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.
Definition Let the lifetime T {\displaystyle T} be a continuous random variable describing the time to failure. If T {\displaystyle T} has cumulative distribution function F ( t ) {\displaystyle F(t)} and probability density function f ( t ) {\displaystyle f(t)} on the interval [ 0 , ∞ ) {\displaystyle [0,\infty )} , then the survival function or reliability function is:
S ( t ) = ∫ t ∞ f ( u ) d u = Pr ( T > t ) = 1 − F ( t ) = 1 − ∫ 0 t f ( u ) d u {\displaystyle S(t)=\int _{t}^{\infty }f(u)\,du=\Pr(T>t)=1-F(t)=1-\int _{0}^{t}f(u)\,du}
Examples of survival functions 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.
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 (AC) system were recorded. The time in hours, t, between successive failures are 1, 3, 5, 7, 11, 11, 11, 12, 14, 14, 14, 16, 16, 20, 21, 23, 42, 47, 52, 62, 71, 71, 87, 90, 95, 120, 120, 225, 246 and 261. The mean time between failures is 59.6. The figure below shows the distribution of the time between failures. The blue tick marks beneath the graph are the actual hours between successive AC failures.
In this example, a curve representing the exponential distribution overlays the distribution of AC failure times; the exponential distribution approximates the distribution of AC failure times. This particular exponential curve is specified by the parameter lambda, λ:
The distribution of failure times is the probability density function (PDF), since time can take any positive value. In equations, the PDF is specified as fT. If time can only take discrete values (such as 1 day, 2 days, and so on), the distribution of failure times is called the probability mass function. Most survival analysis methods assume that time can take any positive value, and fT is the PDF. If the time between observed AC failures is approximated using the exponential function, then the exponential curve gives the probability density function, fT, for AC failure times. Another useful way to display the survival data is a graph showing the cumulative failures up to each time point. These data may be displayed as either the cumulative number or the cumulative proportion of failures up to each time. The graph below shows the cumulative probability (or proportion) of failures at each time for the air conditioning system. The stairstep line in black shows the cumulative proportion of failures. For each step there is a blue tick at the bottom of the graph indicating an observed failure time. The smooth red line represents the exponential curve fitted to the observed data.
A graph of the cumulative probability of failures up to each time point is called the cumulative distribution function (CDF). In survival analysis, the cumulative distribution function gives the probability that the survival time is less than or equal to a specific time, t. Let T be survival time, which is any positive number. A particular time is designated by the lower case letter t. The cumulative distribution function of T is the function
F ( t ) = Pr ( T ≤ t ) , {\displaystyle F(t)=\Pr(T\leq t),}
where the right-hand side represents the probability that the random variable T is less than or equal to t. If time can take on any positive value, then the cumulative distribution function F(t) is the integral of the probability density function f(t). For the air-conditioning example, the graph of the CDF below illustrates that the probability that the time to failure is less than or equal to 100 hours is 0.81, as estimated using the exponential curve fit to the data.
… excerpt ends here. Continue reading the full article.
