{"id":241,"date":"2017-04-18T03:20:46","date_gmt":"2017-04-18T03:20:46","guid":{"rendered":"http:\/\/sites.warnercnr.colostate.edu\/gwhite\/?page_id=241"},"modified":"2017-04-18T03:20:46","modified_gmt":"2017-04-18T03:20:46","slug":"multi-state-models-live-dead-encounters","status":"publish","type":"page","link":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/multi-state-models-live-dead-encounters\/","title":{"rendered":"Multi-State Models with Live and Dead Encounters"},"content":{"rendered":"

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

\n

Multi-State Models with Live and Dead Encounters<\/b><\/span><\/p>\n

The multi-state model with live and dead encounters (Barker et al. 2005)<\/a> is a generalization of the multi-state<\/a> model that allows inclusion of recoveries of marks from dead animals.\u00a0 Format of the encounter history data is LDLD…, where the state is identified in the L portion of the encounter history, and only the value ‘1’ is used in the D portion to sigfify the death of the animal.\u00a0 See Data type<\/a> for more details on coding multi-state encounter histories.<\/span><\/p>\n

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

In addition to the usual assumptions of the multi-state model, this model assumes that apart from group and time effects, the reporting rate of marks from dead animals depends only on the state that the animal was in at the immediately preceeding live-capture occasion.\u00a0 In some applications, it may be reasonable to also assume that the state of the animal at the time of the dead recovery can be used to determine the state of the animal at the previous live-recapture occasion.\u00a0 This assumption is not included in the model so any such information is ignored.<\/span><\/p>\n

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

If there are S states (identified by state labels<\/a>), define:<\/span>
\nf<\/b><\/span>h <\/span>is an SxS matrix with s,t’th element = Pr(animal alive at time h in state s is alive at time h+1 in state t),<\/span>
\ny<\/b><\/span>h<\/span> is an SxS matrix of transition probabilities with s,t’th element = Pr(animal moves from s to t | alive at h and h+1),<\/span>
\nP<\/b><\/span>h<\/span> is an (Sx1) matrix with s’th element = Pr(animal alive at time h in state s is captured),<\/span>
\nS<\/b><\/span>j <\/span>is an (Sx1) vector with s’th element = Pr(animal alive at time j in state s is alive at time j+1),<\/span>\u00a0<\/span>
\nr <\/b><\/span>j<\/span> is an (Sx1) vector with s’th element = Pr(animal in state s that dies between j and j+1 is found and reported),<\/span>
\nD(<\/span>x<\/span>) <\/span>= a diagonal matrix with vector x along the diagonal,<\/span>
\n1<\/b><\/span> = a (sx1) vector of ones,<\/span>
\nY<\/span>h<\/span> is an indicator variable that = 1 if the animal was caught at time h and 0 otherwise.<\/span><\/p>\n

Note that<\/span>f<\/b><\/span>h<\/span> =\u00a0<\/span> D(S<\/b><\/span>h<\/span>)<\/span>y<\/b><\/span>h.<\/span><\/p>\n

The animals in the study can be categorized according to whether their last encounter was as a live recapture or as a dead recovery<\/span>
\n.\u00a0\u00a0<\/span>
\nAnimals last encountered by dead recovery<\/i><\/span>
\nFor an animal first released in state s at time i, that was found dead\u00a0 between samples j and j+1, and was last captured alive at in state t at time k the likelihood, conditional on the first release, is factored into two parts:<\/span><\/p>\n

(1) Pr(encounter history between i and (including) k | first released at time i in state s) is the s,t’th element of the matrix<\/span>
\nformed by taking the product from h=i to h=k-1:\u00a0\u00a0<\/span>
\n\u00a0\u00a0\u00a0\u00a0 <\/span>P<\/span> Y<\/span>h<\/span>f<\/b><\/span>h<\/span>D(P<\/b><\/span>h+1<\/span>) +<\/span>(1-Y<\/span>h<\/span>)<\/span>f<\/b><\/span>h<\/span>D(1<\/b>–P<\/b><\/span>h+1<\/span>).\u00a0\u00a0<\/span>
\nWe take the s,t’th element because we know that the animal was in state s at time i and in state t at time k.<\/span>
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n(2) Pr(not caught between k and (including) j and found dead between j and j+1 | released at time k in state t) is the sum<\/span>
\nacross the t’th row of the matrix formed by taking the product from h=k to h=j-1:\u00a0\u00a0<\/span>
\n\u00a0\u00a0\u00a0\u00a0 <\/span>{P<\/span> f<\/b><\/span>h<\/span>D(1<\/b>–P<\/b><\/span>h+1<\/span>)<\/span>}<\/span>D(1<\/b>–S<\/b><\/span>j<\/span>)D(r <\/b><\/span>j<\/span>).\u00a0\u00a0<\/span>
\nAlthough we know that the animal was in state t at time k, we do not know which state the animal was in at time j.<\/span>
\nHowever it must have been in one of the states and therefore we can find the probability we require by taking the sum across<\/span>
\n\u00a0 the t’th row of this matrix.\u00a0\u00a0<\/span><\/p>\n

Animals last encountered by live recapture<\/i><\/span>
\nFor an animal first released in state s and sample i and last encountered by live-recapture in state t and sample j, the likelihood, conditional on the first release, is factored into the two parts:<\/span>
\n(1) Pr(encounter history between i and (including) j | first released at time i in state s) is the s,t’th element of the matrix<\/span>
\n\u00a0\u00a0 f<\/b><\/span>j-1<\/span>D(P<\/b><\/span>j<\/span>)<\/span>{P<\/span> Y<\/span>h<\/span>f<\/b><\/span>h<\/span>D(P<\/b><\/span>h+1<\/span>) +<\/span>(1-Y<\/span>h<\/span>)<\/span>f<\/b><\/span>h<\/span>D(1-P<\/b><\/span>h+1<\/span>)<\/span>}<\/span>
\nwhere the product is taken from h=i to h=j-2.<\/span><\/p>\n

(2) Pr(Not encountered again | released alive at j in state t).\u00a0 This is found by finding the probability that the animal i<\/span>
\nencountered at least once after sample j using the above expressions, and then subtracting this probability from 1.<\/span><\/p>\n

Parameter Identifiability<\/b><\/span><\/p>\n

If the capture occasions are indexed up to sample t and the dead recovery occasions up to sample i, then in addition to the parameters that can be estimated using the multi-state model, we can also estimate <\/span>y<\/b><\/span>t-1, <\/span>P<\/b><\/span>t, <\/span>S<\/b><\/span>t-1 <\/span>and<\/span> r <\/b><\/span>j<\/span> (j=1,…,t).\u00a0 If i > t then (complicated) confounded products of state-specific survival and reporting rates can also be estimated.<\/span><\/p>\n

Parameter Constraints<\/b><\/span>
\nThe sum of the <\/span>y<\/b><\/span> parameters for transition from one state to all the others for a particular time must sum to <= 1.\u00a0 The MLogit link function is a powerful technique to enforce this constraint.\u00a0 See the link function<\/a> help file for more information on using this technique.\u00a0<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

Contents – Index Multi-State Models with Live and Dead Encounters The multi-state model with live and dead encounters (Barker et al. 2005) is a generalization of the multi-state model that allows inclusion of recoveries of marks from dead animals.\u00a0 Format …<\/p>\n

Multi-State Models with Live and Dead Encounters<\/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-241","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/pages\/241","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=241"}],"version-history":[{"count":1,"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/pages\/241\/revisions"}],"predecessor-version":[{"id":242,"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/pages\/241\/revisions\/242"}],"wp:attachment":[{"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/media?parent=241"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}