Lecture 06 Multinomials and CMR Models for closed populations

Transcription

1 WILD Analyss of Wldlfe Poulatons of 4 Lecture 06 Multnomals and CMR Models for closed oulatons Readngs Wllams, B, JD Nchols, and MJ Conroy 00 Analyss and Management of Anmal Poulatons Academc Press San Dego, Calforna Other resources Ots, D L, P Burnham, G C Whte, and D R Anderson 978 Statstcal nference from cature data on closed anmal oulatons Wldlfe Monograh 6 35 Pollock, H, J D Nchols, C Browne, and J E Hnes 990 Statstcal nference for caturerecature exerments Wldlfe Monograhs Whte, G C, D R Anderson, P Burnham, and D L Ots 98 Cature-recature and removal methods for samlng closed oulatons Los Alamos Natonal Laboratory, Re LA-8787-NERP 35 Chater, Chater 8 Multnomal Dstrbuton and Lkelhoods A fundamental arecaton for the multnomal coeffcent, multnomal dstrbuton, and multnomal lkelhood s of great mortance to understandng and analyzng cature-markrecature data The multnomal coeffcent s an extenson of the bnomal coeffcent wth more than two ossble mutually exclusve outcomes As the bnomal coeffcent was ntroduced by way of con tossng, the multnomal coeffcent s nearly always ntroduced by way of de tossng The multnomal coeffcent or the number of ossble outcomes for de tossng s wrtten: y Deendency among counts y y n y y = y y! y n!! y! y =! y! y! n! y! = One roerty of multnomal data s that there s a deendency among the counts For examle, f a de s thrown and t s not a,, 3, 4, or 5, then t must be a 6 Thus n the table below f we know any 5 of the varables and the total number of trals (tosses) we know the 6 th Ths deendency s a characterstc of the bnomal coeffcent as well If we know number of trals (n) and the number of heads (y), we then know the number of tals (n - y ) Face Number Varable 0 y y 3 3 y y y y 6 TOTAL 60 n

2 WILD Analyss of Wldlfe Poulatons Probablty dstrbuton functon We can also wrte the robablty an outcome or seres of outcomes from the multnomal n the form of a robablty statement: n f ( y n ) y = Note that = Why? For examle, the robablty of rollng a far de ( = /6) sx tmes (n) and turnng u each face only once (n = ) s wrtten: 6! f (,,,,, 6 6, 6, 6, 6, 6, 6) = = 00543!!!!!! of 4 Table Vertcal fle Tag No Date //99 3//993 /6/99 3 3// //99 4 /0/993 Another examle, the robablty of rollng s, 3 3s, and 4 s: 6! f ( 0,, 3,, 0, 0 6 6, 6, 6, 6, 6, 6) = = ( 0! )(! )( 3! )(! )( 0! )( 0! ) Lkelhood As you mght have exected, the lkelhood of the multnomal s of greater nterest to us, snce n ecology we frequently have data (n, y m ) and are seekng to determne the model ( ) The lkelhood for our examle wth the de s: L( n n y ) = y n ln( L( ny) = ln + yln + yln + y3ln y + m n = ln + yln ( ) y = y y Ths lkelhood has all of the same roertes we dscussed for the bnomal case so as you mght have exected, we are nterested n the log-lkelhood: whch s frequently abbrevated as ln( L( ny y3 3 y4 4 y5 5 y6 6 ( ) ( ) ( 3) ymln ( m) m ) ln ln = ( C) + ( y ( ) ) = The multnomal coeffcent lke the bnomal coeffcent can be gnored so what we end u wth s:

3 WILD Analyss of Wldlfe Poulatons 3 of 4 m ( ( ) ) = ln( L( ny) = yln ( data ln( robablty) ) = 4 Cature hstores & multnomals Tycally the rocedure that s followed n CMR studes s to cature a samle of anmals, mark them so that they are unquely dentfable, release them, and record ether recoveres of dead anmals or recatures and observatons (resghtngs) of lve anmals These data m ay be recorded n the form of a vertcal fle that may nclude ancllary nformaton (e, covarates) lke Table These observatons are tycally converted to the form of a cature-hstory matrx where each row reresents an ndvdual and each column reresents a samlng occason On each occason each ndvdual s assgned a f encountered (catured) or a 0 f not encountered Table Cature hstores Occason Ultmately for the closed oulaton estmators dscussed n ths secton these are converted to LLLL data For examle, n a lverelease Tag No 3 study of wth one ntal cature and three samlng 0 0 occasons, the cature hstores of ten ndvduals catured and released on occason mght look lke Table For ndvduals n a oulaton samled on 3 occasons 8 (= m = ) ossble cature hstores The next table demonstrates how these observatons can be summarzed by the number of 5 0 ndvduals (y ) characterzed by each 6 0 ossble cature hstory Table 3 Summarzed hstores 7 These data can be used to estmate a number of useful arameters such as Cature oulaton sze, survval, re-sghtng y Hstory rate deendng uon the data and the assumtons we are wllng to 0 0 make m = N y

4 WILD Analyss of Wldlfe Poulatons 4 of 4 5 How does all of ths relate back to Multnomals and Lkelhoods? Remember the table from the examle of the roll of the de Face Number (y ) TOTAL 60 The lkelhood for such a data set could be constructed as: ( data ln( robablty) ) In CMR studes, each cature hstory s a ossble outcome, analogous to one face of the de (n ) Our data consst of the number of tmes each cature hstory aears (y ) y Cature Hstory Each encounter has an assocated robablty ( ) For examle, the cature hstory [ 0 ] could be nterreted as: Pr{Gven release at tme, not recatured on occason, and catured on occason 3} or Pr{R (- ) (- ) ( 3 )}, Thus, we could construct the lkelhood for our data of m cature hstores of the form: m ( data ln( robablty) ) = 6 Deendency n the multnomal dstrbuton:

5 WILD Analyss of Wldlfe Poulatons 5 of 4 Because of the numercal deendences n the multnomal dstrbuton some values necessary for estmatng arameters can be determned by subtracton How many ndvduals were never recatured? How many were never catured? Probablty of never catured R N m j= m = y j R { } Pr cature hstores Closed Poulaton Recature Models CMR technques have been used to estmate the sze of human oulatons snce the md-600s Frst used n ecology by CGJ Petersen to estmate abundance of a fsh oulaton n 896 Frst used n wldlfe scence by Lncoln to estmate the abundance of ducks usng band-return data n 930 The fundamental rncle s smle and requres only two samlng occasons Frst, a samle of anmals (sze M - marked) catured from and then marked and released back nto the oulaton of nterest (sze N) Then, another samle of ndvduals s catured (sze C - catured) and the number of ndvduals marked n C s counted (R - recatured) If the second samle was random, the roorton of marked ndvduals n the second samle reflects the roorton marked n the entre oulaton Thus, the estmator s based on the roorton: N ˆ C = M R Therefore, the Lncoln-Petersen estmator for N s: CM N ˆ = R However, ths ntutve estmator tends to be based and can overestmate oulaton sze, esecally wth small samles Seber (98) recommended a dfferent form of the estmator, whch s unbased f (M + C) > N and nearly unbased f R > 7: N ˆ = ( C + )( M + ) R +

6 WILD Analyss of Wldlfe Poulatons 6 of 4 Thnk of the classcal ball and urn model of Feller (950) Ths model s qute dfferent from what s often encountered n the real world n that: a Cature robabltes often vary (e, anmals rarely assort randomly after the frst cature) b Closure ether demograhc or geograhc s rarely comlete ) Demograhc closure no brths or deaths ) Geograhc closure no emgraton or mmgraton A large number of researchers and statstcans have contrbuted to the refnement of these estmators Ots et al (978) rovde an extensve treatment of the use of CMR data Hs work dealt manly wth examnng the sources of varaton n cature robabltes Whte et al (98) s another excellent source of nformaton regardng the analyss of CMR data The object of these treatses was the estmaton of oulaton sze, and a fundamental assumton of the aroach was oulaton closure As such survval or recrutment rates were consdered nusance arameters and of lttle nterest Sgnfcant advances of closed models n recent years: a Can be used to examne removal or lve-release data b Wth more than two samlng occasons, the assumton of constant recature robablty over tme can be tested c Develoment of MLEs d Heterogenety of recature robablty can be examned (e, models where cature robablty vares among ndvduals) e Heterogenety models (e, comarsons of grous) f Indvduals covarates va lnk functons g Mxture models used to model undentfed sources of heterogenety 3 Lncoln-Peterson Estmator Deste the weaknesses of ths smle estmator t forms the bass for vrtually all of the CMR methods a ey assumtons: b Poulaton s closed demograhcally and geograhcally c Marks are not mssed or lost d Cature robabltes are equal among ndvduals

7 WILD Analyss of Wldlfe Poulatons 7 of 4 e Probablty dstrbuton (, n, m N,, ) P n = m!( n m ( ) ( ) ( ( )) (( ) ) (( )( )) n m n m N n + n m )!( n N! m )!( N n n )! N oulaton sze m number marked anmals caught n second erod n n number caught and marked n frst erod number caught n second erod robablty of cature n frst erod robablty of cature durng second erod L-P s ostvely (+) based and magntude of bas s nversely related to the samle sze f Assumtons ) Closure trang mortalty further ostvely bases the estmate of N ) Cature robablty a) Can dffer between occasons b) Should be equal among ndvduals c) tra hay negatvely bases ˆN d) tra shy ostvely bases ˆN e) Tags lost or mssed ostvely bases ˆN 4 -samle Cature-Recature Models These models dffer from the Lncoln-Peterson estmator and Seber s estmator n that they nvolve more than two samlng occasons a Samlng scheme/data structure ) Assumtons a) Closure demograhc and geograhc b) Tags are not lost or mssed c) Cature robabltes are arorately modeled (the model s correct) ) Data structure b The data consst of cature hstores dentcal to those used n the oen models for recovery and recature data An encounter hstory s develoed for each unquelymarked ndvdual consstng of s (encounter) or 0s (not encountered) for each otental cature occason (survey, trang event, etc) 5 Modelng aroach These data follow a multnomal dstrbuton:

8 WILD Analyss of Wldlfe Poulatons 8 of 4 P N! ( n N, ) = = n! = where n and are the number of observatons and robablty of each cature hstory Also, lke the oen oulaton models, closed oulatons can be constraned to reduce the number of arameters that are estmated Addtonally, constrants can roduce models that estmate the robablty of cature for unmarked ( c ) and marked anmals ( r ) Thus reducng the total number of arameters estmate from the full n the general model Model Name Eght models ossble n -Samle Recature Models n Sources of cature robablty varaton Estmated arameters M 0 Constant cature robablty N, M t Temoral varaton N, ( = ) M b Behavoral resonse (tra-hay or tra shy) N, c, M h M tb M bh M th M tbh Indvdual heterogenety Temoral and behavoral Behavoral resonse and ndvdual heterogenety Temoral varaton and ndvdual heterogenety Temoral varaton, behavoral resonse and ndvdual heterogenety N, ( = N) N, c, r ( = ) N, c, r ( = N) N, j ( = ) (j = N) N, cj, rj ( = ) (j = N) Only the frst 3 models, M 0, M t, and M b, can be drectly estmated va the lkelhood For the other 5 models, alternatve assumtons and constrants are requred For the model M tb, an MLE can be derved by assumng a relatonsh between the tme-secfc cature and recature robabltes For models M h, M bh, and M th, the estmates can be derved under the assumton that cature robabltes these are random samles of sze N from an underlyng dstrbuton of robabltes, or by usng the concet of coverage However, the fnte mxture models descrbed by Pledger (000) resent a better aroach based on random effects Table 4 from Wllams et al 00 Cature Probablty

9 WILD Analyss of Wldlfe Poulatons 9 of 4 hstory M 0 M t M b 3 3 c r 0 (-) (- 3 ) c r (- r ) 0 (-) (- ) 3 c r (- r ) 00 (-) (- )(- 3 ) c (- r ) 0 (-) (- ) 3 (- c ) c r 00 (-) (- ) (- 3 ) (- c ) c (- r ) 00 (-) (- )(- ) 3 (- c ) c 6 Estmatng oulaton sze 000 (-) 3 (- )(- )(- 3 ) (- c ) 3 a Constant cature robablty Model M 0 Because all of the cature robabltes are constraned to be equal, the robablty dstrbuton reduces to: P N! W xw! ( N M w= n N n ( x N, ) = ( ) w where n s the total number of catures and M + s the total number of unmarked anmals catured + b Temoral varaton n cature robablty Model M t Ths model can be thought of as an extenson of the Lncoln-Peterson ndex wth the beneft of the addtonal nformaton from multle samlng occasons The number of estmated arameters (N, ) s equal to +, one more than the number of samlng occasons The robablty dstrbuton of ths model s: P ( x N, ) w = W w= N! x! )! = ( N M )! w + n ( ) N n Note that when =, t can be shown that ths s the Lncoln-Peterson estmator The oulaton sze can be estmated by maxmzng wth resect to N, or usng:

10 WILD Analyss of Wldlfe Poulatons 0 of 4 M + n = N = N Darroch (958), whch setts the robablty of not beng caught durng the study equal to the roduct of the robablty of not beng caught durng each resectve samlng erod c Behavoral tra resonse Model M b Ths model descrbes a change n cature robablty after the frst encounter wth an anmal Tycal notaton s robablty of cature for unmarked ndvduals - c and marked anmals - r Ths behavoral resonse can be ostve (tra-hay), or negatve (tra-shy) Thus, the model estmates only three arameters N, c, r The robablty functon for ths model s: where P ( x N,, ) w c r = = ( ) m r N! xw! ( ) = ( N M ) N M c m = m j, + + M M m r! M c + the number of recatures durng the study, and m j s the number of marked anmals caught on occason j Also, f M j s the total number of marked anmals n the oulaton at j then M j = M =, the total number of anmals avalable for recature, and M + s the total number of anmals marked n the study The robablty of ntal cature s the total number of frst catures dvded by the total number of frst catures ossble: ˆ M + c = ( N M ), and the robablty of recature s ˆ r = m/ M, the number of recatures dvded by the number avalable for recature Ths model s equvalent to estmaton under a removal (harvest) model d Heterogenety among ndvduals Model M h

11 WILD Analyss of Wldlfe Poulatons of 4 Ths model s truly dfferent concetually from other models n that cature robabltes do not vary temorally or based on behavor, but each ndvdual s assumed to have a unque cature robablty Concetually, ths model s arameterzed as a random samle of cature robabltes ( N ) from some underlyng dstrbuton F() Gven a cell robablty, π j, whch s bascally the average robablty that an ndvdual s catured j tmes: π j = 0!! j! ( j) j ( ) j df( ) The model s descrbed based on f j, the number of anmals caught on j occasons: P ( f f F ) N!! ( N M ) N M + j = π π f + = These models are not MLEs Thus, t s not ossble to use AIC or LRTs for model selecton Nonarametrc MLEs for ths model have been roosed by Norrs and Pollock (995, 996) Pledger s (000) mxture models are MLEs based on a fnte number of grous wth unque cature robabltes The MLE models roosed by Huggns (989, 99) and Alho (990) estmate cature robabltes usng the famlar logt functons of ndvdual covarates:! 0 = f X β e = X β + e These aroaches do not nclude N n the lkelhood and estmates of abundance are based on the Horvtz and Thomson (95) estmator, whch s the sum of the nverse of the robablty of cature at least once durng the study across ndvduals: ˆ = M + = N, * ˆ where ( ˆ ) ˆ * = j = e Combned models of cature robablty It s ossble to cast models that nclude or all 3 of the sources of varaton descrbed above These models requre addtonal constrants and assumtons, and Pledger (000) used mxture models that are MLEs to descrbe the models M tb, M bh, M th, and M tbh 7 Testng model assumtons a Closure All 8 of the above models assume no change n N durng the study Two general aroaches have been roosed for testng ) The null hyothess test n rogram CAPTURE (Ots et al 978) comares the j (robablty of cature for ndvdual at tme j) for all ndvduals catured

12 WILD Analyss of Wldlfe Poulatons of 4 more than twce The alternatve hyothess suggests that ror to frst and subsequent to last cature some ndvduals had j = 0, suggestng that they were recruted to the oulaton sometme after the frst or before the last cature erod ) Pollock et al (974) suggested 4 hyothess tests about tme secfc varaton n the j : a) No mortalty and no recrutment b) Mortalty, but no recrutment c) Recrutment, but no mortalty, and d) Recrutment and mortalty Stanley and Burnham (999) used ths aroach to test for overall oulaton closure whch bascally comares a model of comlete closure to a model for an oen oulaton They resent a seres of contngency tables that decomose closure volatons nto the sources of volatons (e, recrutment or losses), whch corresond to the 4 tests descrbed by Pollock et al (974 above) b Tag loss Poulaton estmates are based hgh by tag loss The excetons are model M b and M bh snce they do not rely on nformaton on recatures Otherwse tag loss has been nvestgated wth double-taggng studes 8 Model selecton It s useful to choose models that balance the recson of estmates delvered by models wth fewer arameters aganst the otental bas of more general models Thus, we are seekng to select the smlest model that fts our data However, many of the 8 models mentoned above are not based on lkelhoods; t s not ossble to base model selecton on AIC (but see Pledger 000) Ots et al (978) and Rexstad and Burnham (99) descrbe an alcable aroach based on goodness of ft and between-model tests a Goodness of ft The multnomal dstrbutons can be used to calculate exected values that are comared to the observed values as we saw n the bootstra GOF and RELEASE GOF for the CJS models Program CAPTURE, whch wll run under MAR, comutes these tests for 4 of the models M b, M t, M h, and M tb We wll dscuss goodness of ft along wth oen models n the next lecture b Between model tests Where MLEs can be comuted for the models lkelhood rato tests can be used comare nested models (eg, M b and M tb ) These tests are condtonal on the more general (e, arameterzed) model fttng the data and ask whether the less general model adequately reresents the data CAPTURE comares models M 0 versus M b and M t and M h and M h versus M bh c Dscrmnant Analyss Because AIC s not avalable for model selecton, CAPTURE uses dscrmnant functon analyss to comare models based on test statstcs and robabltes

13 WILD Analyss of Wldlfe Poulatons 3 of 4 Stanley and Burnham (998) descrbe a margnally mroved aroach that ncororates lnear and multnomal dscrmnant functons They also descrbe a method to ncororate model averagng Agan the mxture models of Pledger (000) are caable of emulatng all 8 of the closedoulatons models and several other varants n a lkelhood framework that allows not only the use of AIC for model selecton, but model averagng as well 9 Study desgn suggestons: a Mnmze volaton of model assumtons ) Closure a) ee t short to mantan closure and mnmze gans and losses b) Tmng s everythng eg, avod erods of seasonal movements or dsersal c) Tra mortalty bases ˆN If substantal use removal models (behavoral resonse) ) Use ndvdual marks methods really don t work wth batch marks b Precson of estmates ) Hgh cature robabltes ncrease recson and reduce bas ) Increasng occasons (> 5) ncreases recson 3) Parsmonous models ncrease recson 4) Behavoral resonse roblematc to closure a) Pre-batng can mnmze effects b) Mnmze trang deaths extreme behavoral resonse c) Mnmze handlng tme to reduce tra shyness 5) Heterogenety a) Stratfy by mortant covarates (locaton, age, sex, etc) b) Covarates use Huggns model c) Dstrbuton of trang effort () Trang densty v home range () Satal arrangement What we won t cover about closed models but you should know about Densty Estmaton wth Cature-Recature a Grds b Nested-grds c Tra webs Removal methods 3 Change n rato methods Dscusson

14 WILD Analyss of Wldlfe Poulatons 4 of 4 Lncoln-Peterson two-samle method bass for cature-recature and cature removal models -samle method a Comlete set of models for all three sources of cature robablty varaton b Some models not based on MLE c Mxture models are romsng soluton 3 Densty estmaton a Some methods estmate N frst and then calculate densty b Gradent samlng methods are desgned to estmate densty 4 Consderatons a CMR - Best methods for dffcult to observe anmals b Observatonal methods should be referable for obvous reasons c Removal methods useful for harvested oulatons General mrecse unless harvest s hgh relatve to N