Biology Reference
In-Depth Information
The probability that the individual was female is then one minus the probability that they
were male.
To illustrate the importance of estimating the sex ratio (the demographic parameter)
d
what we just did above
d
before generating individual sex estimates, imagine that we had fol-
lowed tradition and charged headlong into estimating sex individually for each cranium in
our target sample first. One way to do this is to use numbers directly from our original data,
so that for each additional cranium we score as having a “male” brow, we would say that
there is a 51/(51
þ
16)
¼
0.7612 chance that the individual was actually a male. But this is
a classic inverse probability problem in that we are calculating probabilities given known
sex that the brow will be “sexed” as “male,” “indeterminate,” or “female,” when what we
really want is the probability of sex given our observation on the brow. We could consequently
apply Bayes' theorem:
Pð
}
M
}
jMÞPðMÞ
Pð
}
M
}
jMÞPðMÞþPð
}
M
}
jFÞPðFÞ
PðMj
}
M
}
Þ¼
;
(11.5)
where we use quotation marks to represent “brow sex” and unquoted characters to repre-
sent actual sex. P(F) and P(M) are the prior probabilities of being female versus male.
WemightassumethatP(M)
¼
P(F)
¼
0.5, (because we might assume that there is a 50/50
chance of a skeleton being male versus female) in which case the prior probabilities cancel
and we have
Pð
}
M
}
jMÞ
Pð
}
M
}
jMÞþPð
}
M
}
jFÞ
PðMj
}
M
}
Þ¼
(11.6)
51=ð51 þ 4 þ 5Þ
51=ð51 þ 4 þ 5Þþ 16=ð16 þ 6 þ 32Þ
¼
:
From Equation
11.6
, the probability that a cranium was from a male if we decide that the
brow
“male” is 0.7415, which is slightly different
the probability we estimated earlier
(0.7612). Our estimate of the proportion of individuals who are male (p
m
), which is also
the prior probability that someone was male from our sample of 100 crania (P(M)), was
0.0982, so we should instead have
51=ð51 þ 4 þ 5Þ0:0982
51=ð51 þ 4 þ 5Þ0:0982 þ 16=ð16 þ 6 þ 32Þð1 0:0982Þ
PðMj
}
M
}
Þ¼
;
(11.7)
or 0.2380. Thus, the probability that an individual with a “male brow” ridge in our
example was actually a male is 0.2380 and the probability that the same individual was
female is 0.7620 (1
e
0.2380). The highly skewed sex ratio in favor of females in our example
(target sample) has decreased the probability that an individual we classified as a “male”
based on brow morphology was actually male. This all happened because our na¨ve
assumption of a 50/50 sex ratio that led to Equation
11.6
was far off the mark. In fact,
now that we have an estimate of the percentage of males in the target sample which is
significantly less than 50%, when we see a male-looking brow in the target sample the
individual is less likely to be an actual male than if they had come from a target sample
where males were more common.
