Biology Reference
In-Depth Information
Simulating Data for Sex Ratio Estimation and Extending MLE to Other Traits
The example of 100 crania scored on brow morphology as 35 “male,” 11 “indeterminate,”
and 54 “female” used above is a simulated dataset based on an ordinal categorical trait with
three categories. The trait is femaleness of the brow, with “male” as the least female, “inde-
terminate” as intermediate, and “female” as the most female (or alternatively we could say
maleness of the brow with “female” as the least male, “indeterminate” as intermediate, and
“male” as the most male). We deterministically simulated this dataset by assuming that out of
a sample of 100 crania 10 were from males and 90 were from females, so that we had
2
4
3
5
¼
2
4
3
5
z
2
4
3
5
:
10
90
51=60
16=54
35:1667
10:6667
54:1667
35
11
54
4=60
6=54
(11.8)
5=60
32=54
Note that our estimated proportion of males (0.0982) is very close to the actual proportion
(0.1). Had we not insisted on using integer counts for the number of crania, we would
have precisely recovered the proportion of males.
The types of demographic analysis exemplified so far (estimation of the sex ratio and the
probability of being male or female) can be extended to other single ordinal categorical traits
by simple substitution. Extending this type of demographic analysis to multiple ordinal cat-
egorical traits generally requires adoption of a cumulative probit model, as discussed by
Konigsberg and Hens (1998)
. But what about using measurements on one or more skeletal
traits that display sexual dimorphism to estimate the sex ratio or the probability of being
male or female?
To illustrate this, next we provide an example of a univariate (single) metric trait applica-
tion using
Black's (1978)
published summary statistics for circumference of the femur at the
midshaft from 63 males and 51 females from the Libben Site, Ohio. Among these 114 individ-
uals who had been sexed on the basis of Phenice characteristics, the male mean femoral
circumference was 63 (
4.1) millimeters and the female mean was 51 (
4.2), where the
parenthetical terms give the standard deviations.
Figure 11.1
shows these parameters as
normal distributions. To simulate an example dataset we use the random normal generator
in “R” and simulate data from 100 males and from 900 females using Black's means and stan-
dard deviations, with the simulated values rounded to the nearest millimeter.
A simple, albeit incorrect, method to estimate the sex ratio from this example dataset is to
use the measurement halfway between the female and male means (51 and 63 mm, respec-
tively) as a sectioning point (see
Figure 11.1
). Thus, we sex individuals with measurements
below 57 mm as female, those above 57 mm as male, and divide those at 57 mm equally
between male and female. From our simulated dataset 820 of the females have values below
57 mm, 29 are at 57 mm, and 51 are above 57 mm. For males 7 are below 57 mm, 3 are at 57
mm, and 90 are above 57 mm. Our estimated proportion of males in the sample using this
(incorrect) method is (51
þ
14.5
þ
1.5
þ
90)/1000
¼
0.157, which is higher than the actual propor-
tion of 0.1. Why did we overestimate the proportion of males?
The calculationof 57mmas the sectioningpoint restedon two assumptions, only one ofwhich
is supportable. The first assumption is that the male and female standard deviations are equal,
which they are nearly so. The second assumption is that the sex ratio is 1:1, which it is not. If
we knew a priori thattherewasonlyonemaleforeveryninefemalesthenwecouldsolve
