Survival analysis

Survival analysis is a branch of statistics and epidemiology which deals with death in biological organisms. This topic is called duration analysis or duration modeling in economics or sociology. More generally, survival analysis involves the modeling of time to event data; in this context, death or failure is considered an "event" in the survival analysis literature – traditionally only a single event occurs, after which the organism dead.

More recently, many concepts in survival analysis have been explained by Counting Process Theory, which adds flexibility in that it allows modeling multiple (or recurrent) events. This type of modeling fits very well in many situations, when the event is significant but does not end the lifespan of the subject – e.g. people can go to jail multiple times, alcoholics can start and stop drinking multiple times, and people can get married and divorced multiple times.

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?

To answer such questions, it is necessary to define "lifetime". In the case of biological survival, death is unambiguous, but for mechanical reliability, failure may not be well-defined, for there may well be mechanical systems in which failure is partial, a matter of degree, or not otherwise localized in time. Even in biological problems, some events (for example, heart attack or other organ failure) may have the same ambiguity. The theory outlined below assumes well-defined events at specific times; other cases may be better treated by models which explicitly account for ambiguous events.

The theory of survival presented here also assumes that death or failure happens just once for each subject. Recurring event or repeated event models relax that assumption. The study of recurring events is relevant in systems reliability, and in many areas of social sciences and medical research.

This article is phrased primarily in terms of biological survival, but this is just for convenience. An equivalent formulation in terms of mechanical failure can be made by replacing every occurrence of death with failure.

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


 * $$S(t) = \Pr(T > t)$$

where t is some time, T is a random variable denoting 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 t. The survival function is also called the survivor function or survivorship function in problems of biological survival, and the reliability function in mechanical survival problems. In the latter case, the reliability function is denoted R(t).

Usually one assumes S(0) = 1, although it could be less than 1 if there is the possibility of immediate death or failure.

The survival function must be non-increasing: S(u) ≤ S(t) if u ≥ t. This property follows directly from F(t) = 1 - S (t) being the integral of a non-negative function. This reflects the notion that survival to a later age is only possible if all younger ages are attained. Given this property, the lifetime distribution function and event density (F and f below) are well-defined.

The survival function is usually assumed to approach zero as age increases without bound, i.e., S(t) → 0 as t → ∞, although the limit could be greater than zero if eternal life is possible. For instance, we could apply survival analysis to a mixture of stable and unstable carbon isotopes; unstable isotopes would decay sooner or later, but the stable isotopes would last indefinitely.

Lifetime distribution function and event density
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,


 * $$F(t) = \Pr(T \le t) = 1 - S(t)$$

and the derivative of F, which is the density function of the lifetime distribution, is conventionally denoted f,


 * $$f(t) = F'(t) = \frac{d}{dt} F(t).$$

The function f is sometimes called the event density; it is the rate of death or failure events per unit time.

The survival function is often defined in terms of distribution and density functions


 * $$S(t) = \Pr(T > t) = \int_t^{\infty} f(u)\,du = 1-F(t).$$

Similarly, a survival event density function can be defined as


 * $$s(t) = S'(t) = \frac{d}{dt} S(t) = \frac{d}{dt} \int_t^{\infty} f(u)\,du = \frac{d}{dt} [1-F(t)] = -f(t).$$

Hazard function and cumulative hazard function
The hazard function, conventionally denoted $$\lambda$$, is defined as the event rate at time t conditional on survival until time t or later (that is, T ≥ t),


 * $$\lambda(t)\,dt = \Pr(t \leq T < t+dt\,|\,T \geq t) = \frac{f(t)\,dt}{S(t)} = -\frac{S'(t)\,dt}{S(t)}.$$

Force of mortality is a synonym of hazard function which is used particularly in demography and actuarial science, where it is denoted by $$\mu$$. The term hazard rate is another synonym.

The hazard function must be non-negative, λ(t) ≥ 0, and its integral over $$[0, \infty]$$ must be infinite, but is not otherwise constrained; it may be increasing or decreasing, non-monotonic, or discontinuous. An example is the bathtub curve hazard function, which is large for small values of t, decreasing to some minimum, and thereafter increasing again; this can model the property of some mechanical systems to either fail soon after operation, or much later, as the system ages.

The hazard function can alternatively be represented in terms of the cumulative hazard function, conventionally denoted $$\Lambda$$:


 * $$\,\Lambda(t) = -\log S(t)$$

so transposing signs and exponentiating


 * $$\,S(t) = \exp(-\Lambda(t))$$

or differentiating (with the chain rule)


 * $$\frac{d}{dt} \Lambda(t) = -\frac{S'(t)}{S(t)} = \lambda(t).$$

The name "cumulative hazard function" is derived from the fact that


 * $$ \Lambda(t) = \int_0^{t} \lambda(u)\,du$$

which is the "accumulation" of the hazard over time.

From the definition of $$\Lambda(t)$$, we see that it increases without bound as t tends to infinity (assuming that S(t) tends to zero). This implies that $$\lambda(t)$$ must not decrease too quickly, since, by definition, the cumulative hazard has to diverge. For example, $$\exp(-t)$$ is not the hazard function of any survival distribution, because its integral converges to 1.

Quantities derived from the survival distribution
Future lifetime at a given time $$t_0$$ is the time remaining until death, given survival to age $$t_0$$. Thus, it is $$T - t_0$$ in the present notation. The expected future lifetime is the expected value of future lifetime. The probability of death at or before age $$t + t_0$$, given survival until age $$t_0$$, is just


 * $$P(T \le t_0 + t | T > t_0) = \frac{P(t_0 < T \le t_0 + t)}{P(T > t_0)} = \frac{F(t_0 + t) - F(t_0)}{S(t_0)}.$$

Therefore the probability density of future lifetime is


 * $$\frac{d}{dt}\frac{F(t_0 + t) - F(t_0)}{S(t_0)} = \frac{f(t_0 + t)}{S(t_0)}$$

and the expected future lifetime is


 * $$\frac{1}{S(t_0)} \int_0^{\infty} t\,f(t+t_0)\,dt = \frac{1}{S(t_0)} \int_{t_0}^{\infty} S(t)\,dt,$$

where the second expression is obtained using integration by parts.

For $$t_0 = 0$$, that is, at birth, this reduces to the expected lifetime.

In reliability problems, the expected lifetime is called the mean time to failure, and the expected future lifetime is called the mean residual lifetime.

As the probability of an individual surviving until age t or later is S(t), by definition, the expected number of survivors at age t out of an initial population of n newborns is n × S(t), assuming the same survival function for all individuals. Thus the expected proportion of survivors is S(t). If the survival of different individuals is independent, the number of survivors at age t has a binomial distribution with parameters n and S(t), and the variance of the proportion of survivors is S(t) × (1-S(t))/n.

The age at which a specified proportion of survivors remain can be found by solving the equation S(t) = q for t, where q is the quantile in question. Typically one is interested in the median lifetime, for which q = 1/2, or other quantiles such as q = 0.90 or q = 0.99.

One can also make more complex inferences from the survival distribution. In mechanical reliability problems, one can bring cost (or, more generally, utility) into consideration, and thus solve problems concerning repair or replacement. This leads to the study of renewal theory and reliability theory of aging and longevity.

Censoring
Censoring is a form of missing data problem which is common in survival analysis. Ideally, both the birth and death dates of a subject are known, in which case the lifetime is known.

If it is known only that the date of death is after some date, this is called right censoring. Right censoring will occur for those subjects whose birth date is known but who are still alive when they are lost to follow-up or when the study ends.

If a subject's lifetime is known to be less than a certain duration, the lifetime is said to be left-censored. Left-truncated data is common in actuarial work for life insurance and pensions (Richards, 2010).

It may also happen that subjects with a lifetime less than some threshold may not be observed at all: this is called truncation. Note that truncation is different from left censoring, since for a left censored datum, we know the subject exists, but for a truncated datum, we may be completely unaware of the subject. Truncation is also common. In a so-called delayed entry study, subjects are not observed at all until they have reached a certain age. For example, people may not be observed until they have reached the age to enter school. Any deceased subjects in the pre-school age group would be unknown.

Fitting parameters to data
Survival models can be usefully viewed as ordinary regression models in which the response variable is time. However, computing the likelihood function (needed for fitting parameters or making other kinds of inferences) is complicated by the censoring. The likelihood function for a survival model, in the presence of censored data, is formulated as follows. By definition the likelihood function is the conditional probability of the data given the parameters of the model. It is customary to assume that the data are independent given the parameters. Then the likelihood function is the product of the likelihood of each datum. It is convenient to partition the data into four categories: uncensored, left censored, right censored, and interval censored. These are denoted "unc.", "l.c.", "r.c.", and "i.c." in the equation below.


 * $$ L(\theta) = \prod_{T_i\in unc.} \Pr(T = T_i|\theta)

\prod_{i\in l.c.} \Pr(T < T_i|\theta) \prod_{i\in r.c.} \Pr(T > T_i|\theta) \prod_{i\in i.c.} \Pr(T_{i,l} < T < T_{i,r}|\theta) .$$

For an uncensored datum, with $$T_i$$ equal to the age at death, we have


 * $$ \Pr(T = T_i|\theta) = f(T_i|\theta) .$$

For a left censored datum, such that the age at death is known to be less than $$T_i$$, we have


 * $$ \Pr(T < T_i|\theta) = F(T_i|\theta) = 1 - S(T_i|\theta) .$$

For a right censored datum, such that the age at death is known to be greater than $$T_i$$, we have


 * $$ \Pr(T > T_i|\theta) = 1 - F(T_i|\theta) = S(T_i|\theta) .$$

For an interval censored datum, such that the age at death is known to be less than $$T_{i,r}$$ and greater than $$T_{i,l}$$, we have


 * $$ \Pr(T_{i,l} < T < T_{i,r}|\theta)

= S(T_{i,l}|\theta) - S(T_{i,r}|\theta) .$$

An important application where interval censored data arises is current status data, where the actual occurrence of an event $$ T_i$$ is only known to the extent that it known not to occurred before observation time and to have occurred before the next.

Non-parametric estimation
The Nelson–Aalen estimator can be used to provide a non-parametric estimate of the cumulative hazard rate function.

Distributions used in survival analysis

 * Exponential distribution
 * Weibull distribution
 * Exponential-logarithmic distribution