WILD Estimating Abundance for Closed Populations with Mark-Recapture Methods

Transcription

1 WILD 50 - Esiaig Abudace for Closed Populaios wih Mark-Recapure Mehods Readig: Chaper 4 of WNC book (especially secios 4. & 4.) Esiaig N is uch ore difficul ha you igh iiially expec A variey of ehods ca be used: o Cesus assue cou all aials i he populaio o Saple plos assue cou all aials o plos o Trasec ehods esiae deecio probabiliy as a fucio of disace fro a lie rasec or poi o aials o Capure-recapure esiae capure probabiliy our focus Or, ca abado esiaio ad use a idex ofe doe, seldo esed Esseially coes dow o dealig wih couig aials ad relaig he cou o he uber i he populaio soehow. Basic cocep or odel ha is ofe releva whe esiaig N: Expeced Cou = N x p, where N = pop size & p = prob of deecio Esiaor: N C p Tha is, you go ou ad cou aials ad wa o relae ha cou (C) o N. Thus, ca see ha rigorous esiaio of p is key o good esiaio of N. Cesus assue p = for he eire sudy area A closed populaio has o birhs, deahs, iigraio, or eigraio durig he sudy period. Thus, N (a sae variable) really is a logical focus for a populaio sudy durig such a period! Ad rae variables such as survival are o of ieres wih closure. (We ll see how o work wih boh closed ad ope periods soo). Backgroud: Ball ad ur sudies: Reach i o a ur wih 00 whie balls ad reove a saple ( = 30). Mark he balls i he saple ad replace he. Take aoher rado saple ( = 36) ad cou he uber of arked ( = 0) ad uarked balls (u = 6). We expec he proporio of arked balls o oal balls i he populaio o be he sae as he proporio of arked ad uarked balls observed i our saple. Esiaig Abudace for Closed Populaio Page of 9

2 N OR N N, This is he Licol-Peerse Esiaor for rappig occasios (see pg 90 of WNC) N 08 0 You ca re-arrage his equaio o see ha N C. p Cosider o be he cou saisic ad esiae p = = N N p, or N Probabiliy disribuio for occasios: N! P(,, N, p, p ) ( p p ) ( p q ) ( q p ) ( q q ) N r!( )!( )!( N r)! Where qi = -pi, r = + - = uber of uique aials capured i sudy, & N - r is he uber of aials ever caugh. Closed-for MLEs: p / N p / N Bias-adjused esiaor: ( )( ) N ; ucodiioally ubiased N. var (N ) = ( +)( +)( )( ) ( +) ( +) Esiaig Abudace for Closed Populaio Page of 9

3 Exaple: N = 50, p = 0.6, p = 0.3. Expeced frequecy for each ecouer hisory Ecouer hisory Expeced frequecy Probabiliy 45 Npp Np( p) ( 0.3) 0 30 N( p) p 50 ( 0.6) (o see) 70 N( p)( p) 50 ( 0.6) ( 0.3) =50 =75 =45 p 45 / or... p / N 50 / p 45 / or... p / N 75 / Assupios: ) Populaio is closed o addiios (birh & iigraio) ad o losses (deah & eigraio) ) Marks are o los or overlooked or isread by researchers 3) All aials are equally likely o be capured i each saple (his is he assupio ha we will alos always eed o relax). We ll relax i o be capure probabiliies are appropriaely odeled. For os real biological applicaios, we eed o: ) Cosider possible heerogeeiy i capure probabiliy, ) Carefully cosider he cocep of closure, ad 3) Use ore ha occasios. Paraeers of ieres: ) N populaio size ) D populaio desiy (possible hough o our focus) 3) p, c capure & recapure probabiliies Esiaig Abudace for Closed Populaio Page 3 of 9

4 Saisics: ) j - # capured o he j h occasio, j =,,,. ). oal # of capures i he sudy = j 3) uj - # of ew aials capured o he j h occasio, j =,,,. 4) fj capure frequecies = # idividuals capured exacly j ies i days of rappig. 5) M+ # of differe idividuals caugh i eire sudy. 6) Mj uber arked aials i he populaio a he ie of he j h saple, Noe ha M0 = 0. 7) M. Su of he Mj, o icludig M+. 8) j - # of arked aials capured i he j h saple. 9). su of he j. 0) he uber of rappig occasios i he sudy. ) = he se of possible ecouer hisories i he sudy. { } X The probabiliy disribuio for he se of possible ecouer hisories uder cosa capure probabiliy is: N! P X N p p p X! ( N M )!. N. ({ }, ) ( ) The MLE for cosa capure probabiliy is: N! L N p X p p X! ( N M )!. N. (, ) ( ) To help you see wha s goig o wih he ulioial coefficie, cosider : 3 aials wih EH, aials wih 0 EH, & 3 aials wih 0 EH. Thus, =, = 6, = 5, ad. =. N! N L( N, p X) p ( p) 3!! 3!( N 8)! Subsiuig values of N ad p io he equaio allows esiaio of he os likely esiaes for hese paraeers give he daa se ad he odel. Of course, his odel is siple ad akes he resricive assupios ha p = c ad is cosa across occasios & aials. As we ll discuss i class, uch of he work ha has bee doe for his class of odels has focused o easig his assupio ad odelig variaio i p. j Esiaig Abudace for Closed Populaio Page 4 of 9

5 Basically, 3 broad sources of variaio are cosidered possible: ) eporal variaio i capure probabiliy ( ), ) behavioral resposes o rappig ( b ): p c, ad 3) heerogeeiy i capure probabiliies for differe idividuals ( h ). By cosiderig all possible cobiaios of hese 3 facors, 8 odels are obaied: M(0), M(), M(b), M(h), M(b), M(h), M(bh), ad M(bh). MARK also has a variey of oher odels ha use a codiioal likelihood approach i which N is o acually cosidered as a paraeer. This is doe by codiioig o oly aials ecouered (r). This ca be hady because i les us brig idividual covariaes o bear. NOTE: Idividual covariaes are ypically o used i closed odels hough soe approaches do allow heir use (see pages i WNC book). This is because, we are ow esiaig capure probabiliy for all he aials i he populaio bu do have covariae values for he aials ha were ever capured. We ca use group-level covariaes, which ca be useful. Ad, if ie peris we will discuss odels ha allow use of idividual covariaes. The figure below is fro chaper 4 of CW, which was wrie by Paul Lukacs, ad shows he various odels ha exis. We ll focus o he full likelihood odels for p ad c. We ll discuss he ideas of ixures ad is-ideificaio if ie allows laer i he seeser. Esiaig Abudace for Closed Populaio Page 5 of 9

6 Likelihoods for soe of he basic odels: To give you a beer feel for how he paraeers are esiaed i he various odels, here are he likelihood equaios for M(), M(b), ad M(b). M() (+ paraeers): N! j L( N, p j X) p j ( p j) j X! ( N M )! N j If =, he MLE for N for M() is: N, i.e., he Licol-Peerse esiaor M(b) (3 paraeers): As oed o page 99 of your exbook, i he behavioral respose odel, capure probabiliy ca vary as a resul of previous capure such ha p c, ad he respose ca be rap happiess (c > p) or rap shyess (p > c). L( N, p, c X) N! p ( p) c ( c) X! ( N M )! M NM M.. M.. A oeworhy propery of his odel is ha he recapure iforaio is used oly i he esiaio of c (a uisace paraeer). Tha is, recapures do provide iforaio wih respec o esiaio of p or N. Thus, oce M(b) has bee ideified as appropriae, he esiaio is doe as if he sudy were a reoval sudy. Thus, i is criical i his ype of sudy ha depleio of uarked aials is achieved, i.e., he uber of ew aials capured o each occasio should decrease wih each successive occasio. If his does o occur sufficiely, he odel ca fail o produce resuls. Valid esiaes are obaied if: j ( j) ( ) 0 Failure ca readily occur if you have eporal chages i p such ha few idividuals are caugh early i he sudy ad ay ew idividuals are caugh lae i he sudy. Idividual heerogeeiy i p causes uderesiaio of N wih M(b): he agiude of bias depeds upo he uber of aials ha are esseially ucachable. j j Esiaig Abudace for Closed Populaio Page 6 of 9

7 M(h) (N+ paraeers): As esiaio is o possible uder he geeral forulaio where every aial has is ow p (p, p,, pn = {pi}), i is useful o hik of {pi} as a rado saple (size of saple is N) coig fro soe probabiliy disribuio F(p), where each pi falls bewee 0 &. A axiu likelihood esiaor is possible if he paricular faily of disribuios of which F(p) is a eber is specified. Oe possibiliy is he Bea disribuio (he bea disribuio is specified by paraeers (alpha ad bea) ad hece, he ae. The bea disribuio is a coiuous disribuio wih all of is o-zero probabiliy bewee 0 ad..803 dbea( x 3 7) 3 Bea disribuio wih: alpha=3, bea= x. The This bea is a disribuio appealig idea is a ha coiuous is useful disribuio for cocepualizig wih all how of is he p s igh be ozero disribued probabiliy i he populaio, bewee bu 0 ad has. o proved useful i pracice. A variey of aleraives exis (See figure above fro chaper 4 of CW by P. Lukacs) icludig ixure odels, as well as several approaches available i Progra CAPTURE, which ca be called fro wihi Progra MARK. Noe: Models coaiig heerogeeiy are probably appropriae for ay sudies. Thus, i is worh udersadig he cocep ad kowig soe of he ways you ca obai esiaes for M(h) odels. M(bh) (N + paraeers- each aial has is ow p ad a c): The esiaor of N for his odel is based o he firs-capure daa (siilar o wha s doe wih M(b), i.e., a reoval esiaor). Bu, wih heerogeeiy ore us be doe; we eed a geeralized reoval esiaor. Wih heerogeeiy, you expec he average probabiliy of firs capure o decrease wih each occasio & you expec he os rapid decrease over he firs few occasios. O he firs occasio, o aial has bee caugh before so he average p for j= is high relaive o p6. Why? Because by occasio 6 os of he aials ha are easy o cach have already bee caugh! So, he rick o esiaio is o see if you ca fi he daa usig oly a few average p s. The idea is o fi a sequece of icreasigly geeral odels o he average value for he values of pj ad o use he siples odel ha fis. Sep : Evaluae he fi of usig a sigle average p for all ocassios p p p 3... p, his is he sae as M(b) Esiaig Abudace for Closed Populaio Page 7 of 9

8 Sep : If he odel o he previous sep does fi, geeralize & evaluae agai. Sep 3: Geeralize ore as eeded: Ec. Ec. p p p 3... p p p p 3... p Now ha you re o ryig o fi a p for every aial i he populaio, you CAN build hese odels i MARK. This is very useful because for odels i MARK we ca copare odels wih our ypical AIC procedures. I lab his week, you ll see how o build each of he odels described i he seps above. Okay, we wo go over he oher odels i ay ore deail. Bu, I hope his iroducio o he odels alog wih readigs has give you a dece idea of he key coceps of esiaio for hese odels. Also, realize ha here is a solid body of lieraure available o his opic. Fially, if you are workig wih his odel ype i your research, you will eed o becoe ore failiar wih soe of he ore coplex odels ad he aribues of heir esiaes. I lab, you will becoe failiar wih how o: () produce esiaes for all odels excep M(bh) ad () coduc siulaios ha evaluae esiaor perforace uder differe saplig codiios. As I hik you ca surise, i is bes if you ca be jusified i usig sipler odels. Thus, you eed o cosider ways of havig as few facors as possible affec he capure probabiliies i your sudies. I ers of heerogeeiy, wih he odels we are reviewig,you are liied i your use of idividual covariaes. Tha is, you ca oly use covariaes ha ca be eered usig he desig arix, e.g., covariaes ha apply o a group of aials, e.g., rappig effor o each occasio, sex, size class, ec. Breakig he daa up by group covariaes, e.g., sex, size class, ec., is a ehod ha ca be used o reduce heerogeeiy wihi each group. Of course, i also eas ha you will eed o have adequae daa for esiaio for each group! Ad, you ca work o lear abou oher approaches such as Huggis Models lised uder Codiioal Likelihood o he righ side of he cocepual diagra of Closed Models i Chaper 4 of CW. Over-Paraeerized Closed Capures Models As icely explaied i secio 4.3. of he CW Chaper by P. Lukacs, whe a odel is specified i Progra MARK wih uideifiable p esiaes, he esiaes of N are jus M +. Cocepually, usig a odel wih ie-varyig capure probabiliies, here s why. The esiae of populaio size is basically, M N ( p ) ( p )...( p ) Esiaig Abudace for Closed Populaio Page 8 of 9

9 where he ueraor is he uber of idividuals capured ad he deoiaor is he probabiliy ha a aial was capured durig he sudy ( ius all he probabiliies of o beig capured!). Tha is, he probabiliy of o beig iiially capured o he firs occasio is -p, o beig capured o he secod occasio is -p, ad so o o -p. The produc of hese ers is he probabiliy of ever beig capured durig he sudy. Thus, ius his produc is he probabiliy of beig capured a leas oce durig he sudy. Therefore, he deoiaor is a correcio o iflae he ueraor. Whe a closed-capures odel is over-paraeerized, he las er i he deoiaor becoes zero because p is esiaed as (for reasos I wo go io). This akes he produc porio of he deoiaor = 0 ad he eire deoiaor =. This resuls i N = M+. Whe you use closed-capures odelig, you should always kow wha he value of M+ is ad be sure ha your resuls for N have o collapsed o he value of M+ because of over-paraeerizaio. The value of M+ is i he full oupu for each odel ad so very easy o check. Noe: Progra MARK acually paraeerizes he likelihood i ers of he uber of idividuals ever caugh, f0, such ha f 0 = N M +, where he oaio f0 is a referece o he frequecy, or cou, of aials observed 0 ies. Thus, MARK provides esiaes of f0 i he real paraeers lis, ad provides esiaes of N uder he derived paraeers lis. MARK uses a log lik for f0, which keeps values of f0 bewee 0 ad ifiiy. N is esiaed as M+ + f 0, ad so he variace for N is he variace for f 0 as M+ is kow. A addiioal proble wih closed capure odels i Progra MARK is ha ofe he uber of paraeers is o correcly copued because values of close o M+ are assued o o be esiaed causig a error i he algorih o deerie he uber of paraeers acually esiaed, which causes errors i he resulig AIC value. So, as always pay aeio o paraeer cous ad adjus cous as eeded. N Esiaig Abudace for Closed Populaio Page 9 of 9