Biology Reference
In-Depth Information
This section on simulating ages-at-death may seem excessive, but the ability to simulate
ages-at-death is a useful tool for demonstrating, and particularly for testing, demographic
methods. While simulations can always be criticized for being unrealistic, they have the
advantage that we can control various aspects of the data production so that in the end we
can compare our analysis to the results we “should have gotten” given that we know the
way that the data were simulated. In the next section we examine the simulation of “age indi-
cators” in order to produce test cases for demographic analysis. To keep the example simple,
we use only the exponential hazard model and limit it to immature individuals, specifically
children ages 0 to 12 years. While age estimation at the younger ages is generally not prob-
lematic, the simulations and methods we present can be used in applications for older
individuals.
Simulating Long Bone Growth
We use quotation marks around the term “age indicators,” because we need to
decouple the processes of growth and development and of progressive skeletal change
in adults from the idea of “indicators,” particularly when dealing with senescent
processes. There is a long history of using skeletal markers or indicators as predictors
for age-at-death, and a consequent tendency for researchers to use models that make
age dependent on the markers or skeletal variables.
Bocquet-Appel and Masset (1982)
point out the error in this approach for ordered skeletal traits (such as Todd phases),
and
Aykroyd et al. (1999)
and
Konigsberg et al. (1997)
point out the problem for contin-
uous traits. The idea that age is the dependent variable (rather than the skeletal variables
being the dependent variables depending on age) is logically inconsistent, and runs
counter to studies of growth and development (
McCammon, 1970; Roche et al., 1988;
Cameron, 2002
). For children it is very unusual to see age treated as the dependent vari-
able, although a few publications do present regressions of fetal gestational ages in weeks
on long bone lengths (
Scheuer et al., 1980; Sherwood et al., 2000
).
In order to simulate records for children between birth and 12 years of age, we chose
length of the femur diaphysis as an “age indicator,” and drew summary data from
Maresh's
(1970)
radiographic longitudinal growth study. From his study we use the means and stan-
dard deviations of bone lengths for males observed at two months, four months, six months,
and then at six month intervals up until age 12 years (these are given in Maresh's Table F-7).
In order to simulate bone lengths at any given age on a continuum, we need to smooth Mar-
esh's tabular interval data. We accomplish this using fractional polynomials, which do not
require the powers in the polynomial to be integers and which can produce simpler equa-
tions (
Royston and Altman, 1994; Sauerbrei et al., 2006
). Polynomial regression is a commonly
applied tool that can be used to fit long bone length to powers of age. As an example,
Figure 11.6
shows a third-degree polynomial that regresses long bone length on age, the
square of age, and the cube of age as a dashed line, and the fractional polynomial as a solid
line, both fit to the means from Maresh. While the dashed line appears to fit reasonably well,
the solid line also fits well.
In the case of
the Maresh data,
the best fitting fractional polynomial model
is
ag
p
, so the power is 0.5. The best fitting fractional polynomial
model for the standard deviations from Maresh is actually the linear model, resulting in
femur length
¼ 3:6 þ 9:78
