Geoscience Reference
In-Depth Information
Assuming that
ˆ
1
and
ˆ
2
are estimated by the proportions seen in samples
of size
n
1
and
n
2
, taken at times 1 and 2, respectively, the variances of
N
ˆ
1
and
ˆ
1
, respectively, are given approximately by the equations
ˆ
{
}
Var(
ˆ
)
(
ˆ
)/(
Var(
NNpNVarp pp
)
≈
2
+
2
−
)
2
,
(5.10)
1
1
1
2
2
1
2
and
{
}
Var(
ˆ
)
)Var(
ˆ
)( )Var(
ˆ
)/(
2
2
2
x Np
≈
(
p Np
+
p
p
−
p
)
,
(5.11)
1
1 2
1
2 1
2
1
2
where
{
}
Var(
ˆ
)
p
=
p pn nN
(1)/(1/)
−
−
i
.
(5.12)
i
i
i
i
i
These variances can be estimated by substituting estimates for true param-
eter values, as necessary. The variance of
N
ˆ
1
is the same as that of
N
ˆ
2
, and the
variance of
ˆ
1
is the same as that of
ˆ
2
. In general, the change-in-ratio method
gives estimates with good precision if the number of animals removed is
large and highly selective for one of the subclasses. Equivalently, in a cap-
ture-and-marking study, a large number of animals in one of the subclasses
must be marked. Spurious results can be obtained if the ratios of the two
subclasses are little changed between the two sample surveys.
Approximate confidence limits can be calculated in the usual way as
N
ˆ
1
±
·
ˆ
·
ˆ
SE(
ˆ
)
1 /2 1
, where the estimated standard errors are the
square roots of the estimated variances, and
z
α/2
is the value exceeded with
probability α/2 for the standard normal distribution. Better confidence lim-
its for small samples were reviewed by Seber (1982, p. 363).
z N
SE(
1
and
)
xz x
±
α
/2
α
EXAMPLE 5.2 Estimating a Mule Deer Population Size
Rasmussen and Doman (1943) described how a mule deer population
near Logan, Utah, suffered a severe loss during the 1938-1939 winter.
Counts before the loss gave an estimated proportion
ˆ
1
= 0.4536 of fawns,
and after the loss the estimated proportion was
ˆ
2
= 0.3464. A complete
survey of the area disclosed
r
x
= 248 dead fawns and
r
y
= 60 dead adults.
From these data, Equation (5.7) yields
N
ˆ
1
= (248 − 308 × 0.3464)/(0.4536 − 0.3464) = 1318.2 deer
and Equation (6.8) yields
ˆ
1
= 0.4536 × 1318.2 = 597.9 fawns.
