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 ${\displaystyle T}$ be a continuous random variable describing the time to failure. If ${\displaystyle T}$ has cumulative distribution function ${\displaystyle F(t)}$ and probability density function ${\displaystyle f(t)}$ on the interval ${\displaystyle [0,\infty )}$, then the survival function or reliability function is:

$${\displaystyle S(t)=\Pr(T>t)=1-F(t)=1-\int _{0}^{t}f(u)\,du}$$

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: The median survival is the point where the survival function intersects the value 0.5. For example, for survival function 2, 50% of the subjects survive 3.72 months. Median survival is thus 3.72 months.

Median survival cannot always be determined from the graph alone. For example, in survival function 4, more than 50% of the subjects survive longer than the observation period of 10 months.