the probability of surviving up to time \(t\),

\[S(t) = 1 - F(t) = pr(T > t), \hspace{3mm} 0 < t < \infty.\]

The hazard function is the instantaneous death rate given survival up to time \(t\),

\[h(t) = \lim_{\delta \rightarrow 0}{\frac{pr(t < T < t + \delta|T > t)}{\delta}}.\]

The survival function and the hazard function are related. The probability of dying during the interval \((t, t + \delta)\), \(f(t) = F'(t)\), is the probability of dying during the interval given survival up to point \(t\) times the probability of surviving up to point \(t\), \(f(t) = h(t) S(t)\).

\(S(t)\) is also the exponent of the negative cumulative hazard function,

\[S(t) = e^{-H(t)}.\]

You can use the survival function to estimate the mean and median survival times. The mean survival time is \(E(T) = \int S(t)dt\). The median survival time is \(S(t) = 0.5\).