Geoscience Reference
In-Depth Information
or
{
}
ˆ
(Total catch)/ 1 1 ˆ)
n
N
=
−
−
p
(5.4)
for
N
.
Soms (1985) has shown that Equations (5.3) and (5.4) yield estimates that are
about as good as those obtained from the more complicated method devel-
oped by Zippin (1956, 1958). Soms also provided the following equations for
the variances of the regression estimators, noting that they appear to give
satisfactory results providing that the population size
N
is about 200 or more:
n
∑
{
}
ˆ
)
n
n
2
n
n
2
Var(
NNq
≈
/(1 )1 /(1)
−
q
+
nq
−
q
c
/
s
,
(5.5)
i
i
i
=
1
and
n
∑
Var(
ˆ
)
2
2
pq cNs
≈
/(
)
,
(5.6)
i
i
i
1
=
where
q
= 1 −
p
,
c
i
= {
i
− (
n
+ 1)/2}/{
n
(
n
² − 1)/12}, and
s
i
=
p
(1 −
p
)
i
-1
.
Zippin (1956) included an example in which small mammals were marked
in a 3-night trapping program; the captures were
z
1
= 165,
z
2
= 101, and
z
3
= 54.
Based on approximations for the maximum likelihood estimates available in
the 1950s, he estimated
N
ˆ
= 400, with standard error 26.3, a 95% confidence
interval of 347 to 453, and an estimated probability of capture of
ˆ
= 0.4253.
With Soms's method, a linear regression line is fitted to the data (
y
i
= log
e
(
z
i
),
x
i
=
i
− 1): (5.106, 0), (4.615, 1), and (3.989, 2). The equation of the line is
y
= 5.1285 − 0.5585
x
, so that the estimated proportion of animals removed
each night is
ˆ
= 1 − exp(−0.5585) = 0.4279, which is close to Zippin's (1956)
estimate. A total of 320 animals were captured, and the total population size
was estimated to have been 320, divided by the probability that an animal
will be captured during the three trapping nights,
N
ˆ
= 320/{1 − (1 −
ˆ
)
3
} =
320/(0.8128) = 394. Using Equation (5.5), the estimated variance of
N
ˆ
is 812.37,
so that the estimated standard error is
812.37 28.5
, and the approximate
95% confidence interval, 394 ± (1.96)(28.5), has a lower limit of 338 and upper
limit of 450.
Using the interactive program CAPTURE, the maximum likelihood esti-
mates of the proportion of animals removed each night are
ˆ = 0.4253, and
=
=
N
ˆ
394
with standard error = 22.6, with a 95% confidence interval for
N
of
362 to 453. The population estimates from Zippin's (1956) article (400) and
the program CAPTURE (394) agree favorably with those provided by Soms's
linear regression method (394); however, the confidence limits provided by
