208 INTRODUCTION TO SURVIVAL ANALYSIS0 1 2 3
Time (years)Figure 14.6 Hazard functions, h(t).length of time from the first to the second myocardial infarction can be approximated
by a constant hazard function. When the hazard function is constant, a distribution
called the exponential distribution can be assumed. The exponential is the simplest
type of theoretical distribution used in survival analysis.
Numbers 2 and 3 in Figure 14.6 show a decreasing and an increasing hazard
function. A decreasing hazard function is found when as time passes since entry into
the study the patient is more apt to live (at least for a short interval). This could occur
because an operation is successful in treating the condition and the patients who die
due to the operation do so soon after the operation. An increasing hazardfunction
may be found when the treatment is not successful and as time passes the patient is
more likely to die.
Number 4 shows a linear increasing hazard function. This is the hazard function
from the death density and survival function given in Figures 14.3 and 14.5. That
is, if we took a particular value of t and divided the height of f(t) in Figure 14.3
by the height of S(t) at the same value oft, we would obtain a point on the linearly
increasing line labeled 4. In interpreting h(t) and S(t), it is useful to keep in mind
that they are inversely related. That is, if S(t) quickly drops to a low value, we would
expect to have a high initial h(t).
14.3 COMPUTING ESTIMATES OF f(t), S(t), AND h(t)
In this section we show how to estimate the death density, survival function, and
hazard function using clinical life tables. In making life tables, the data are grouped
in a manner somewhat similar to what was done in Section 4.1. We also present an
estimation of the survival function using the Kaplan-Meier method. The Kaplan-