{"id":344,"date":"2017-05-05T19:24:04","date_gmt":"2017-05-05T19:24:04","guid":{"rendered":"http:\/\/sites.warnercnr.colostate.edu\/gwhite\/?page_id=344"},"modified":"2017-05-05T19:24:04","modified_gmt":"2017-05-05T19:24:04","slug":"heterogeneity-open-models","status":"publish","type":"page","link":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/heterogeneity-open-models\/","title":{"rendered":"Heterogeneity Open Models"},"content":{"rendered":"

Contents<\/a> – Index<\/a><\/span><\/p>\n

\n

Heterogeneity Open Models<\/b><\/span><\/p>\n

Cormack-Jolly-Seber Data Type with Mixtures<\/b><\/span><\/p>\n

Pledger et al. (2003)<\/a><\/span> proposed a <\/span>mixture model<\/a><\/span> to account for heterogeneity in the Cormack-Jolly-Seber (CJS) <\/span>data type<\/a><\/span>.\u00a0 This model incorporates a single mixture parameter (pi) to model heterogeneity in both phi and p<\/i>.\u00a0 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.\u00a0<\/span><\/p>\n

This data type can be accessed by the <\/span>PIM<\/a><\/span> | <\/span>Change Data Type<\/a><\/span> menu choices.\u00a0 So, you can create a normal CJS MARK file, and then compare the usual models with these models that include heterogeneity.<\/span><\/p>\n

The data type has 3 parameters: pi, phi, and p<\/i>.\u00a0 The following example illustrates the model with 5 occasions and 2 mixtures for a single group.\u00a0 The time-specific <\/span>PIMs<\/a><\/span> look like the following.<\/span><\/p>\n

pi PIM<\/span><\/p>\n

1<\/span><\/p>\n

phi PIM<\/span><\/p>\n

2 3 4 5<\/span>
\n\u00a0 3 4 5<\/span>
\n\u00a0\u00a0 4 5<\/span>
\n\u00a0\u00a0\u00a0 5<\/span>
\n6 7 8 9<\/span>
\n\u00a0 7 8 9<\/span>
\n\u00a0\u00a0 8 9<\/span>
\n\u00a0\u00a0\u00a0 9<\/span><\/p>\n

p<\/i> PIM<\/span><\/p>\n

10 11 12 13<\/span>
\n\u00a0 11 12 13<\/span>
\n\u00a0\u00a0 12 13<\/span>
\n\u00a0\u00a0\u00a0 13<\/span>
\n14 15 16 17<\/span>
\n\u00a0 15 16 17<\/span>
\n\u00a0\u00a0 16 17<\/span>
\n\u00a0\u00a0\u00a0 17<\/span><\/p>\n

The structure of the phi and p PIMs is just the upper-triangular array typical of a CJS model duplicated for the 2 mixtures.\u00a0 Normally, as suggested by Pledger et al. (2003), the parameter estimates would be additive across the mixtures.\u00a0 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).\u00a0 The model name might be {pi phi(t+h2) p(t+h2)}, where h2 indicates the set of 2 mixtures.<\/span><\/p>\n

pi phi int phi t1 phi t2 phi t3 phi mix p<\/i> int p<\/i> t1 p<\/i> t2 p<\/i> t3 p<\/i> mix<\/u><\/span>
\n1 0 0 0 0 0 0 0 0 0 0<\/span>
\n0 1 1 0 0 1 0 0 0 0 0<\/span>
\n0 1 0 1 0 1 0 0 0 0 0<\/span>
\n0 1 0 0 1 1 0 0 0 0 0<\/span>
\n0 1 0 0 0 1 0 0 0 0 0<\/span>
\n0 1 1 0 0 0 0 0 0 0 0<\/span>
\n0 1 0 1 0 0 0 0 0 0 0<\/span>
\n0 1 0 0 1 0 0 0 0 0 0<\/span>
\n0 1 0 0 0 0 0 0 0 0 0<\/span>
\n0 0 0 0 0 0 1 1 0 0 1<\/span>
\n0 0 0 0 0 0 1 0 1 0 1<\/span>
\n0 0 0 0 0 0 1 0 0 1 1<\/span>
\n0 0 0 0 0 0 1 0 0 0 1<\/span>
\n0 0 0 0 0 0 1 1 0 0 0<\/span>
\n0 0 0 0 0 0 1 0 1 0 0<\/span>
\n0 0 0 0 0 0 1 0 0 1 0<\/span>
\n0 0 0 0 0 0 1 0 0 0 0<\/span><\/p>\n

If you desire a model where there is no heterogeneity on phi, but still on p<\/i>, then all you would have to do is delete the phi mix column in the above design matrix.\u00a0 The result would be that phi parameters 2-5 would be identical to 6-9.\u00a0 This model might be named {pi phi(t) p(t+h2)}.<\/span><\/p>\n

The Pledger mixture model has a consistent likelihood with the regular CJS model, so you can compare these models with other CJS models using <\/span>AICc<\/a><\/span>.<\/span><\/p>\n

Cormack-Jolly-Seber Data Type with Random Effects<\/b><\/span><\/p>\n

Gimenez and Choquet (2010)<\/a><\/span> proposed an extension of the CJS data type where individual random effects are modeled.\u00a0 Each animal is assumed to have its own random offset from the population mean.\u00a0 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.\u00a0 With this structure, Gaussian-Hermite quadrature can be used to integrate out the random effects and approximate the capture-recapture model likeliood.\u00a0 This same approach is used in the <\/span>mark-resight data types<\/a><\/span> (<\/span>McClintock and White 2009<\/a><\/span>, <\/span>McClintock et al. 2009a<\/a><\/span>) with individual random effects. The number of nodes can be set in the <\/span>File | Preferences<\/a><\/span> window.<\/span><\/p>\n

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<\/i>‘s.\u00a0 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 <\/span>AIC<\/a><\/span> can be used to compare models.<\/span><\/p>\n

The CJS data type with random effects is available from the CJS data type through the PIM | <\/span>Change Data Type<\/a><\/span> menu choice.<\/span><\/p>\n

Pradel Data Type with Mixtures<\/b><\/span><\/p>\n

Pradel et al. (2009)<\/a><\/span> proposed a similar <\/span>mixture model<\/a><\/span> for the <\/span>Pradel data type<\/a><\/span>.\u00a0 However, the mixtures only apply to p<\/i>, and not to phi, gamma, lambda, or f<\/i>.\u00a0 All three parametrization (i.e., seniority with gamma, population change with lambda, and recruitment with f<\/i>) are implemented.\u00a0 The likelihood for the mixture models is consistent with the usual Pradel models, so <\/span>AICc<\/a><\/span> can be used to compare the mixture models to the regular models.\u00a0 You use the PIM | <\/span>Change Data Type<\/a><\/span> menu choices to change the <\/span>data type<\/a><\/span> to these <\/span>mixture models<\/a><\/span>.<\/span><\/p>\n

The <\/span>Pradel data type<\/a><\/span> does not use the upper-triangular PIM structure, because inferences are being made to animals not yet captured.\u00a0 Thus,, the PIMs are much simpler than the CJS Pledger mixture data type described above.<\/span><\/p>\n

Link-Barker Data Type with Mixtures<\/b><\/span><\/p>\n

The equivalent <\/span>mixture model<\/a><\/span> on p was also incorporated into the Link-Barker data type.\u00a0 This data type is equivalent to the Pradel recruitment parametrization, except that the Link-Barker data type correctly handles losses on capture.\u00a0 Thus, you normally get identical -2log likelihood values for the Pradel f<\/i> and Link-Barker models, except when there are losses on capture.<\/span><\/p>\n

Link-Barker Data Type with Random Effects<\/b><\/span><\/p>\n

Gimenez and Choquet (2010)<\/a><\/span> proposed an extension of the CJS data type where individual random effects are modeled.\u00a0 Each animal is assumed to have its own random offset from the population mean.\u00a0 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.\u00a0 With this structure, Gaussian-Hermite quadrature can be used to integrate out the random effects and approximate the capture-recapture model likeliood.\u00a0 This same approach is used in the <\/span>mark-resight data types<\/a><\/span> (<\/span>McClintock and White 2009<\/a><\/span>, <\/span>McClintock et al. 2009a<\/a><\/span>) with individual random effects.\u00a0 I have implemented this same approach with the <\/span>Link-Barker<\/a><\/span> data type. The number of nodes can be set in the <\/span>File | Preferences<\/a><\/span> window.<\/span><\/p>\n

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<\/i>‘s, and sigmaf models the individual heterogeneity of the f<\/i>‘s.\u00a0 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 <\/span>AIC<\/a><\/span> can be used to compare models.<\/span><\/p>\n

The Link-Barker data type with random effects is available from the <\/span>Pradel<\/a><\/span> data type through the PIM | <\/span>Change Data Type<\/a><\/span> menu choice.<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

Contents – Index 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.\u00a0 This model incorporates a single mixture parameter (pi) to model heterogeneity …<\/p>\n

Heterogeneity Open Models<\/span> Read More »<\/a><\/p>\n","protected":false},"author":117,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-344","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/pages\/344","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/users\/117"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/comments?post=344"}],"version-history":[{"count":1,"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/pages\/344\/revisions"}],"predecessor-version":[{"id":345,"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/pages\/344\/revisions\/345"}],"wp:attachment":[{"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/media?parent=344"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}