{"id":288,"date":"2017-04-18T03:54:08","date_gmt":"2017-04-18T03:54:08","guid":{"rendered":"http:\/\/sites.warnercnr.colostate.edu\/gwhite\/?page_id=288"},"modified":"2017-04-18T03:54:08","modified_gmt":"2017-04-18T03:54:08","slug":"recruitment-parameters-jolly-seber-models","status":"publish","type":"page","link":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/recruitment-parameters-jolly-seber-models\/","title":{"rendered":"Recruitment Parameters in Jolly-Seber Models"},"content":{"rendered":"

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

\n

Recruitment Parameters in Jolly-Seber Models<\/b><\/span><\/p>\n

The Jolly-Seber models estimate the recruitment rate into the population, as well at the apparent survival rate, phi, and the probability of capture given that the animal is available for capture (p<\/i>).\u00a0 Because the parametrization of the Jolly-Seber model is not unique, multiple parametrizations have been developed and are provided in MARK.\u00a0\u00a0<\/span><\/p>\n

Pradel Models<\/b><\/span><\/p>\n

Pradel’s (1996) original paper defined the parameter gamma(i<\/i>) as the probability that an animal at time i<\/i> had not entered the population between time i<\/i> and i<\/i>-1.\u00a0 In terms of Jolly’s original model, gamma(i+1<\/i>) = 1 – B<\/i>(i<\/i>)\/N<\/i>(i<\/i>+1).<\/span><\/p>\n

Lambda is the rate of change of the population, so lambda(i<\/i>) = N<\/i>(i<\/i>+1)\/N<\/i>(i<\/i>).\u00a0 The default link function for lambda is the <\/span>log link<\/a><\/span>.\u00a0 That is, unless a parameter-specific link is used, lambda will be computed with the log link.<\/span><\/p>\n

f<\/i> is the fecundity rate of the population, so that f<\/i>(i<\/i>) is the number of new animals in the population at time i<\/i> per animal in the population at time i<\/i>-1, or N<\/i>(i<\/i>+1) = N<\/i>(i<\/i>) f<\/i>(i<\/i>) + N<\/i>(i<\/i>) phi(i<\/i>) .\u00a0 The default link function for f<\/i> is the <\/span>log link<\/a><\/span>.\u00a0 That is, unless a parameter-specific link is used, f<\/i> will be computed with the log link.<\/span><\/p>\n

The following table provides the relationships between these 3 parameters.<\/span><\/p>\n

\u00a0 gamma(i<\/i>+1) = phi(i<\/i>)\/[f<\/i>(i<\/i>) + phi(i<\/i>)]<\/span><\/p>\n

\u00a0 gamma(i+1<\/i>) = phi(i<\/i>)\/lambda(i<\/i>)<\/span><\/p>\n

\u00a0 lambda(i<\/i>) = phi(i<\/i>)\/gamma(i<\/i>+1)<\/span><\/p>\n

\u00a0 lambda(i<\/i>) = f<\/i>(i<\/i>) + phi(i<\/i>)<\/span><\/p>\n

\u00a0 f<\/i>(i<\/i>) = lambda(i<\/i>) – phi(i<\/i>)<\/span><\/p>\n

\u00a0 f<\/i>(i<\/i>) = phi(i<\/i>) { [ 1 – gamma(i<\/i>+1) ] \/ gamma(i<\/i>+1) }<\/span><\/p>\n

Because all of the Pradel models are based on the same likelihood (including the <\/span>mixture models<\/a><\/span>), the AIC<\/a> values of these models are comparable.\u00a0 However, none of the other models in the set of models discussed here, except the Link-Barker<\/a> data type,have the same likelihood, so AIC<\/a> values are not comparable.<\/span><\/p>\n

Pradel et al. (2009) proposed using the Pledger mixture model with the Pradel model.\u00a0 The mixture distribution is only applied to the p<\/i> parameter, as described <\/span>here<\/a><\/span>.<\/span><\/p>\n

Individual covariates can be used to model phi and p<\/i> in the Pradel models.\u00a0 However, the biological meaning of modeling lambda as a function of an individual covariate is not clear.\u00a0 Intuitively, it makes more sense to model f<\/i> and gamma as functions of individual covariates, even though these parameters can be combined with phi to provide a derived estimate of lambda.<\/span><\/p>\n

Pradel’s original paper included parameters to handle losses on capture, or animals removed from the population.\u00a0 However, this parametrization is not what is coded in MARK, because normally the cost of all the extra parameters is a nuisance.\u00a0 If your data has many losses on capture, then you should be using the <\/span>Link-Barker<\/a><\/span> parametrization.<\/span><\/p>\n

Burnham’s Jolly-Seber Model<\/b><\/span><\/p>\n

Burnham developed a parametrization of the Jolly-Seber model that provides estimates of the rate of population change (lambda) and the population size on the first trapping occasion (N(1)).\u00a0 However, the likelihood of this model has been consistently difficult to optimize because of the penalty constraints<\/a> required to keep the parameters consistent with each other.\u00a0 Thus, its use is not suggested.<\/span><\/p>\n

Simulated annealing<\/a><\/span> (alternate optimization method) does sometimes produce valid estimates with this data type.<\/span><\/p>\n

POPAN<\/b><\/span><\/p>\n

Schwarz and Arnason (1996) parametrized the Jolly-Seber model in terms of a super population (N), and the probability of entry (pent in MARK, beta in the paper).\u00a0 The POPAN<\/a> data type implements this model.\u00a0 The MLogit link function<\/a> provides a constraint that makes the sum of the pent parameters <= 1, with the probability of occurring in the population on the first occasion as 1 – sum(pent(t)).<\/span><\/p>\n

The POPAN data type is the only open model in MARK that will generally produce a population estimate for the super population or the population size at each occasion.\u00a0 The Burnham model will sometimes produce estimates of population sizes.<\/span><\/p>\n

Link-Barker<\/b><\/span><\/p>\n

Link and Barker (2003) reparametrized the POPAN model from the probability of entry (pent) to the recruitment parameter (f<\/i>), and is available as the Link-Barker<\/a> data type.\u00a0 The reason for this reparametrization was to provide a more biologically meaningful interpretation of the parameters of the model, as part of a hierarchical modeling approach.\u00a0 Both the Pradel recruitment and the Link-Barker model have the f<\/i> parameter.\u00a0 However, these parameters are not exactly equivalent, although they generate identical estimates if there are not losses on capture, i.e., all animals captured are also released.\u00a0 One advantage of the Link-Barker model is that f<\/i> is not affected by losses on capture, whereas the Pradel parametrization is affected by losses on capture.\u00a0 Thus, if you have losses on capture, then you should be using the Link-Barker model in MARK rather than the Pradel f<\/i> model.<\/span>
\nA version of the <\/span>Link-Barker data type with individual random effects<\/a><\/span> has also been implemented.<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

Contents – Index Recruitment Parameters in Jolly-Seber Models The Jolly-Seber models estimate the recruitment rate into the population, as well at the apparent survival rate, phi, and the probability of capture given that the animal is available for capture (p).\u00a0 …<\/p>\n

Recruitment Parameters in Jolly-Seber 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-288","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/pages\/288","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=288"}],"version-history":[{"count":1,"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/pages\/288\/revisions"}],"predecessor-version":[{"id":289,"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/pages\/288\/revisions\/289"}],"wp:attachment":[{"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/media?parent=288"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}