Germán Rodríguez
Generalized Linear Models Princeton University

7. Survival Models

Our final chapter concerns models for the analysis of data which have three main characteristics: (1) the dependent variable or response is the waiting time until the occurrence of a well-defined event, (2) observations are censored, in the sense that for some units the event of interest has not occurred at the time the data are analyzed, and (3) there are predictors or explanatory variables whose effect on the waiting time we wish to assess or control. We start with some basic definitions.

7.1 The Hazard and Survival Functions

Let $$T$$ be a non-negative random variable representing the waiting time until the occurrence of an event. For simplicity we will adopt the terminology of survival analysis, referring to the event of interest as ‘death’ and to the waiting time as ‘survival’ time, but the techniques to be studied have much wider applicability. They can be used, for example, to study age at marriage, the duration of marriage, the intervals between successive births to a woman, the duration of stay in a city (or in a job), and the length of life. The observant demographer will have noticed that these examples include the fields of fertility, mortality and migration.

7.1.1 The Survival Function

We will assume for now that $$T$$ is a continuous random variable with probability density function (p.d.f.) $$f(t)$$ and cumulative distribution function (c.d.f.) $$F(t) = \Pr\{ T < t \}$$, giving the probability that the event has occurred by duration $$t$$.

It will often be convenient to work with the complement of the c.d.f, the survival function

$\tag{7.1}S(t) = \Pr\{ T \ge t \} = 1 - F(t) = \int_t^\infty f(x) dx,$

which gives the probability of being alive just before duration $$t$$, or more generally, the probability that the event of interest has not occurred by duration $$t$$.

7.1.2 The Hazard Function

An alternative characterization of the distribution of $$T$$ is given by the hazard function, or instantaneous rate of occurrence of the event, defined as

$\tag{7.2}\lambda(t) = \lim_{dt\rightarrow0} \frac{\Pr\{t \le T < t + dt | T \ge t \} }{dt}.$

The numerator of this expression is the conditional probability that the event will occur in the interval $$[t,t+dt)$$ given that it has not occurred before, and the denominator is the width of the interval. Dividing one by the other we obtain a rate of event occurrence per unit of time. Taking the limit as the width of the interval goes down to zero, we obtain an instantaneous rate of occurrence.

The conditional probability in the numerator may be written as the ratio of the joint probability that $$T$$ is in the interval $$[t,t+dt)$$ and $$T \ge t$$ (which is, of course, the same as the probability that $$t$$ is in the interval), to the probability of the condition $$T \ge t$$. The former may be written as $$f(t)dt$$ for small $$dt$$, while the latter is $$S(t)$$ by definition. Dividing by $$dt$$ and passing to the limit gives the useful result

$\tag{7.3}\lambda(t) = \frac{f(t)}{S(t)},$

which some authors give as a definition of the hazard function. In words, the rate of occurrence of the event at duration $$t$$ equals the density of events at $$t$$, divided by the probability of surviving to that duration without experiencing the event.

Note from Equation 7.1 that $$-f(t)$$ is the derivative of $$S(t)$$. This suggests rewriting Equation 7.3 as

$\lambda(t) = - \frac{d}{dt}\log S(t).$

If we now integrate from 0 to $$t$$ and introduce the boundary condition $$S(0) = 1$$ (since the event is sure not to have occurred by duration 0), we can solve the above expression to obtain a formula for the probability of surviving to duration $$t$$ as a function of the hazard at all durations up to $$t$$:

$\tag{7.4}S(t) = \exp \{ - \int_0^t \lambda(x)dx \}.$

This expression should be familiar to demographers. The integral in curly brackets in this equation is called the cumulative hazard ( or cumulative risk) and is denoted

$\tag{7.5}\Lambda(t) = \int_0^t \lambda(x) dx.$

You may think of $$\Lambda(t)$$ as the sum of the risks you face going from duration 0 to $$t$$.

These results show that the survival and hazard functions provide alternative but equivalent characterizations of the distribution of $$T$$. Given the survival function, we can always differentiate to obtain the density and then calculate the hazard using Equation 7.3. Given the hazard, we can always integrate to obtain the cumulative hazard and then exponentiate to obtain the survival function using Equation 7.4. An example will help fix ideas.

Example: The simplest possible survival distribution is obtained by assuming a constant risk over time, so the hazard is $\lambda(t) = \lambda$

for all $$t$$. The corresponding survival function is

$S(t) = \exp \{ -\lambda t \}.$

This distribution is called the exponential distribution with parameter $$\lambda$$. The density may be obtained multiplying the survivor function by the hazard to obtain

$f(t) = \lambda \exp\{-\lambda t\}.$

The mean turns out to be $$1/\lambda$$. This distribution plays a central role in survival analysis, although it is probably too simple to be useful in applications in its own right.$$\Box$$

7.1.3 Expectation of Life

Let $$\mu$$ denote the mean or expected value of $$T$$. By definition, one would calculate $$\mu$$ multiplying $$t$$ by the density $$f(t)$$ and integrating, so

$\mu = \int_0^\infty t f(t) dt.$

Integrating by parts, and making use of the fact that $$-f(t)$$ is the derivative of $$S(t)$$, which has limits or boundary conditions $$S(0) = 1$$ and $$S(\infty) = 0$$, one can show that

$\tag{7.6}\mu = \int_0^\infty S(t) dt.$

In words, the mean is simply the integral of the survival function.

7.1.4 A Note on Improper Random Variables*

So far we have assumed implicitly that the event of interest is bound to occur, so that $$S(\infty) = 0$$. In words, given enough time the proportion surviving goes down to zero. This condition implies that the cumulative hazard must diverge, i.e. we must have $$\Lambda(\infty) = \infty$$. Intuitively, the event will occur with certainty only if the cumulative risk over a long period is sufficiently high.

There are, however, many events of possible interest that are not bound to occur. Some men and women remain forever single, some birth intervals never close, and some people are happy enough at their jobs that they never leave. What can we do in these cases? There are two approaches one can take.

One approach is to note that we can still calculate the hazard and survival functions, which are well defined even if the event of interest is not bound to occur. For example we can study marriage in the entire population, which includes people who will never marry, and calculate marriage rates and proportions single. In this example $$S(t)$$ would represent the proportion still single at age $$t$$ and $$S(\infty)$$ would represent the proportion who never marry.

One limitation of this approach is that if the event is not certain to occur, then the waiting time $$T$$ could be undefined (or infinite) and thus not a proper random variable. Its density, which could be calculated from the hazard and survival, would be improper, i.e. it would fail to integrate to one. Obviously, the mean waiting time would not be defined. In terms of our example, we cannot calculate mean age at marriage for the entire population, simply because not everyone marries. But this limitation is of no great consequence if interest centers on the hazard and survivor functions, rather than the waiting time. In the marriage example we can even calculate a median age at marriage, provided we define it as the age by which half the population has married.

The alternative approach is to condition the analysis on the event actually occurring. In terms of our example, we could study marriage (perhaps retrospectively) for people who eventually marry, since for this group the actual waiting time $$T$$ is always well defined. In this case we can calculate not just the conditional hazard and survivor functions, but also the mean. In our marriage example, we could calculate the mean age at marriage for those who marry. We could even calculate a conventional median, defined as the age by which half the people who will eventually marry have done so.

It turns out that the conditional density, hazard and survivor function for those who experience the event are related to the unconditional density, hazard and survivor for the entire population. The conditional density is

$f^*(t) = \frac{f(t)}{1-S(\infty)},$

and it integrates to one. The conditional survivor function is

$S^*(t) = \frac{S(t)-S(\infty)}{ 1-S(\infty) },$

and goes down to zero as $$t \rightarrow \infty$$. Dividing the density by the survivor function, we find the conditional hazard to be

$\lambda^*(t) = \frac{ f^*(t) }{ S^*(t) } = \frac{ f(t) }{ S(t)-S(\infty) }.$

Derivation of the mean waiting time for those who experience the event is left as an exercise for the reader.

Whichever approach is adopted, care must be exercised to specify clearly which hazard or survival is being used. For example, the conditional hazard for those who eventually experience the event is always higher than the unconditional hazard for the entire population. Note also that in most cases all we observe is whether or not the event has occurred. If the event has not occurred, we may be unable to determine whether it will eventually occur. In this context, only the unconditional hazard may be estimated from data, but one can always translate the results into conditional expressions, if so desired, using the results given above.

Math rendered by