Heterogeneity Closed Captures Mtbh
This data type provides the most general closed captures model, and is available as both the full-likelihood and the Huggins version. Each of the initial capture probability, p, and recapture probability, c, parameters are modeled as mixtures of values. Details of these models are provided by Norris and Pollock (1995) and Pledger (1998, 2000). For example, with only one group and 5 occasions and using a mixture of 2 distributions, the following PIM's would be created:
pi
1
p
2 3 4 5 6
7 8 9 10 11
c
12 13 14 15
16 17 18 19
N
20
Not all 20 parameters are estimable without further constraints. A useful model is to make the recaptures equal to the initial captures giving model Mth, or making the p's constant and the c's constant, giving model Mbh. The PIMs for model Mbh could be:
pi
1
p
2 2 2 2 2
3 3 3 3 3
c
4 4 4 4
5 5 5 5
N
6
For 3 mixtures with the above example, the general PIMs would be:
pi
1 2
p
3 4 5 6 7
8 9 10 11 12
13 14 15 16 17
c
18 19 20 21
22 23 24 25
26 27 28 29
N
30
Note that the design matrix can also be used to construct the constraints that result in useful models. In particular, designs that maintain additive effects between the mixtures across time are reasonable. Consider the following PIMs:
pi
1
p
2 3 4 5 6
7 8 9 10 11
c
12 13 14 15
16 17 18 19
N
20
Then the following design matrix would allow an additive effect of occasion and an additive effect for recapture probability. Column 1 is for a single pi, column 2 is the intercept for the first p mixture, column 3 is the intercept for the second p mixture, columns 4 to 7 are 4 time effects, column 8 is the recapture effect, and column 9 is for the single N.
Row 1 2 3 4 5 6 7 8 9
|
1 | 1 0 0 0 0 0 0 0 0
2 | 0 1 0 1 0 0 0 0 0
3 | 0 1 0 0 1 0 0 0 0
4 | 0 1 0 0 0 1 0 0 0
5 | 0 1 0 0 0 0 1 0 0
6 | 0 1 0 0 0 0 0 0 0
7 | 0 0 1 1 0 0 0 0 0
8 | 0 0 1 0 1 0 0 0 0
9 | 0 0 1 0 0 1 0 0 0
10 | 0 0 1 0 0 0 1 0 0
11 | 0 0 1 0 0 0 0 0 0
12 | 0 1 0 0 1 0 0 1 0
13 | 0 1 0 0 0 1 0 1 0
14 | 0 1 0 0 0 0 1 1 0
15 | 0 1 0 0 0 0 0 1 0
16 | 0 0 1 0 1 0 0 1 0
17 | 0 0 1 0 0 1 0 1 0
18 | 0 0 1 0 0 0 1 1 0
19 | 0 0 1 0 0 0 0 1 0
20 | 0 0 0 0 0 0 0 0 1