Huggins Closed Captures Models
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 = f0/C + M(t+1)
Upper = f0*C + M(t+1)
C = exp(1.96 sqrt(log(1 + CV(f0-hat))^2)), where CV(f0-hat) = SE(f0-hat)/f0-hat
Huggins Closed Captures Data Type with Pledger Mixtures
Huggins models with mixture distributions are available, both the simple Mh model with just pi and p, and the more complex time-varying models Mth and Mtbh with pi, p, and c.
Huggins Closed Captures Data Type with Random Effects
Gimenez and Choquet (2010) proposed an extension of the CJS data type where individual random effects are modeled. Each animal is assumed to have its own random offset from the population mean. These random effects are assumed to be on the logit or log scale, so that the random effect is additive, with a normal distribution with mean zero and standard deviation sigma assumed. With this structure, Gaussian-Hermite quadrature can be used to integrate out the random effects and approximate the capture-recapture model likelihood. This same approach is used in the mark-resight data types (McClintock and White 2009, McClintock et al. 2009a) with individual random effects. The number of nodes can be set in the File | Preferences window.
For the Huggins closed captures with random effects data type, an additional parameter is used: sigmap models the individual heterogeneity of the p‘s. For sigmap = 0, you obtain the same likelihood as the basic Huggins data type, so the likelihoods of the random effects data type are compatible with the basic model, and thus AIC can be used to compare models.
The random effect is integrated for the probability of observing an encounter history, and for p*. Then 1 – p* is used to correct the encounter history probability for the conditioning out of the all zero history.
A similar model is used to model heterogeneity in the open models.
Recognize that more than 2 occasions are necessary to detect individual heterogeneity. What these models are doing is modeling the extra-binomial variation in the data (overdispersion). These models are useful for determining whether the data are overdispersed, i.e., whether parameter heterogeneity exists.
Median chat Procedure
The median chat procedure can be used with Huggins closed captures data. This is because the Huggins model conditions on the number of unique animals captured, M(t + 1). So to generate data with overdispersion, the median chat procedure generates encounter histories for exactly M(t + 1) animals. However, the median chat procedure cannot be used with the robust design data types with Huggins closed captures because this conditioning on M(t + 1) for the closed captures portion of the likelihood means that the survival portion of the likelihood is no longer valid.