Geoscience Reference
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and the time-varying M
t
is given by
logit(
P
ij
) =
a
+
c
j
,
with
c
k
= 0. For the logit models corresponding to the remaining six mod-
els, see the work of Huggins (1991) and Chao and Huggins (2005b). Under
Huggins's method, closed-population models have a much broader range
than the Otis et al
.
(1978) models, as any individual or occasional covariate
accounting for individual heterogeneity and time effects can be included in
the models covered by Equation (7.5). In fact, it is customary in the mark-
recapture literature of closed populations to denote by M*
0
, M*
t
, . . . , M*
tbh
the
set of models involving the effects accounted for in the Otis et al
.
models,
together with possibly one or more individual or occasional covariates.
As information criteria indices are available for the generalized linear
models fitted, model selection is practicable. Finally, supposing that a total
of
n
individuals were captured, the population size is estimated by the
Horvitz-Thompson estimator,
∑
n
∏
k
ˆ
ˆ
−
1
N
=
{1
−
(1
−
P
)}
( 7. 6 )
HT
ij
i
=
1
j
=
1
and the corresponding standard error of
N
ˆ
HT
is approximated by the asymp-
totic equations given by Huggins (1989, 1991).
Example 7.3 A Hypothetical Study
A hypothetical data set was built based on an example code included in
the documentation of the R package mra (McDonald, 2012). The result-
ing encounter history for 29 animals captured or recaptured on five sam-
pling occasions is summarized in Table 7.3, while Table 7.4 shows part
TABLE 7.3
Summary Statistics of the Hypothetical Encounter History of 29 Animals for
t
= 5
Sampling Occasions
j
u
j
m
j
n
j
M
j
f
j
1
14
0
14
0
3
2
7
5
12
14
12
3
4
11
15
21
9
4
2
14
16
25
5
5
2
15
17
27
0
M
6
= 29
Note: u
j
, number of first captures at time
j
,
j
= 1, . . . , 5;
m
j
, number of recaptures at time
j
;
n
j
,
u
j
+
m
j
= number of animals captured at time
j
;
M
j
,
u
1
+
u
2
+ . . . +
u
j
-1
= number of marked
animals just before the
j
th occasion;
M
6
, total number of distinct animals recorded in the
study;
f
j
, number of animals captured exactly
j
times throughout the whole study.
