Geoscience Reference
In-Depth Information
as those given by Equation (7.3). For this reason, Equation (7.3) should work
well for small samples as well as large samples and when the sample sizes
are not fixed. Also, it can be argued that the standard error conditional on
the observed values of
n
1
and
n
2
is the one that is of interest.
EXAMPLE 7.1 The Size of a Deer Mouse Population
As an example, consider part of the results of an experiment on the effects
of controlled burning on populations of the deer mouse (
Peromycus
maniculatus
) as discussed by Skalski and Robson (1992, p. 126). On one
unburned site, 3 days of trapping and marking produced
n
1
= 49 marked
mice. Three more days of trapping then produced
n
2
= 82 mice, of which
m
2
= 26 were marked. For these data, the nearly unbiased estimate from
Equation (7.2) is
N
ˆ
*
= (49 + 1)(82 + 1)/(26 + 1) − 1 = 152.7
with
·
N
ˆ
*
SE(
)
= √[50 × 83 × (49 − 26) × (82 − 26)/(27
2
× 28)] = 16.2
This suggests that the population size was probably within the range
152.7 ± 1.96 × 16.2, or 121 to 184, taking the estimate plus and minus 1.96
standard errors as an approximate 95% confidence interval with round-
ing to integers.
7.3.2 Sample Size Recommendations
To avoid wasting effort on a study that is not able to give a reasonable level
of accuracy, it is important to plan sample sizes in advance, albeit with the
recognition that planned sample sizes and obtained sample sizes may differ
considerably. Robson and Regier (1964) have given recommendations for the
sample sizes that will yield estimates of
N
using Chapman's modification of
the Peterson-Lincoln index, which are within π% of the true value with 95%
confidence. They considered values of π = 50%, 25%, and 10% and provided
sample size charts in which, for given pairs of (
n
1
,
n
2
), one is able to determine
the necessary precision with 95% confidence.
After reviewing a number of different combinations of pairs (
n
1
,
n
2
), a gen-
eral pattern appears. Fairly large samples are required to obtain accurate
population size estimates from the Petersen-Lincoln equation. As a general
rule, there should be at least 10 marked animals in the second sample, and for
a large population far more than this is needed. For example, if the popula-
tion size is 100 and
n
1
and
n
2
are to be equal, then to obtain a 95% confidence
interval of about 90 to 110, it is necessary to have
n
1
=
n
2
= 65, leading to about
42 marked animals in the second sample. For a second example, suppose
that
N
is 10,000. Then, taking
n
1
=
n
2
= 800 will give a 95% confidence interval
