Biology Reference
In-Depth Information
FIGURE 11.4
Log haz-
ards for each of the three
components in a Siler model
fit to
Coale and Demeney's
(1983)
Model West 1 for
females. The unlabeled
dashed line is the total log
hazard.
residual
0
20
40
60
80
Age
also be used to simulate hazard models not available in “R.” To use inversion, we simulate
survivorship values and then “look these up” to see what ages they translate into. This
“looking up” of ages forms the inversion step, since we are simulating survivorship and
converting it to age, rather than following the actual process where one's survivorship
depends on both age and random events. If U is a random uniform number (i.e., a random
number that is uniformly distributed such that
0 < U 1
), then we can solve Equation
11.12
for age to find that
t ¼
lnðUÞ
l
:
(11.14)
Equation
11.14
was used to simulate the ages in
Figure 11.3
, although the function “rexp” in
“R” could be used directly. For single component hazard models such as the Gompertz
model, one can also solve the survivorship function for age in order to simulate ages. The
Gompertz model has survivorship to age t of
SðtÞ¼expða
3
=
b
3
ð1 expðb
3
tÞÞÞ;
(11.15)
where we number the a and b parameters with a “3” to represent the third component of
mortality in the Siler model. Equation
11.15
can be solved for age at a given survivorship,
just as we did for the exponential hazard, so that ages at death can be simulated from
