ESTIMATION OF SURVIVAL RATES FROM A TAG-RECAPTURE STUDY WITH TAG LOSS. Walter K. Kremers. Biometrics Unit, Cornell, Ithaca, New York ABSTRACT

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1 ESTIMATION OF SURVIVAL RATES FROM A TAG-RECAPTURE STUDY WITH TAG LOSS BU-846-M by Walter K. Kremers Bometrcs Ut, Corell, Ithaca, New York July, 1984 ABSTRACT A probablty model s descrbed for a tag-recapture study where brds are released aually ad brds may be recovered ad detfed by a permaet bad or resghted ad detfed by a o-permaet collar. Survval, capture, ad sghtg probabltes are assumed to deped o the year. Collar reteto probabltes are assumed to deped o oly the age of the collar. Closed form maxmum lkelhood estmates do ot exst for the geeral model but umercal solutos are easly obtaed by the EM algorthm.

2 INTRODUCTION Survval rates ca be estmated through the use of may dfferet tag-recapture models. If survval rates are of prmary terest apart from populato sze a seres of models developed by Browe et.al.(l978), ad Seber(l971) are applcable. These works are based o recoveres, that s, o reobservatos where the brd (amal) s caught ad permaetly removed from the populato. Browe's work s also exteded to resghtgs or orecovery observatos, where capture probabltes are replaced by probabltes of observato wthout reobservato thereafter. Geerally for tag-recapture studes reteto of detfyg bads (or marks) a assumed to be permaet or that reteto rates are corporated to survval rates. Here we cosder a model smlar to that of Browe ad Seber but based o two types of observatos ad two types of marks. Oe mark, a detfyg "bad", s assumed to be permaet ad the secod, a detfyg "collar", s ot. The frst type of observato s a recovery from whch we observe the bad ad f retaed, the collar. The secod type of observato s a resghtg of the brd by the collar. Heurstcally we estmate collar loss from those brds whch are recovered ad the estmate survval from brds resghted by accoutg for collar loss, ad by brds recovered. Followg the coveto of Browe et.al. we gore termedate observatos ad cosder lkelhood fuctos for recaptures ad resghtgs usg oly the "fal" observato. By cosderg oly the fal observato ad gorg termedate observatos for each brd the lkelhood becomes markedly smpler though wth ths smplfcato s a loss of formato. Closed form Maxmum Lkelhood Estmates (MLE's) do ot exst for the model proposed here but umercal MLE's are easly obtaed by the use of the EM algorthm (Dempster,ec.a1.,1977) or other

3 -2- umercal procedures. Varaces ad covaraces must be estmated from estmates of the formato matrx. MODEL & NOTATION Brds are released oce a year, ad observed for ~ years. The probablty of survval from the tme of release oe year to the tme of release the ext year s assumed to be the same for all brds alve at begg of ths perod, ad s allowed to deped o the year. If a amal survves from oe year to the ext we assume the probablty of collar reteto to deped o age of the collar, but ot o the year. We also assume that for every year, each brd s recovered wth the same probablty, ad each collared brd s sghted but ot recovered wth the same probablty. Collar loss ad mortalty betwee the tme of reobservato ad most recet release are assumed eglgble. Wth otato smlar to that of Browe et.al. we cosder the followg radom varables. Xcl(,j)= umber of brds released the 'th year, recovered the j'th year wth collars. XcO(,j) umber of brds released the,th year, recovered the j'th year wthout collars. Xsl(,j) =umber of brds released the 'th year, sghted (wth collars) the j'th year ad ot observed thereafter. X(,j) Xcl(,j) + XcO(,j) + Xsl(,j) Cc(j) =!j =l Xcl(,j) + XcO(,j) Csl(j) =!j l Xsl(,j) R() =!!= X(m,j)

4 -3- T() ~ X(m,) ~S,~+1 D() ~ (Xc1(m,) + Xs1(m,)) ~,:-m~ EcO() t- 2;. 1 XcO(m,m+), As, ad addto to, the otato of Browe et.al. we defe the followg parameters. S= probablty that a brd survves utl the (+1)'th release gve t has survved utl the 'th release. P probablty that a collar s retaed years gve the brd bearg the collar has retaed ts collar ad survved -1 years sce ts release. f =probablty that a brd s recovered the 'th year gve t s alve at the tme of the 'th release. g ~probablty that a brd s resghted but ot recovered the 'th year gve t s alve ad has ts collar at the tme of the 'th release. For ths model we assume that the sghtg of a brd year j wthout ts recovery year j s ot related to the survval of the brd durg year j, except for the fact that the brd was ot recovered. That s, because survval rates are the same for sghted ad osghted brds, P[Survves year jlsghted but ot captured] P[Survves year jlot captured] =P[Survves year j ad s ot captured]/p[ot captured] P[Survves]/P[ot captured] Smlarly the probablty of a brd beg see year j gve t s ot recovered year j ad has a collar s P[see year j lalve,collar preset & ot recovered] g./(1-f.) J J

5 -4- Let x,j be the probablty of a brd ot beg observed after the j'th year gve the brd s sghted but ot recovered the j'th year ad was released the 'th year. Because the probablty of reteto of the collar s depedet o the age of the collar x,j s ot a fucto of j aloe. I terms of the capture, sghtg, survval, ad reteto probabltes x,j 1-[Sj/((1-fj)J ~ m-1 m- x~~ Wm J "+l{[ j+1s (1-g -f )/(1-f )]x[f m +g m j - + 1P ]}, j where j la s uderstood to be equal to oe. + Smlarly let (1-A) be the probablty of a brd released the 'th year ever beg observed. The (1-A) l-~~ {[llm-ls (1-f -g )/(1-f )]x[f +g llm-1 1 P ]} ~- = m m Each brd released may be last observed oly oe year, or ot observed at all. The probablty that a brd released the 'th year s last sghted the j'th year, s the probablty that a brd survves ad retas the collar utl the j'th release, s sghted the j'th year, ad s ot observed thereafter. Ths probablty s THE LIKELIHOOD Let C be the costat N() } ( Xcl(,),..,Xc1(,~),Xc0(,+l),.,Xc0(,~),Xs1(,),..,xsl(,~) where ( A } A!/(al! a2!. a![a-~ 1a]!) a 1,a 2,,a Sce a brd may be last observed oly oe year the probablty dstrbuto fucto or lkelhood s that of a multomal dstrbuto

6 -5- ad after some smplfcato, s foud to be L Cx{ll~ ls~() p~() (1-Pl. P-l)EcO() (l-a )N()-R()} {ll~ Xsl(,j)} x,j=ix,j fcc() Csl() g For the aalogous model by Browe, where survval ad recovery rates are depedet o year, the dmesoalty of the suffcet statstc s the same as the parameter space ad closed form solutos are derved. Here the dmesoalty of the suffcet statstc s greater tha the parameter space ad closed form MLE's are ot kow (to the author). To obta MLE's we cosder the E-M algorthm. Let "ghost" sghtgs be sghtgs of those brds whch we have released but have lost ther collars. Because these brds have lost ther collars we are uaware of the resghtg. However f estmates of the parameters are gve we may calculate the expected umber of ghost resghtgs ad clude ths the lkelhood. Whe ths s doe we may obta closed form "MLE's" from ths adjusted lkelhood. The ew estmates are the used to estmate the expected umber of ghost resghtgs ad the procedure cotues teratvely utl a maxmum s acheved. Specfcally the procedure s as follows. Let!=the probablty that a brd sghted the j'th year s ot reobserved ad ether s ts ghost 1-[S./((1-fj))]x'r,t j+l{[llm-jl+ls (1-g -f )/(1-f )]x [f +g ]} J 'm= ~ m m Let G probablty that a brd (or ts ghost) s resghted but ot recovered the 'th year ad ot reobserved thereafter= g!.

7 -6- Let (1-~~)= probablty that a brd released the 'th year s ever observed ad ether s ts ghost 1- m-1 = 1- ~~.{[TI s (1-f -g )/(1-f )] X [f +g ]} ~=1 m m Execute the expectato procedure by lettg XsO(,j) be the expected umber of brds last resghted as ghosts the j'th year of those released the 'th year!~:!+1 { Xs1(,j) r;:~ Sm] Adjust the Xs1(,j) for the resghtgs of ghosts by multplyg by the probablty of o reobservato by the collar or of the ghost gve there s o reobservato by the collar. That s let Xs1*(,j) (!j/x,j)xsl(,j). Redefe all radom varables terms of the Xsl*(,j), ad addto defe Cs(j) RR() TT() EscO() = Csl(j) + I. 1 XsO(,j),. R() +!.. XsO(,j) ]"'"1 = T() + Lro$,~+l XsO(m,),_ = EsO() +!;. 1 XsO(m,m+) Execute the maxmzato procedure by observg that the "lkelhood" s proportoal to TI~=l {S~T() P~() (1-Pl.. P-l)EscO() (l-~~ )N()-RR()} ad "solve for the MLE's" whch are S() (RR()/N()) ((TT()-Cc()-Cs())/TT()) (N(+l)/RR(+l)) P(l) (D(1)-D(2))/(D(l)-D(2)+Ecs0(1))

8 -7- P() = (D()-D(+l))/[(D()-D(+l)-EscO())P(l). P(-1)] f() (RR()/N()) (Cc()/TT()) G() (RR()/N()) (Cs()/TT()) Retur to the expectato procedure ad terate utl a stable set of parameter estmates are obtaed. NUMERICAL EXAMPLE The data Table I are cotrved so that the MLE's are s 1 -s 2 =0.8, s 3 0.7, P 1 =0.8, P 2 P 3 0.9, g 1 g 2 0.6, g 3 =g ad f 1 f N 1 =.. N For a varety af startg values covergece s reached to wth two sgfcat dgts of the (correct) MLE's wth ffty teratos. DISCUSSION We have cosdered the model where all brds have the same survval ad reobservato probabltes. These assumptos ca be relaxed just as the work of Browe et.al. If collar reteto probabltes are depedet o oly the age of the collar estmates of parameters may be estmated usg the EM algorthm as we have doe here ad models may be compared usg the lkelhood-rato test. Future work mght cosder models where collar reteto probabltes are ot determed by the age of the collar. A shortcomg of ths model s that collar loss ad mortalty betwee the tme of reobservato ad most reset release are assumed eglgble. Ths assumpto may be reasoable for brds tagged ad released September ad huted or sghted October but s ot reasoable f the brds are released stead March. Natural extesos of ths ths model

9 -8- would be to relax ths assumpto. I practces where resghtg rates are hgh ad recovery rates are low most of the formato for survval rates wll be from resghtgs apart from a costat of proportoalty determed by the collar reteto probabltes. Wthout recovery formato the model s overparamterzed as S ad Pj always occur together. If all S are multpled by a costat ad all Pj dvded by the same costat the lkelhood s uchaged. However apart from ths costat parameters may be estmated. I partcular the S/Sj are estmable ad applcatos where treds survval probabltes are of prmary terest these treds may be estmated from resghtg data aloe wthout kowg collar reteto probabltes REFERENCES Browe, C.,D.R.Aderso,K.P.Burham,ad D.S.Robso (197~). Scac2sc2cal Iferece From Bad Recovery Oaca-A Hadbook. U.S. Fsh ad Wldlfe Servce, Resource Publcato #131. Dempster,A.p.,N.M.Lard,ad D.B.Rub (1977). Maxmmum lkelhood from complete data va the EM algorthm. J..N.Scac2sc.Soc.B 39:1-38. Seber, G.A.F. (1970). Estmatg tme-specfc survval ad reportg rates for adult brds from bad returs. B2omecr2ka 57:

10 -9- Table I Cotrved data for a tag-recapture study wth J, 4 year of observato Xc1(,j) y 4 20 e a XcO(,j) r f r e 1 4 e a Xsl(,j) s e