Basic Statistics

SURVIVAL FUNCTIONS 207

The survival function is useful in estimating the proportion of patients who will

survive until time t. For example, patients with a life-threatening disease might want

to know what proportion of patients treated in a similar fashion live 1 year. From the

survival function in Figure 14.5, we can see that about 40% survive at least 1 year.

This can be seen by finding 1 year on the horizontal axis and drawing a vertical straight

line that crosses the curve at a little less than 0.4 on the vertical axis. Or a physician

might want to know how long the typical patient survives. This can be accomplished

by finding the place on S(t) where the height is .5 and then looking down to see the

corresponding time.

If one treatment resulted in an appreciably higher survival function than another,

that would presumably be the preferred treatment. Also, the shape of the survival

function is often examined. A heroic treatment that results in an appreciable number

of patients dying soon, followed by a very slowly declining survival function, can be

contrasted with a safer but ineffective treatment that does not result in many immediate

deaths but whose survival rate keeps declining steeply over time.

Statistical survival programs commonly include a plot of the survival function

in their output and are used to evaluate the severity of illnesses and the efficacy of

medical treatments.

14.2.4 The Hazard Function

One other function commonly used in survival analysis is the hazard function. In
understanding the risk of death to patients over time, we want to be able to examine
the risk of dying given that the patient has lived up to a particular time. With a severe
treatment, there may be a high risk of dying immediately after treatment. Or, as in
some cancers, there may be a higher risk of dying two or more years after operation
and chemotherapy.
The hazard function gives the conditional probability of dying between time t and
tplus a short interval called At given survival at least to time t, all divided by At, as
At approaches 0. The hazard function is not the chance or probability of a death but
instead, is a rate. The hazard function must be >0, but there is no fixed upper value
and it can be > 1. It is analogous to the concept of speed. Mathematically, the hazard
function is equal to f(t)lS(t), the death density divided by the survival function. It
is also called the force of mortality, conditional failure rate, or instantaneous death
rate. We denote the hazard function by h(t).
One reason h(t) is used is that its shape can differ markedly for different diseases.
Whereas all survival functions, S(t), are decreasing over time, the hazard function
can take on a variety of shapes. These shapes are used to describe the risk of death to
patients over time. Figure 14.6 shows some typical hazard functions. The simplest
shape, labeled number 1, is a horizontal straight line. Here, the hazard function is
assumed to be constant over time. A constant hazard function occurs when having
survived up to any time t has no effect on the chance of dying in the next instant. Some
authors call this the no memory model. It assumes that failures occur randomly over
time. Although this assumption may appear unrealistic, a constant hazard rate has
been used in some biomedical applications. For example, it has been shown that the