Survival analysis

Survival analysis is a branch of statistics which deals with death in biological organisms and failure in mechanical systems. Death or failure is called an "event" in the survival analysis literature, and so models of death or failure are generically termed time-to-event models.

Survival analysis attempts to answer questions such as: what is the fraction of a population which will survive past a certain time? Of those that survive, at what rate will they die or fail? Can multiple causes of death or failure be taken into account? How do particular circumstances or characteristics increase or decrease the odds of survival?

Table of contents

1 General formulation 2 Some survival distributions 3 References

General formulation

The object of primary interest is the survival function, conventionally denoted S, which is defined as

where t is some time, T is the time of death, and "Pr" stands for probability. That is: the survival function is the probability that the time of death is later than some specified time.

Related quantities are defined in terms of the survival function. The lifetime distribution function, conventionally denoted F, is defined as the complement of the survival function,

and the derivative of F (i.e., the density function of the lifetime distribution) is conventionally denoted f,

The hazard function, conventionally denoted , is defined as the event rate at time t conditional on survival until time t or later,

Force of mortality is a synonym of hazard function which is used particularly in demographics. The hazard function can alternatively be represented in terms of the cumulative hazard function, conventionally denoted :

so

Future lifetime at a given time is denoted by the time remaining until death, thus future lifetime is in the present notation. The expected future lifetime is the expected value of future lifetime. Now the event density given survival until or later, given survival until , is just

,

so the expected future lifetime is given by

For , i.e., at birth, this reduces to the expected lifetime.

Some survival distributions

Survival models are constructed by choosing a basic survival distribution. It is straightforward to phrase model fitting and analysis in general terms, using the concepts outlined in under "General formulation", above. Thus it is relatively easy to substitute one distribution for another, in order to study the consequences of different choices.

The choice of survival distribution expresses some particular information about the relation of time and any exogenous variables to survival, and as such, it is analogous to the choice of link function in generalized linear models. There are several distributions commonly used in survival analysis, which are listed in the table below. Additional types of distributions can be found in the references.

Here indicates the standard normal cumulative distribution function. See normal distribution.

References

Regina Elandt-Johnson and Norman Johnson. Survival Models and Data Analysis. New York: John Wiley & Sons. 1980/1999.

Terry Therneau. "A Package for Survival Analysis in S". http://www.mayo.edu/hsr/people/therneau/survival.ps, at: http://www.mayo.edu/hsr/people/therneau.html