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Huggins Closed Captures Models

The closed captures data type consist of 2 versions of 3 different parameterizations of p and c: the full likelihood version and Huggins version.  For the Huggins (1989, 1991) version, the population size (N) is conditioned out of the likelihood.  An example is the best way to illustrate this concept.  Consider the 8 possible encounter histories for 3 occasions with the p, c data type:

Encounter History Probability
111  pcc
110  pc(1 - c)
101  p(1 - c)c
011  (1 - p)pc
100  p(1 - c)(1 - c
010  (1 - p)p(1 - c)
001  (1 - p)(1 - p)p
000  (1 - p)(1 - p)(1 - p)

For each of the encounter histories except the last, the number of animals with the specific encounter history is known.  For the last encounter history, the number of animals is N - M(t + 1), i.e., the population size minus the number of animals known to have been in the population.  The approach described by Huggins (1989, 1991) was to condition this last encounter history out of the likelihood by dividing the quantity 1 minus this last history into each of the others.  The result is a new multinomial distribution that still sums to one.

The derived parameter N is then estimated as M(t + 1)/[1 - (1 - p)(1 - p)(1 - p)] for data with no individual covariates.  A more complex estimator is required for models that include individual covariates to model the p parameters.

Confidence intervals for N are computed using a lognormal distribution and the number of animals never seen, f0 = N-hat - M(t+1), where M(t+1) is the number of marked animals in the population at time t + 1 (i.e., the number of animals marked during the study, and hence known to be in the population).  See page 212 of Burnham et al. (1987) for the explanation of this lognormal formula.
Confidence intervals (95%) for N are computed with a lognormal distribution with M(t+1) as a lower bound.
Lower = N/C + M(t+1)
Upper = N*CM(t+1)
C = exp(1.96 sqrt(log(1 + CV(f0-hat))^2)), where CV(f0-hat) = SE(f0-hat)/f0-hat