{"id":243,"date":"2017-04-18T03:22:20","date_gmt":"2017-04-18T03:22:20","guid":{"rendered":"http:\/\/sites.warnercnr.colostate.edu\/gwhite\/?page_id=243"},"modified":"2017-04-18T03:22:20","modified_gmt":"2017-04-18T03:22:20","slug":"multi-state-model-live-recaptures","status":"publish","type":"page","link":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/multi-state-model-live-recaptures\/","title":{"rendered":"Multi-State Model for Live Recaptures"},"content":{"rendered":"

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

\n

Multi-State Model for Live Recaptures<\/b><\/span><\/p>\n

The multi-state model (also known as multi-strata model) of Brownie et al. (1993)<\/a> and Hestbeck et al. (1991)<\/a> allows animals to move between states with transition probabilities.\u00a0 These models are an extention of the Cormack-Jolly-Seber model (CJS)\u00a0 (Cormack 1964, Jolly 1965, Seber 1965)<\/a> live recapture model extended to multiple areas or states.\u00a0 A multi-state model with live and dead encounters<\/a> can also be fitted in MARK, as well as the robust-design multistate model<\/a>.\u00a0 In addition, the set of psi parameters (see below) can be redefined as to which transitions are estimated with the Change PIM Definitions<\/a> menu choice.<\/span><\/p>\n

Model Structure<\/b><\/span><\/p>\n

Define the probability of survival for an animal during interval i<\/i> on state j<\/i> as S<\/i>(ij<\/i>), and the probability that the animal is captured during encounter occasion i<\/i> on state j<\/i> as p<\/i>(ij<\/i>).\u00a0 In addition, the probability that an animal moves from state j<\/i> to state k<\/i> during interval i<\/i> is psi<\/i>(ijk<\/i>).\u00a0 If the<\/span>
\npsi<\/i>(ijk<\/i>) have the constraint that<\/span><\/p>\n

sum from k<\/i>=1 to K <\/i>of psi<\/i> (ijk<\/i>) = 1,<\/span><\/p>\n

estimation is possible.\u00a0 The effect of this constraint is that animals that move off all the states in the study, i.e., move outside the study area, cause the estimates of survival to be biased in the sense that “apparent survival” is estimated.\u00a0 That is, emigration off all the states in the study results in “apparent survival” being survival times the probability that the animal remains on the study area.\u00a0 Note that a special link function<\/a>, the multinomial logit link function, has been included as an option if parameter-specific link functions are used to enforce the above constraint.<\/span><\/p>\n

A simple example will make this model clearer.\u00a0 Assume that 3 strata are sampled: A, B, and C (these are the <\/span>state labels<\/a><\/span>.\u00a0 Encounter histories must include the information of which state an animal was captured in.\u00a0 Thus, instead of using a “1” to indicate capture, we use the <\/span>state labels<\/a><\/span>.\u00a0 For 5 encounter occasions, a history such as<\/span><\/p>\n

BCACC<\/span><\/p>\n

could result.\u00a0 That is, the animal was initially captured in state B, captured in state C during the second occasion, captured in state A on the third occasion, captured in state C on the fourth occasion, and then in state A on the fifth occasion.\u00a0 The cell probability describing this encounter history is<\/span><\/p>\n

[S<\/i>(1B) psi<\/i>(1BC) p<\/i>(2C)] [S<\/i>(2C) psi<\/i>(2CA) p<\/i>(3A)] [S<\/i>(3A) psi<\/i>(3AC) p<\/i>(4C)] [S<\/i>(4C) (1 – psi<\/i>(4CA) – psi<\/i>(4CB))\u00a0 p<\/i>(5A)],<\/span><\/p>\n

where encounter occasions are separated within brackets.\u00a0 Note that for the fourth interval, the probability of remaining in state C is just 1 minus the sum of the probabilities of leaving state C.\u00a0 This cell probability demonstrates a key assumption of this model: survival is modeled with the survival rate for the state where the animal was captured, and then transition to a new state takes place.\u00a0 That is, all mortality takes place before movement.\u00a0 An animal cannot transition to a new state where a different survival rate pertains, and then die.\u00a0 If it dies, it must do so in the current state.\u00a0 If it lives, then it can transition to a new state.\u00a0 This assumption is critical if survival rates are different between the states.\u00a0 If survival is constant across the states, then the assumption is not important.\u00a0 Biologically, this assumption is difficult to accept, and limits the usefulness of the model.<\/span><\/p>\n

The nastiness that results when an animal is not captured (i.e., a 0 is in the encounter history) is difficult to demonstrate without matrix algebra.\u00a0 However, to give you a feeling for what happens, consider the encounter history\u00a0\u00a0<\/span><\/p>\n

BC0CA .<\/span><\/p>\n

The 0 on occasion 3 can be explained by 3 possibilities: the animal remained in C and was not captured:<\/span><\/p>\n

S<\/i>(2C) [1 – psi<\/i>(2CA) – psi<\/i>(2CB)] [1 – p<\/i>(3C)] S<\/i>(3C) [1 – psi<\/i>(2CA) – psi<\/i>(2CB)] p<\/i>(4C)\u00a0 ,<\/span><\/p>\n

or the animal transitioned to state A and was not captured, and then moved back to C:<\/span><\/p>\n

S<\/i>(2C) psi<\/i>(2CA) [1 – p<\/i>(3A)] S<\/i>(3A) psi<\/i>(2AC) p<\/i>(4C) ,<\/span><\/p>\n

or the animal transitioned to state B and was not captured, and then moved back to C:<\/span><\/p>\n

S<\/i>(2C) psi<\/i>(2CB) [1 – p<\/i>(3B)] S<\/i>(3B) psi<\/i>(2BC) p<\/i>(4C)\u00a0 .<\/span><\/p>\n

For both of the cases where the animal moved, it has to return to state C, because it was captured in state C during the fourth occasion.<\/span><\/p>\n

The number of parameters in this simple example is already large.\u00a0 There are 3 state-specific survival rates for each interval, and 3 state-specific capture probabilities for the last 4 occasions.\u00a0 In addition, each interval i<\/i> has the transition probabilities psi<\/i>(i<\/i>AB), psi<\/i>(i<\/i>AC), psi<\/i>(i<\/i>BC), psi<\/i>(i<\/i>BA), psi<\/i>(i<\/i>CA), and psi<\/i>(i<\/i>CB).\u00a0\u00a0 Thus, a total of 12 PIMs<\/a> are created in Program MARK to estimate these parameters.\u00a0 Note that these PIMs can be redefined with the Change PIM Definition<\/a> menu choice to select a different psi as the value obtained by subtraction.<\/span><\/p>\n

A trick to remember with the multi-state models is that the sum of the psi values from a state must be <= 1.\u00a0 The multinomial logit link, mlogit<\/a>, is available to help enforce this constraint.<\/span><\/p>\n

Another difficulty with the multi-state models is that there may be multiple maximums in the likelihood funciton.\u00a0 To be more sure that the solution you have found for the estimates is the global maximum of the likelihood, consider checking the solution with the simulated annealing<\/a> algorithm.<\/span><\/p>\n

At this time, only the movement model without memory is implemented in Program MARK.\u00a0 Brownie et al. (1993) describe more complex models where the animal remembers where it was on the previous occasion.\u00a0 This memory model requires a very large amount of data to provide reasonable estimates because the number of parameters grows quickly, even more so than the model considered above.<\/span><\/p>\n

The multi-state data type has also been extended to incorporate the robust design<\/a>, including all 12 of the possible closed captures data types<\/a>, plus dead recoveries can be included in the multi-state data type with live and dead encounters<\/a>.\u00a0 With the closed captures robust design extention of the multi-state models, you can obtain estimates of the population size of each state.<\/span><\/p>\n

More details on the theory of the multi-state model are available at the WWW site <\/span>http:\/\/warnercnr.colostate.edu\/~gwhite\/fw663\/Mark.html<\/a><\/span>.<\/span><\/p>\n

Each state occurring in the data must be identified with a single character <\/span>state labels<\/a><\/span> that identifies the state in the encounter histories.\u00a0 In addition, each state can have a more detailed descriptor (the <\/span>state name<\/a><\/span>) that identifies the state in the output.<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

Contents – Index Multi-State Model for Live Recaptures The multi-state model (also known as multi-strata model) of Brownie et al. (1993) and Hestbeck et al. (1991) allows animals to move between states with transition probabilities.\u00a0 These models are an extention …<\/p>\n

Multi-State Model for Live Recaptures<\/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-243","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/pages\/243","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=243"}],"version-history":[{"count":1,"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/pages\/243\/revisions"}],"predecessor-version":[{"id":244,"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/pages\/243\/revisions\/244"}],"wp:attachment":[{"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/media?parent=243"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}