Heterogeneity Open Models
Heterogeneity Open Models
Cormack-Jolly-Seber Data Type with Mixtures
Pledger et al. (2003) proposed a mixture model to account for heterogeneity in the Cormack-Jolly-Seber (CJS) data type. This model incorporates a single mixture parameter (pi) to model heterogeneity in both phi and p. Normally, I would not think that heterogeneity in phi is an issue, but to be true to the original paper, I have implemented the model in MARK as Pledger et al. (2003) described it.
This data type can be accessed by the PIM | Change Data Type menu choices. So, you can create a normal CJS MARK file, and then compare the usual models with these models that include heterogeneity.
The data type has 3 parameters: pi, phi, and p. The following example illustrates the model with 5 occasions and 2 mixtures for a single group. The time-specific PIMs look like the following.
pi PIM
1
phi PIM
2 3 4 5
3 4 5
4 5
5
6 7 8 9
7 8 9
8 9
9
p PIM
10 11 12 13
11 12 13
12 13
13
14 15 16 17
15 16 17
16 17
17
The structure of the phi and p PIMs is just the upper-triangular array typical of a CJS model duplicated for the 2 mixtures. Normally, as suggested by Pledger et al. (2003), the parameter estimates would be additive across the mixtures. The following design matrix would generate a time-specific model, but where the time-specific values for mixture A (with probability pi) are additive with mixture B (with probability 1 – pi). The model name might be {pi phi(t+h2) p(t+h2)}, where h2 indicates the set of 2 mixtures.
pi phi int phi t1 phi t2 phi t3 phi mix p int p t1 p t2 p t3 p mix
1 0 0 0 0 0 0 0 0 0 0
0 1 1 0 0 1 0 0 0 0 0
0 1 0 1 0 1 0 0 0 0 0
0 1 0 0 1 1 0 0 0 0 0
0 1 0 0 0 1 0 0 0 0 0
0 1 1 0 0 0 0 0 0 0 0
0 1 0 1 0 0 0 0 0 0 0
0 1 0 0 1 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 1 1 0 0 1
0 0 0 0 0 0 1 0 1 0 1
0 0 0 0 0 0 1 0 0 1 1
0 0 0 0 0 0 1 0 0 0 1
0 0 0 0 0 0 1 1 0 0 0
0 0 0 0 0 0 1 0 1 0 0
0 0 0 0 0 0 1 0 0 1 0
0 0 0 0 0 0 1 0 0 0 0
If you desire a model where there is no heterogeneity on phi, but still on p, then all you would have to do is delete the phi mix column in the above design matrix. The result would be that phi parameters 2-5 would be identical to 6-9. This model might be named {pi phi(t) p(t+h2)}.
The Pledger mixture model has a consistent likelihood with the regular CJS model, so you can compare these models with other CJS models using AICc.
Cormack-Jolly-Seber 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 asumed 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 likeliood. 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 CJS data type, 2 additional parameters are used: sigmaphi models the individual heterogeneity of the phi’s, and sigmap models the individual heterogeneity of the p‘s. For sigmaphi = 0 and sigmap = 0, you obtain the same likelihood as the basic CJS 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 CJS data type with random effects is available from the CJS data type through the PIM | Change Data Type menu choice.
Pradel Data Type with Mixtures
Pradel et al. (2009) proposed a similar mixture model for the Pradel data type. However, the mixtures only apply to p, and not to phi, gamma, lambda, or f. All three parametrization (i.e., seniority with gamma, population change with lambda, and recruitment with f) are implemented. The likelihood for the mixture models is consistent with the usual Pradel models, so AICc can be used to compare the mixture models to the regular models. You use the PIM | Change Data Type menu choices to change the data type to these mixture models.
The Pradel data type does not use the upper-triangular PIM structure, because inferences are being made to animals not yet captured. Thus,, the PIMs are much simpler than the CJS Pledger mixture data type described above.
Link-Barker Data Type with Mixtures
The equivalent mixture model on p was also incorporated into the Link-Barker data type. This data type is equivalent to the Pradel recruitment parametrization, except that the Link-Barker data type correctly handles losses on capture. Thus, you normally get identical -2log likelihood values for the Pradel f and Link-Barker models, except when there are losses on capture.
Link-Barker 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 asumed 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 likeliood. This same approach is used in the mark-resight data types (McClintock and White 2009, McClintock et al. 2009a) with individual random effects. I have implemented this same approach with the Link-Barker data type. The number of nodes can be set in the File | Preferences window.
For the Link-Barker data type, 3 additional parameters are used: sigmaphi models the individual heterogeneity of the phi’s, and sigmap models the individual heterogeneity of the p‘s, and sigmaf models the individual heterogeneity of the f‘s. For sigmaphi = 0, sigmap = 0, and sigmaf = 00, you obtain the same likelihood as the basic Link-Barker 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 Link-Barker data type with random effects is available from the Pradel data type through the PIM | Change Data Type menu choice.