Survival function

The survival function is the probability that a patient, device, or other object of interest will survive beyond a specified 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. Another name for the survival function is the complementary cumulative distribution function.

Let T be a continuous random variable with cumulative distribution function F(t) on the interval [0,∞). Its survival function or reliability function is:

In some cases, such as the air conditioner example, the distribution of survival times may be approximated well by a function such as the exponential distribution. Several distributions are commonly used in survival analysis, including the exponential, Weibull, gamma, normal, log-normal, and log-logistic. These distributions are defined by parameters. The normal (Gaussian) distribution, for example, is defined by the two parameters mean and standard deviation. Survival functions that are defined by parameters are said to be parametric.

In the four survival function graphs shown above, the shape of the survival function is defined by a particular probability distribution: survival function 1 is defined by an exponential distribution, 2 is defined by a Weibull distribution, 3 is defined by a log-logistic distribution, and 4 is defined by another Weibull distribution.

For an exponential survival distribution, the probability of failure is the same in every time interval, no matter the age of the individual or device. This fact leads to the "memoryless" property of the exponential survival distribution: the age of a subject has no effect on the probability of failure in the next time interval. The exponential may be a good model for the lifetime of a system where parts are replaced as they fail. It may also be useful for modeling survival of living organisms over short intervals. It is not likely to be a good model of the complete lifespan of a living organism. As Efron and Hastie (p. 134) note, "If human lifetimes were exponential there wouldn't be old or young people, just lucky or unlucky ones".

A key assumption of the exponential survival function is that the hazard rate is constant. In an example given above, the proportion of men dying each year was constant at 10%, meaning that the hazard rate was constant. The assumption of constant hazard may not be appropriate. For example, among most living organisms, the risk of death is greater in old age than in middle age – that is, the hazard rate increases with time. For some diseases, such as breast cancer, the risk of recurrence is lower after 5 years – that is, the hazard rate decreases with time. The Weibull distribution extends the exponential distribution to allow constant, increasing, or decreasing hazard rates.

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