Survival function

The survival function is a function that gives the probability that a patient, device, or other object of interest will survive past a certain time. The survival function is also known as the survivor function or reliability function. The term reliability function is common in engineering while the term survival function is used in a broader range of applications, including human mortality. The survival function is the complementary cumulative distribution function of the lifetime. Sometimes complementary cumulative distribution functions are called survival functions in general. Let the lifetime T be a continuous random variable with cumulative hazard function F(t) and hazard function f(t) on the interval [0,∞). Its survival function or reliability function is: The graphs below show examples of hypothetical survival functions. The x-axis is time. The y-axis is the proportion of subjects surviving. The graphs show the probability that a subject will survive beyond time t. For example, for survival function 1, the probability of surviving longer than t = 2 months is 0.37. That is, 37% of subjects survive more than 2 months. For survival function 2, the probability of surviving longer than t = 2 months is 0.97. That is, 97% of subjects survive more than 2 months. Median survival may be determined from the survival function. For example, for survival function 2, 50% of the subjects survive 3.72 months. Median survival is thus 3.72 months. In some cases, median survival cannot be determined from the graph. For example, for survival function 4, more than 50% of the subjects survive longer than the observation period of 10 months. The survival function is one of several ways to describe and display survival data. Another useful way to display data is a graph showing the distribution of survival times of subjects. Olkin, page 426, gives the following example of survival data. The number of hours between successive failures of an air-conditioning system were recorded.