1.5 Relation between functions

For parametric survival models, time is assumed to follow some well-known distribution whose probability density function \(f(t)\) can be expressed in terms of unknown parameters. Once a probability density function is specified for survival time, the corresponding survival and hazard functions can be determined.

For example, the survival function can be ascertained from the probability density function by integrating over the probability density function from time \(t\) to infinity, or by calculating the difference between one and the cumulative distribution function \(F(t)\). The hazard can then be found by dividing the negative derivative of the survival function by the survival function. Note that the functions \(f(t)\), \(F(t)\), \(h(t)\), and \(H(t)\) are all related.

1. Assume that \(T\) is non-negative and continuos:

   • Probability density function:

     • \(f(t) = F'(t) = \frac{dF(t)}{dt}\)

   • Cumulative distribution function:

     • \(F(t) = P(T \le t) = \int_0^t{f(u)}{du}\)

   • Survival function

     • \(S(t) = 1 - F(t)\)

     • \(S(t) = P(T > t) = \int_t^{+\infty}{f(u)}{du}\)

     • \(S(t) = exp \left( - \int_0^t h(u) du \right)\)

     • \(S(t) = \exp(-H(t))\)

   • Hazard function

     • \(h(t) = \frac{ f(t)}{S(t)}= \frac{ -d[S(t)]/dt}{S(t)}\)

   • Cumulative hazard function

     • \(H(t) = \int_0^t h(u) du\)

1. Assume that \(T\) is non-negative and discrete,

   • Probability mass function:
     • \(p(t_i) = P(T = t_i)\)
     • \(p(t_i) = S(t_{i-1}) - S(t_i)\)
     • \(p(t_i) = F(t_i) - F(t_{i-1})\)
   • Cumulative distribution function:
     • \(F(t) = P(T \le t) = \sum_{t_i \le t}{p(t_i)}\)
   • Survival function
     • \(S(t) = \prod_{t_i \le t} \left( 1 - h(t_i) \right)\)
   • Hazard function
     • \(h(t) = \frac{ p(t_i)}{S(t_{i-1})}= \frac{ -d[S(t)]/dt}{S(t)}\)
     • \(h(t) = 1- \frac{ S(t_i)}{S(t_{i-1})}\)
   • Cumulative hazard function
     • \(H(t) = \sum_{t_i \le t} h(t_i)\)