Geoscience Reference
In-Depth Information
be made using the same fixed population size, and it is appropriate to think
of producing some average of them to give the best combined estimate.
A number of methods have been proposed for producing a single com-
bined estimate of the size of a closed population from a series of independent
estimates. If all of the independent estimates are about equally precise, then
the simple average
ˆ
ˆ
ˆ
ˆ
*
*
*
*
NNNNk
=+++
(
)/
…
1
2
k
is appropriate, with a standard error that can be estimated by the usual equa-
tion
·
ˆ
ˆ
*
*
. However, if this is not the case because the
surveys were more intensive on some occasions than others, then a more
complicated type of combined estimate should be used. White and Garrott
(1990, Chapter 10) discussed six possible combined estimates and concluded
that a joint maximum likelihood estimator is generally best, followed by the
simple average.
SE(
N
)
=Σ
{ Var(
N k
)/ }
i
i
8.6 Flexible Modeling Procedures
An approach to modeling mark-recapture data that has been advocated by
Lebreton et al. (1992) represents a generalization of the work of Cormack
(1964), Jolly (1965), and Seber (1965). First, a model is proposed to obtain esti-
mates of capture and survival rates. The likelihood function for a set of data
is then constructed by multiplying together the probabilities of observing
the recapture patterns, given the time of first capture times. This function is
then maximized with respect to the unknown parameters in the model. For
example, consider a study of an animal population that lasts 7 years and sup-
pose that the captures and recaptures of an animal are recorded as 0110100,
where 1 indicates a capture and 0 indicates no capture. This animal is then
recaptured in years 3 and 5 after its first capture in year 2. For such an ani-
mal, the probability of the recapture pattern is assumed to be
ϕ
2
p
3
ϕ
3
(1 -
p
4
)ϕ
4
p
5
[(1 - ϕ
5
) + ϕ
5
(1 -
p
6
)(1 - ϕ
6
) + ϕ
5
(1 -
p
6
)ϕ
6
(1 -
p
7
)],
where ϕ
i
is the probability of surviving from year
i
to year
i
+ 1,
p
i
is the
probability of being captured in year
i
, and the three terms that are added
within the square brackets are (1) the probability of dying in the year fol-
lowing the last sighting; (2) the probability of surviving to year 6, not being
captured in that year, and dying in the following year; and (3) the probability
of surviving to year 7 but not being captured in either year 6 or year 7. For a
given model, probabilities can be obtained for all other recapture patterns in
