Basic Statistics

SURVIVAL FUNCTIONS 205

Figure 14.3 Death density function, f(t).

of interest. In most statistical books, the notation used to indicate the death density
is f(t), where t is used to denote time.
Note that the death density function in Figure 14.3 is skewed to the right. This is
a common shape for such data. Unlike other continuous data that we have analyzed
so far, survival data are almost never assumed to be normally distributed.
In Figure 14.3, a vertical line has been drawn at a particular time t. For any given
t, the area to the left oft represents the proportion of patients in the population who
have died before time t. This proportion has been shaded gray. This area is called
the cumulative death distribution function and is usually denoted by F(t). The area
to the right oft is 1 - F(t) since the total area under the curve in Figure 14.3 is 1.
The area to the right represents the proportion of patients who have not died yet or,
in other words, the proportion surviving at least to time t. The proportion surviving
at least to time t is denoted by S(t).

14.2.2 The Cumulative Death Distribution Function

In Figure 14.4, the cumulative death distribution function F(t) is displayed. Note that
at time zero it is zero and it increases up to 1 at about time 3. The height of the curve
at time t gives the proportion of patients who have died before time t. By time t = 3,
all the patients have died. In Figure 14.4, t is approximately .8 and a little < .5 of
the patients have died by this time. Note that F(t) in Figure 14.4 looks rather similar
to Figure 6.1 1, even though Figure 6.1 1 is the cumulative function for a symmetric
normal curve. This illustrates the difficulty in distinguishing the shape of the death
density function simply by looking at the cumulative death distribution function.
Figure 14.4 also illustrates the relationship between F(t) and S(t), the survival
function. At any time t, the proportion dying before t plus the proportion surviving
at least to t equals 1.