{"id":233,"date":"2017-04-18T03:16:17","date_gmt":"2017-04-18T03:16:17","guid":{"rendered":"http:\/\/sites.warnercnr.colostate.edu\/gwhite\/?page_id=233"},"modified":"2017-05-05T19:27:31","modified_gmt":"2017-05-05T19:27:31","slug":"huggins-closed-captures-models","status":"publish","type":"page","link":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/huggins-closed-captures-models\/","title":{"rendered":"Huggins Closed Captures Models"},"content":{"rendered":"

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

\n

Huggins Closed Captures Models<\/b><\/span><\/p>\n

The closed captures data type consist of 2 versions of 3 different parameterizations of p<\/i> and c<\/i>: the full likelihood version and Huggins version.\u00a0 For the Huggins (1989, 1991)<\/a> version, the population size (N<\/i>) is conditioned out of the likelihood.\u00a0 An example is the best way to illustrate this concept.\u00a0 Consider the 8 possible encounter histories for 3 occasions with the p<\/i>, c<\/i> data type:<\/span><\/p>\n

Encounter History Probability<\/u><\/span>
\n111\u00a0 pcc<\/i><\/span>
\n110\u00a0 pc<\/i>(1 – c<\/i>)<\/span>
\n101\u00a0 p<\/i>(1 – c<\/i>)c<\/i><\/span>
\n011\u00a0 (1 – p<\/i>)pc<\/i><\/span>
\n100\u00a0 p<\/i>(1 – c<\/i>)(1 – c<\/i>)\u00a0<\/span>
\n010\u00a0 (1 – p<\/i>)p<\/i>(1 – c<\/i>)<\/span>
\n001\u00a0 (1 – p<\/i>)(1 – p<\/i>)p<\/i><\/span>
\n000\u00a0 (1 – p<\/i>)(1 – p<\/i>)(1 – p<\/i>)<\/span><\/p>\n

For each of the encounter histories except the last, the number of animals with the specific encounter history is known.\u00a0 For the last encounter history, the number of animals is N<\/i> – M<\/i>(t <\/i>+ 1), i.e., the population size minus the number of animals known to have been in the population.\u00a0 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.\u00a0 The result is a new multinomial distribution that still sums to one.<\/span><\/p>\n

The derived parameter<\/a> N<\/i> is then estimated as M<\/i>(t <\/i>+ 1)\/[1 – (1 – p<\/i>)(1 – p<\/i>)(1 – p<\/i>)] for data with no individual covariates.\u00a0 A more complex estimator is required for models that include individual covariates to model the p<\/i> parameters.<\/span><\/p>\n

Confidence intervals for N<\/i> are computed using a lognormal distribution and the number of animals never seen, f<\/i>0 = N<\/i>-hat – M<\/i>(t<\/i>+1), where M<\/i>(t<\/i>+1) is the number of marked animals in the population at time t<\/i> + 1 (i.e., the number of animals marked during the study, and hence known to be in the population).\u00a0 See page 212 of <\/span>Burnham et al. (1987)<\/a><\/span> for the explanation of this lognormal formula.<\/span>
\nConfidence intervals (95%) for N<\/i> are computed with a lognormal distribution with M<\/i>(t<\/i>+1) as a lower bound.<\/span>
\nLower = f<\/i>0\/C<\/i> + M<\/i>(t<\/i>+1)<\/span>
\nUpper = f<\/i>0*C<\/i> +\u00a0 M<\/i>(t<\/i>+1)<\/span>
\nC<\/i> = exp(1.96 sqrt(log(1 + CV(f<\/i>0-hat))^2)), where CV(f<\/i>0-hat) = SE(f<\/i>0-hat)\/f<\/i>0-hat<\/span><\/p>\n

Huggins Closed Captures Data Type with Pledger Mixtures<\/b><\/span><\/p>\n

Huggins models with <\/span>mixture distributions<\/a><\/span> are available, both the simple Mh model with just pi and p<\/i>, and the more complex time-varying models Mth and Mtbh with pi, p<\/i>, and c<\/i>.<\/span><\/p>\n

Huggins Closed Captures 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 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.\u00a0 With this structure, Gaussian-Hermite quadrature can be used to integrate out the random effects and approximate the capture-recapture model likelihood.\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 Huggins closed captures with random effects data type, an additional parameter is used: sigmap models the individual heterogeneity of the p<\/i>‘s.\u00a0 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 <\/span>AIC<\/a><\/span> can be used to compare models.<\/span><\/p>\n

The random effect is integrated for the probability of observing an encounter history, and for p*.\u00a0 Then 1 – p* is used to correct the encounter history probability for the conditioning out of the all zero history.<\/span><\/p>\n

A similar model is used to model <\/span>heterogeneity in the open models<\/a><\/span>.<\/span>
\nRecognize that more than 2 occasions are necessary to detect individual heterogeneity.\u00a0 What these models are doing is modeling the extra-binomial variation in the data (overdispersion).\u00a0 These models are useful for determining whether the data are overdispersed, i.e., whether parameter heterogeneity exists.<\/span><\/p>\n

Median chat Procedure<\/b><\/span>
\nThe <\/span>median chat procedure<\/a><\/span> can be used with Huggins closed captures data.\u00a0 This is because the Huggins model conditions on the number of unique animals captured, M<\/i>(t<\/i> + 1).\u00a0 So to generate data with overdispersion, the median chat procedure generates encounter histories for exactly M<\/i>(t<\/i> + 1) animals.\u00a0 However, the median chat procedure cannot be used with the robust design data types with Huggins closed captures because this conditioning on M<\/i>(t<\/i> + 1) for the closed captures portion of the likelihood means that the survival portion of the likelihood is no longer valid.<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

Contents – Index 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.\u00a0 For the Huggins (1989, 1991) version, the population size …<\/p>\n

Huggins Closed Captures 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-233","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/pages\/233","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=233"}],"version-history":[{"count":4,"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/pages\/233\/revisions"}],"predecessor-version":[{"id":349,"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/pages\/233\/revisions\/349"}],"wp:attachment":[{"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/media?parent=233"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}