Biology Reference
In-Depth Information
FIGURE 11.7
Plot of
the predicted mean femoral
length plus and minus two
standard deviations across
age. The open points are as
in Figure 11.6, while the lines
for the mean and plus and
minus two standard devia-
tions were drawn using
fractional polynomials.
0
2
4
6
8
10
12
Age (years)
We treat age in the logarithmic scale and then exponentiate within the likelihood function
itself to reduce numerical difficulties in calculating the likelihood. The integral across age
is easily handled using the “integrate” function in “R.”
From the individual likelihoods we can write the total log-likelihood as
250
ln
ð
f
ðFL
i
jlÞÞ:
(11.18)
i ¼1
We again use “maxLik” in “R” to find the maximum likelihood estimate (and its error) for the
exponential hazard parameter. This results in an estimated exponential hazard parameter of
0.3451 with a standard error of 0.0220.
Figure 11.8
shows a Kaplan
e
Meier plot (
Kaplan and
Meier, 1958
) of the survivorship from the actual (albeit simulated) ages-at-death and the 95%
confidence intervals for survivorship estimated from the simulated femur length data. The fit
is good, which is to be expected given that we used the same models and assumptions to
simulate and analyze the data. Specifically, we simulated the deaths under an exponential
hazard and analyzed the simulated femoral length data assuming that the mortality could
be modeled using an exponential hazard. We also simulated growth and then used the
same underlying equations to model growth.
In an actual analysis of a skeletal sample we will not know the true underlying form of the
hazard of death against age, we may run the risk of applying growth standards that are not
appropriate for the given sample, and we certainly will not have access to the true ages. For
