Geoscience Reference
In-Depth Information
let
X
denote marked animals and
Y
denote unmarked animals. At time 1,
x
1
= 0,
p
1
= 0, and
y
1
=
N
1
because there are no marked animals in the popula-
tion. However, there is an addition of
n
1
marked animals, leading to a nega-
tive removal of
r
x
= −n
1
and a positive removal of unmarked animals,
r
y
=
n
1
.
Finally, the total removed is
r
=
r
x
+
r
y
= −
n
1
+
n
1
= 0. Substituting into the
change-in-ratio formula for the initial population size, we have
N
ˆ
1
= (
r
x
−
r
ˆ
2
)/
(
ˆ
1
−
ˆ
2
) = (−
n
1
)/(−
ˆ
2
) =
n
1
/
ˆ
2
. However, the proportion of marked animals
at time 2 is estimated by
ˆ
2
=
m
/
n
2
, leading to the Lincoln-Petersen estimate
N
ˆ
1
=
n
1
n
2
/
m
. Variances can be computed from either the equations given in
Chapter 7 or the ones presented here.
Pollock et al. (1985) have pointed out that a potential practical problem with
the use of the change-in-ratio method is that the two types of individual may not
be equally likely to be observed in a sample from the population, with the result
that
ˆ
1
and
ˆ
2
will be biased estimators. They noted, however, that if the remov-
als are of only one type of animal, then the number of this type can be estimated
without bias irrespective of whether the two types are equally visible and dis-
cussed how a three-sample design can be used to estimate the numbers of both
types of animal. The estimation equations are quite straightforward if one type
of animal is removed between the times of sample 1 and sample 2, and then
the other type of animal is removed between the times of sample 2 and
sample 3. Removals of some of both types between sample 1 and 2 and then
the further removal of some of both types between sample 2 and sample 3
can also in principle be allowed, although estimation of population sizes is
then somewhat more complicated.
An interesting application of the three-sample change-in-ratio method was
described by Lancia et al. (1988) as part of a harvest strategy for populations
such as deer for which harvests are well controlled and hunting seasons are
short. They showed that taking two separate single-sex harvests together
with three samples of deer provides a means of achieving the harvest quota
and estimating the population size.
Application of change-in-ratio methods to multiple time periods using
a software program called USER was covered by Skalski and Millspaugh
(2006). They discussed multiclass and sequential methods and illustrated
how the USER program can be used to calculate maximum likelihood esti-
mates. Other extensions of the change-in-ratio method to multiclass situ-
ations and a combination of change in ratio with effort information were
covered by Udevitz and Pollock (1991, 1995). Computer programs for analy-
sis of these extended methods are provided using SAS statistical software
(http://alaska.usgs.gov/science/biology/biometrics/cir01/using_sas.php).
