Biology Reference
In-Depth Information
In general, increasing the size of a sample will reduce the size of the standard error of the
estimates, but does not necessarily alter the proportion estimated. We can illustrate this by
generalizing the above example. Equation 11.2 has a maximum that we can find explicitly
without having to resort to numerical maximization. Setting the first derivative equal to
zero and solving for p m , we have the surprisingly ugly
s
346853376U 2 þ 214651801M 2 þ 1107225625F 2
þUð1239427200F 545720448MÞþ975024050FM
20672U þ 120497Mþ 64625F 3
P m ¼
;
65780N
(11.3)
where M is the number of crania with “male” brows (35 in the above example), U is the
number with “indeterminate” brows (11 in the above example), F is the number with
“female” brows (54 in the above example), and N is the total sample size (100 in the above
example). We can also explicitly find the “information” (the negative of the second deriva-
tive) of Equation 11.2 as
4U
89401M
3025F
I ¼
2 þ
2 þ
2 ;
p m
17p m
20
p m
15 þ 1 p m
þ 8ð1 p m Þ
27
12 þ 16ð1 p m Þ
2025
291600
11664
9
27
(11.4)
where p m is the maximum likelihood estimate from Equation 11.3 . The reciprocal of the
information is an estimate of the variance, so the square root of this reciprocal is the stan-
dard error. We can use Equations 11.3 and 11.4 for a new example where we have 10 times
as many crania (N ¼ 1000), but with brow indicators in the same proportions (350 “male,”
110 “indeterminate,” and 540 “female”). From Equation 11.3 we get an estimated propor-
tion of males at 0.0982, identical with our previous example. The addition of 900 crania to
the original 100 does, however, decrease the standard error (from Equation 11.4 ) to 0.0270,
giving a 95% confidence interval of from 0.0453 to 0.1511, or 4.53% to 15.11% male.
Having an estimate of the proportion of the target sample that is male is absolutely
“mission critical” in obtaining individual estimates of sex, as we directly show in the
next section.
Estimating the Sex of Individuals (or the Probability that Individuals were Male)
In demographic terms, estimating the sex of an individual in fact means estimating the
probability that an individual was a male, or the probability that an individual was
a female. This is a different way of thinking about estimating sex compared to traditional
osteological analyses. The results of an osteological analysis might identify individuals as
male, probably male, possibly male, possibly female, probably female, female, and
unknown or indeterminate. These are largely unquantifiable statements, as we have no
way of knowing how much more likely a “probable male” was to have been male than
would be the case for a “possible male.” Instead of producing such qualitative sex esti-
mates, we want to quantitatively estimate the probability that an individual was male.
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