i Trasct Mthods for lat Survs S.T. Bucklad * D.. Borchrs. Johsto.. Hrs 3 ad T.. Marqus 4 tr for Rsarch ito cological ad viromtal Modllig Uivrsit of St. drws Th Osrvator Buchaa Gards St drws KY6 9Z Scotlad Dt of Zoolog Uivrsit of amridg Dowig Strt amridg B 3J glad 3 Dt of Mathmatics ad Statistics acastr Uivrsit acastr 4YF glad 4 tro d statística licaçõs da Uivrsidad d isoa Bloco amo Grad 749-06 isoa ortugal *mail: stv@mcs.st-ad.ac.uk SUMMRY. Itrst i survs for moitorig lat audac is icrasig du i art to th d to quatif th rat of loss of iodivrsit. i trasct samlig offrs a fficit wa to moitor ma scis. Howvr th mthod dos ot work wll i som circumstacs for aml o small surv lots wh th lat scis has a strogl aggrgatd distriutio or wh lats that ar o th li ar ot asil dtctd. W dvlo a crossd dsig togthr with mthods that loit th additioal iformatio from such a dsig to addrss ths rolms. Th mthods ar illustratd usig data o a colo of cowslis. KY WORDS: Biodivrsit moitorig Distac samlig i trasct samlig Markrcatur Samlig immotil octs.. Itroductio t th 00 World Summit o Sustaial Dvlomt i Johasurg olitical ladrs agrd to striv for a sigificat rductio i th currt rat of loss of iological divrsit th ar 00. atural cosquc of this commitmt is a dmad for surv mthods that allow
widsrad moitorig of iological oulatios at low cost. s otd Bucklad t al. 005 th survs must carfull dsigd to sur that th moitorig data ar satiall rrstativ ad masurs of iodivrsit should costructd so that th ar ot iasd wh dtctailit of scis withi surv lots is imrfct. Distac samlig Bucklad t al. 00 004 is idall suitd for moitorig a umr of scis. It rovids a rigorous framwork for surv dsig Bucklad t al. 00:8-33 Stridrg t al. 004 ad licitl modls dtctailit. Much of th currt rsarch dvlomt i distac samlig is to allow rlial alicatio of th mthods to a widr rag of oulatios. For aml aak ad Borchrs 004 rovid comrhsiv mthods for oulatios whr dtctio at th li or oit is ot crtai ukacs t al. 004 cosidr assiv distac samlig mthods i which aimals rcord thir ow distacs from th li or oit trig tras Bucklad t al. 006 dvlo oit trasct mthods for wh a tra or lur is ositiod at ach oit. I this ar w cosidr th rolm of stimatig th audac of lats withi a colo or sit. urrtl thr is littl iodivrsit moitorig of lats which cotrasts with th situatio for mammals irds rtils ad amhiias dsit th fact that lats ar far mor rvasiv. Most lat survs targt rar scis of cosrvatio itrst. tlas survs i which osrvrs rcord a wid rag of scis grid squar ar also oular ut ths survs ar of littl us for moitorig chag i audac istad th ar dsigd to idtif chags i rag of scis. Bcaus atlas survs grall do ot hav a rigorous surv dsig ad ffort ca vr varial atlas data ar of limitd valu for iodivrsit moitorig. Ma lat scis srad across th groud so that idividual lats ar ot radil idtifial. For such scis quadrat samlig of som form is usd i which rct covr of ach scis is assssd withi th samld quadrats ithr dirct stimatio or us of a scal th valus of which icras o-liarl as rct covr icrass. W do ot addrss such survs hr. Rathr w cosidr th cas of scis for which idividual lats ar radil
idtifial. For som lats it ma mor aroriat to stimat audac of idividual flowr siks which ma mor visil ad idtifial tha th rst of th lat. umr of flowr siks ma covrtd to lat audac if th ma umr of flowr siks r lat is stimatd. I li trasct samlig sustatiall mor groud ca covrd i a giv tim tha for quadrat samlig for which all lats withi quadrats must coutd if lat audac is to stimatd without ias. Bcaus lats oft hav a vr atch distriutio th ailit to covr a larg ara of groud with modst rsourcs is a imortat advatag. Furthr dtctailit is modlld i li trasct samlig whras with quadrat samlig or stri trasct samlig it is assumd to crtai ad th assumtio is sldom tstd. I ma circumstacs stadard li trasct samlig is vr ffctiv for lat oulatios. Howvr scial circumstacs oft al which ma that mor sohisticatd mthods ma dd:. Th surv sit ma vr small although th moitorig rogramm ma comris ma such sits withi a rgio. If sit-scific audac or dsit stimats ar rquird stadard li trasct samlig rquirs that thr ar of th ordr of 0 trasct lis or mor at ach sit. If ths ar radoml locatd or mor usuall a grid of sstmaticall-sacd lis is radoml locatd i th sit this surs that lats ar uiforml distriutd to a good aroimatio with rsct to distac from th arst li for a sigl ralisatio of th dsig. Howvr a sigl sit or colo ma too small to allow so ma lis without ovrla of th associatd stris. Furthr caus lats oft hav vr atch distriutios th assumtio of uiformit ma adl violatd for a sigl ralisatio of th dsig although it is satisfid wh avragd across all ossil ralisatios.. For small or crtic scis lats o th li ma ass udtctd violatig th assumtio of crtai dtctio o th li. 3
W cosidr surv dsig ad aalsis mthods to addrss oth issus. Th motivatig aml rlats to hr-rich grasslad at Flcfaulds Madow i Fif Scotlad a Scottish Wildlif Trust rsrv. W us th mthods to stimat audac of cowslis rimula vris o a sctio of th rsrv.. Mthods. Surv dsig W us a crossd dsig comrisig two sstmatic grids of aralll lis rdicular to ach othr Fig.. W will assum that o of ths grids is oritd orth to south /S whil th othr is ast to wst /W. Dtctios withi a distac w of a li ar rcordd so that th stris of Fig. hav width w. Thr ar two imortat advatags to this dsig rlativ to a covtioal dsig:. ocatios of dtctios alog o st of lis rovid iformatio from which th distriutio of lats with rsct to distac from th li for th rdicular st ma stimatd. This allows th dtctio fuctio to stimatd wh th distriutio of lats with rsct to distac from th li is o-uiform.. Whrvr lis from th two grids itrsct th squar of sid w that is ctrd o th oit of itrsctio Fig. is survd twic rovidig doul-osrvr mark-rcatur data from which roailit of dtctio ca stimatd without assumig dtctio is crtai o th li. Morovr caus th squar is survd from ach of its ctrlis o ruig /S ad th othr /W th ddc tw dtctios arisig from htrogit i th dtctio roailitis is rducd makig th task of modllig th htrogit asir.. Stri trasct samlig I stri trasct samlig w assum that all lats withi th stri ar dtctd. Hc this is a form of lot samlig i which th lots ar log ad arrow. stimatio quatios ar 4
giv i W di. Th advatag of th crossd dsig ovr a aralll stri dsig i this cas is that th assumtio that all lats withi th stri ar dtctd ca tstd dtrmiig whthr thr ar lats i th itrsctio squars that ar dtctd i o of th survs ut ot oth if thr ar ma such lats th mor sohisticatd mthods ar dd..3 ovtioal li trasct samlig W ow allow lats to missd withi th covrd stris ut w still assum that all lats o th ctrli ar dtctd. I covtioal li trasct samlig w fit a dtctio fuctio to data scifig a liklihood for th rdicular distacs to dtctd lats coditioal o saml si. osidrig /W data ol w hav f g w whr is distac from th li of th th dtctd lat... f is th roailit dsit fuctio f of distacs valuatd at is th vctor of aramtrs usd to scif f g is th roailit that a lat at distac from th li is dtctd w w 0 g d is th roailit that a lat is dtctd giv that it is somwhr withi th survd stri of half-width w. ot that f g / w for all i [0w]. For a giv st of distacs... w maimi th aov liklihood to otai th maimum liklihood stimator ˆ of w ad hc ˆ w 0 gˆ d. W ca ow stimat lat audac as a ˆ ˆ whr is th si of th surv rgio ad a w is th covrd ara corrsodig to th /W 5
lis with dfid as th total lgth of /W lis Bucklad t al. 00. It is assumd that lats availal for dtctio ar uiforml distriutd with rsct to distac from th li a rquirmt that is achivd o avrag aroriat radomiatio of th surv dsig. W ca otai a scod stimat ˆ from th data from th /S lis. I this cas th liklihood is i g w i whr w w 0 g d i is th distac of th i th dtctd lat from th /S lis i... ad g is th roailit that a lat at distac from a /S li is dtctd. Th otimal wa to ool th data from th two sts of trascts is to form a oit coditioal liklihood. ossil stratg is to form a liklihood for data from th itrsctio squars asd o th roailit that a lat is dtctd from at last o of th lis assig through a squar. Howvr this forcs th assumtio that whthr a lat is dtctd from th /W li is iddt of whthr it is dtctd from th /S li. W rfr to avoid this assumtio as audac stimatio is ssitiv to failurs of it. W ca rlac this assumtio th wakr assumtio that roailit of dtctio of a lat from o li is iddt of its distac from th othr li takig as th oit liklihood W cosidr this issu i gratr dtail i th t sctio. W maimi th liklihood i to giv ˆ ad ˆ ad hc ˆ ad ˆ. Softwar Distac Thomas t al. 005 ma usd for this uros coductig iddt aalss of th /W data ad th /S data. If fild mthods ar idtical for th /S ad th /W lis it is rasoal to assum that g g g sa so that sa. This aalsis ma achivd i Distac trig th /W data as o stratum ad th /S data as a 6
scod stratum ad fittig a oold dtctio fuctio across th two strata. Stratum-scific audac stimats should th avragd ot summd. Horvit-Thomso-lik stimator Borchrs t al. 00:43 ow rovids our stimat of audac: ˆ ˆ a ˆ 3 with a a a w. ot that ach itrsctio squar has survd twic ad its si is icludd i a twic. Th stimat / ˆ corrsods to stimatd audac i th /W stris ad similarl / ˆ is stimatd audac i th /S stris. Ovrall audac is stimatd dividig thir sum th ratio of si of th covrd rgio to th si of th whol surv rgio Borchrs t al. 00:56. If lats ar markdl o-uiform through th surv rgio ad th umr of lis i th dsig is small th followig stimator might rfrrd: ˆ 0.5 q ˆ q ˆ 4 whr q is th roortio of lats dtctd from th /S lis that fall withi th itrsctio squars ad hc withi th /W stris ad covrsl for q. Thus q stimats th roortio of octs i th surv rgio that fall withi th /W stris ad q th roortio that fall withi th /S stris. If chac thr ar roortioall mor lats i th survd stris tha i th gas tw thm for a giv ralisatio of th dsig stimator ˆ will td to ovrstimat audac whras stimator ˆ sstiall uss two ratio stimators to adust for th uvss i distriutio. Th dtctio fuctio ma modlld as a fuctio of covariats i additio to distac from th li usig th coditioal liklihood mthods of Borchrs t al. 998 ad 7
Marqus ad Bucklad 003. If th covariats for dtctio ar dotd th vctor th w ˆ gˆ d w ad ˆ ad similarl for ˆ. I ractic a sigl ˆ 0 modl for th dtctio fuctio is likl to rov adquat: g g g sa. If ˆ ˆ ad ar assumd to iddt th dlta mthod ma usd to stimat a aroimat variac of ˆ : ˆ var ˆ ˆ ˆ var var ˆ a ˆ 5 whr ˆ a vâr cv cv ˆ ˆ ˆ ad similarl for v âr. Th ˆ cofficits of variatio cv ad cv ˆ ma stimatd for aml usig th softwar Distac Thomas t al. 005. ot th us of th fiit oulatio corrctio as outlid Bucklad t al. 00:87. I most alicatios a ad th corrctio is gligil ut this ma ot tru i survs of small oulatios of lats whr th covrd ara a ma a sustatial roortio of th surv rgio. Similarl stimator ˆ has aroimat variac: ˆ ˆ var ˆ ˆ 0.5var var ˆ q ˆ q 6 ˆ q ˆ q a ˆ whr vâr cv cv ˆ cv q ad similarl for v âr whr cv q q ˆ ma stimatd from th tw-li saml variac i q wightd th umr of osrvatios cotriutig to th stimat of q from ach li. If th umr of lis is small th ootstra sgmts asd aroud th itrsctio squars as dfid 8
low ma usd so that th cv is stimatd from th tw-sgmt saml variac. This mthod ma also usd to stimat cv. Th oaramtric ootstra rovids a siml t roust mthod of stimatig variac without havig to assum iddc tw ˆ ad ˆ. Usuall i li trasct samlig th lis would rsamld Bucklad t al. 00:83. Howvr thr ar two rolms with that aroach hr: first th data withi ach itrsctio squar d to kt itact for som of th mthods of this ar for which it would uclar how to rocd if o of th two lis formig a itrsctio aard i a rsaml ut th othr did ot ad scod if th umr of lis is small as it somtims is for survs of lats ootstra rsamlig of lis ilds oor stimats of variac. advatag of th crossd dsig is that w ca divid th covrd rgio u ito small sgmts that ar sstmaticall sacd through th surv rgio with o itrsctio squar r sgmt. For aml if thr ar ust fiv /W lis ad fiv /S lis this ilds 5 uits which is sufficit for th oaramtric ootstra to rovid rlial stimats of variac. W show i Fig. 3 two was of surimosig a grid of squars ovr a crossd surv dsig so that ootstra rsamls ca gratd samlig th squars ad thir associatd data with rlacmt. Havig gratd a larg umr of rsamls th data from ach rsaml ar aalsd usig th sam rocdurs as for th ral saml. Th saml variac of th ootstra stimats of a giv aramtr of itrst such as rovids a stimatd variac of th corrsodig stimat. siml ut roust mthod of stimatig cofidc limits is to ordr th ootstra stimats of th aramtr ad tract th aroriat rctils from th ordrd list as aroimat limits Bucklad 984. If ootstra rsamls ar costraid so that ach rsaml has th corrct umr of dg squars of ach t s Fig. 3 th total ffort will ot var across rsamls. Howvr this stratg ma giv oor stimats of variac wh th umr of dg squars of ach cofiguratio is small W di. 9
difficult of th aov imlmtatio of th ootstra is that thr is o fiit oulatio corrctio. If th covrd ara a is a sustatial roortio of th surv rgio sa 0% or mor th th ootstra ca usd to stimat cv cv cv ˆ ad cv ˆ ad ths stimats ar th sustitutd ito q 5 if stimator ˆ is usd. For stimator ˆ w also d to otai ootstra stimats of cv q ad cv q so that w ca us q 6. Howvr this stratg has th disadvatag that th stimats ˆ ˆ ad whr rlvat q ad q ar all assumd to iddt. If for aml w wish to allow i q 5 for covariac tw ad ad tw ˆ ad ˆ ths covariacs ma stimatd th saml covariacs of th ootstra stimats ad th covariac trms ca addd to 5 to giv ˆ var ˆ ˆ ˆ ˆ var var cov a ˆ ˆ ˆ ˆ 7 whr vâr ˆ ad vâr ˆ ar as for ad côv a ˆ a ˆ côv ˆ ˆ c ôv. ˆ ˆ ˆ ˆ ˆ ˆ Th aov imlmtatio of th ootstra assums that th rsamlig uits ar radoml distriutd wh i ralit th ar sstmaticall distriutd. Fwstr t al. i r. cosidr ossil solutios to rduc a ias i th rsultig variac stimats..4 i trasct samlig with o-uiform distriutio of octs osidr first th /W lis. Dfi to th df of distacs from th li of all octs withi th covrd stris whthr or ot th ar dtctd with 0 w ad suos th aramtrs of ar. ot that is th distriutio of distacs from th li aftr 0
oolig data across all th /W lis. Similarl w dfi distacs of lats from th arst /S li. aramtrid for If w wr to cosidr data from th /W lis ol th th liklihood of th osrvd coditioal o saml si is: f g 8 whr w 0 g d Borchrs ad Burham 004. If octs ar uiforml distriutd with rsct to distac from th li th w ad 8 simlifis to. Wh / ad g must oth stimatd th aov liklihood is ot sufficit as th two fuctios alwas aar as a roduct. Th itrsctio squars rovid us with th data w d to stimat if w cosidr lats dtctd from th /S lis thir distacs from th /W lis ar a rrstativ saml from rovidd roailit of dtctio from th /S li is iddt of distac alog th /S li ad hc iddt of distac from th /W li. Th liklihood w d will ow comris si comots: o for dtctios i rgios of t Fig. aothr for thos i rgios of t B ad four for thos from itrsctio squars t. osidr th first two comots. Suos w hav rdicular distacs... for dtctios i rgios of t ad distacs i i... B for dtctios i rgios of t B. lig 8 to ths rgios w hav g B B g i i i B whr w 0 g w d ad 0 g d.
ow cosidr th dtctios i th itrsctio squars togthr with th distacs.... W d to cosidr th thr ts of dtctio: lats dtctd from a /W li ol ad thir distacs from th arst /S li ad /W li rsctivl... lats dtctd from a /S li ol ad corrsodig distacs... ad dtctd from oth a /W ad a /S li with distacs.... Thus. Formulatio of a aroriat liklihood for osrvatios i itrsctio squars rquirs som car. I articular withi a itrsctio squar w d to avoid th assumtio that dtctio from th /W li is iddt of ig rcordd from th /S li. larg clarl visil lat is likl to dtctd from oth lis so that a aalsis that assums iddc without modllig such htrogit will ild iasd stimats of audac. Our otatio imlicitl assums that g is iddt of ad g is iddt of which sms vr rasoal. This assumtio surs that... tak with... ar oitl a radom saml of si from. Similarl... ad... ar togthr a radom saml of si from. Ths thrfor cotriut ad to th liklihood. W also kow that... ad... ar oitl a radom saml of si from g /. Similarl... ad... ar oitl a radom saml of si from g /. Thus ths cotriut th followig comots to th liklihood.
g g g g Th ovrall coditioal liklihood for dtctios i itrsctio squars coditioal o th osrvd saml sis B ad is th th roduct of ths four comots. ot that th distacs for dulicat dtctios hav ach usd twic ut ach of ths distacs is osrvd twic: oc from a /W li ad oc from a /S li. osidr a lat at [ ] dtctd from oth lis. Th as osrvd from th /W li is a radom draw from whras as osrvd from th /S li is a radom draw from g /. Th aov roduct of liklihood comots givs th aroriat liklihood rovidd ths draws ar iddt of ach othr ad of th corrsodig draws of. W kow is osrvd from th /W li with roailit g ad from th /S li with roailit g. Th first roailit is a fuctio of alo ad th scod of alo. Thus if w assum that th oit dsit of lat locatios whthr dtctd or ot th th draws ar iddt. This assumtio also surs that is draw iddtl of. s for i most survs it would rasoal to assum that g g g ad giv that th umr of itrsctio squars is likl to quit larg v wh th umr of trascts i ach dirctio is small ad th ar sstmaticall sacd throughout th rgio iddc of th ad locatios for all ad i [ 0 w ] is likl to quit a mild assumtio. Udr ths simlifig assumtios th comid liklihood for all thr rgio ts B ad is: 3
4 B B g g 9 g g g g lthough w hav usd th sam id for ach rssio th saml to which it rfrs is idicatd th ur limit of th rag of. Wh ad ar oth uiform o ] 0 [ w 9 is quivalt to with th addd assumtio that g g g. W maimi th liklihood i 9 to giv ˆ ˆ ad ˆ ad hc ˆ ad ˆ. Horvit-Thomso-lik stimator Borchrs t al. 00:43 rovids our stimat of audac which w ca sarat out ito stimats of audac i rgios of t B ad Fig. : ˆ ˆ B B ˆ ˆ ˆ ˆ ˆ ˆ from which ovrall audac is stimatd as 3 ˆ ˆ ˆ ˆ ˆ B a 0 with w a. Th stimats ˆ B ˆ ˆ ad ˆ corrsod to stimats of audac i sctios of th covrd rgio. lso ˆ ˆ is a stimat of th total umr of lats i th /W stris asd o distacs of dtctd lats from th /W lis ad B ˆ ˆ is th corrsodig stimat for th /S stris. I 0 w divid thir sum a /. If lats ar markdl o-uiform through th surv rgio ad th umr of lis i th dsig is small w could agai us q ad q i lac of this ratio:
ˆ ˆ ˆ ˆ ˆ 4 0.5 q q B Th ootstra ma imlmtd as for to quatif th rcisio of ths stimats. ot that caus ˆ ad ˆ oth stimat th sam quatit w could calculat th diffrc ˆ ˆ i ach ootstra rsaml ad hc otai a ootstra cofidc itrval for th tru diffrc. If this itrval dos ot iclud ro this ma idicat that w hav a iaroriat modl for our data..5 i trasct samlig with ucrtai dtctio o th trackli Th data from th itrsctio squars ca rgardd as two-saml mark-rcatur data th osrvr marks th lats dtctd whil trasitig th first li through a squar ad rcaturs occur durig th scod trasit. So far w hav igord this iformatio caus its us coms at th cost of strog iddc assumtios aout th roailit of dtctio i ach saml ad stimatio is ot roust to failur of ths assumtios. Wh dtctio o th trackli is crtai this fact d ol sulmtd a stimat of th sha of th dtctio fuctio to ild a stimat of th fuctio itslf th oolig roustss rort Bucklad t al. 004:389-39 shows that this is achival wh variatio i dtctio roailit du to factors othr tha distac from th li ar igord. Wh dtctio o th trackli is ucrtai w must rl o th mark-rcatur data to hl stimat th dtctio fuctio ad hc w must cosidr th ffcts of htrogit i th catur roailitis. This htrogit ca modlld ovr all distacs aak ad Borchrs 004 or w ca assum oit iddc aak 999 aak ad Borchrs 004 or oth Borchrs t al. 006. W first cosidr th cas without additioal covariats for modllig htrogit. W follow th covtio of aak ad Borchrs 004 that g rrsts th sha of th dtctio fuctio ut scald so that g 0 whras rrsts th dtctio fuctio itslf. Thus if dtctio o th li is crtai th two fuctios ar quivalt. 5
6 For rgios ad B of Fig. w hav th sam liklihoods as for. Rlacig g : B B i i i B whr w d 0 ad w d 0. W could if w wishd divid to ad ottom of th first liklihood ] 0 [ i which cas th liklihood is agai i trms of g rathr tha th two forms ar quivalt which is wh w ca us this liklihood to ifr ol th sha of. For rgio followig th dvlomt of aak ad Borchrs 004 ad Borchrs t al. 006 w hav: I additio w hav from th rvious sctio ad. W ow d th mark-rcatur comot of th liklihood corrsodig to th itrsctio squars. t th catur histor of th th dtctd lat i rgio. Th r{ ω whr is th roailit that a lat at distac from a /W li is dtctd giv that it was dtctd from a /S li ad ω is th full st of osrvd catur historis. Th coditioal liklihood is thus
7 ω ω B W hav ] [ } 0 r{ ] [ } 0 r{ } r{ If w assum iddc th ad. I this cas ias i th audac stimat ma rducd modllig th dtctio fuctios ad hc th catur histor roailit distriutio as fuctios of additioal covariats aak ad Borchrs 004. I th rsc of additioal covariats ut with th simlifig assumtios of iddc with ad oth uiform o ] 0 [ w ad sa th liklihood rducs to r{ ω whr ] [ } 0 r{ ] [ } 0 r{ } r{ I most survs it will rasoal to assum that ad ar oth uiform o ] 0 [ w ad th crossd dsig rovids data to tst this assumtio. datig th oit iddc mthod of aak ad Borchrs 004 to th crossd dsig w assum iddc of dtctio tw th /W ad /S lis ol at distac ro. I this cas th dtctio fuctio for th /W lis is modlld as 0 g. Hr
is th coditioal dtctio fuctio giv dtctio from th /S lis 0 valuatd at =0 which is qual to th ucoditioal roailit at distac ro: 0 0 assumtio. Dtctio roailitis for th /S lis ar modlld similarl. Th coditioal dtctio fuctios ad ar stimatd from th liklihood comot ω alo. This is quivalt to usig iar rgrssio tratig dtctios i aras of t from th /S lis as trials ad dtctio/o-dtctio of ths from th /W lis as th iar rsos. Th dtctio roailit for th /S lis is stimatd similarl usig dtctios i aras of t from th /W lis as trials ad dtctio/o-dtctio of ths from th /S lis as th iar rsos. s i th rvious sctio ovrall audac ma stimatd usig ithr q 0 or q..6 full liklihood aroach W could add a furthr comot to our liklihoods corrsodig to ifrc aout oulatio si giv th data i our surv stris W di. 3. Flcfaulds Madow Flcfaulds Madow is a Scottish Wildlif Trust SWT rsrv ar rs i Fif Scotlad. SSSI Sit of Scial Scitific Itrst sic 983 th fild cotais scru ad hr-rich grasslad aroimatl ha of which was usd for th surv Fig.. Data wr collctd i Ju 005 o svral scis ad th cowsli was slctd for tstig th mthods. aralll li trascts wr sstmaticall sacd at 9m itrvals with a radom start with ruig aroimatl /S ad aothr /W. stri half-width of w. 5 m was usd. Th 4 trascts gav 9 itrsctio squars i.. aras of 9m whr lats wr ottiall dtctal from oth trascts Fig.. ach trasct was survd o of two osrvrs who walkd alog th /W trascts first coctratig thir sarch ffort o ad clos to th trasct li. Dtctd lats 8
wr ol rcordd oc th osrvr had carfull sarchd for othr ar lats. Distacs of dtctd lats from th li wr th masurd with a.5m rul a lats furthr tha.5m from th li wr ot rcordd. Th trasct li had marks vr mtr allowig distacs alog th li to asil masurd. 30cm rul was usd for smallr scal masurmts o th lat itslf. ll dtctd lats from th /W trascts wr markd writig a uiqu umr o th udrsid of a laf to sur that if th lat was s from o of th /S lis it could idtifid. For ach dtctd lat i additio to distacs alog ad from th li th followig varials wr rcordd: its si dfid as th lgth of th lat multilid its width cm th umr of idividual flowrhads o th lat ad a visiilit cod takig valus from o to four o corrsodig to a lat stadig aov th surroudig vgtatio through to four for a lat surroudd much tallr vgtatio. Data collctd from th /W lis idicatd that cowslis could assumd to uiforml distriutd with rsct to distac from th /S lis Kolmogorov-Smirov ramérvo Miss ad tsts 0. i ach cas ad similarl for th covrs Kolmogorov- Smirov ramér-vo Miss ad tsts 0. i ach cas. Dsit th vr clustrd distriutio of cowslis rsultig i wid cofidc itrvals i Tal lis i ach dirctio wr sufficit to sur that th assumtio that lats wr uiforml distriutd with rsct to distac from th li oc th distac data wr oold across aralll stris was rasoal. half-ormal modl for th dtctio fuctio was foud to fit wll for oth sts of lis ad I was margiall smallr wh th modl was fittd saratl to th two data sts rathr tha to th oold data. Rsultig stimats ar show i Tal. stimators ˆ ad ˆ gav similar rcisio as stimatd from qs 5 ad 6 rsctivl ut stimatd audac was rathr lowr usig stimator ˆ. Th ffct of covariats o dtctailit was lord. I th cas of aalss with 0 I slctd a modl with a osrvr ffct a factor covariat with two lvls for 9
/W lis ad a modl with ffcts of osrvr lat si ad visiilit cod for th /S lis. umr of flowrhads was ot slctd i ithr modl. Rsultig audac stimats ar show i Tal. W allowd for 0 assumig logistic forms for ad whil iitiall rtaiig th assumtio of uiform distriutio of lats with rsct to lis. Slctig varials o th asis of I ld to a modl for ad which icludd distacs ut ot li oritatio /W or /S as laator varials. This modl gav a stimatd 0 of 0.694 with 95% cofidc itrval 0.45 0.839. This imlis that aout 30% of lats o th lis ar missd if w assum 0 audac stimats ar aroud 30% lowr tha if w rla this assumtio. Usig q audac is stimatd to 58 lats whil usig q 0 it is stimatd to 507 lats Tal. I th cas of th aalss with 0 ad stimatd ad I idicatd a dtctio fuctio modl with ffcts for /W vs /S ad distac togthr with a sigl modl for ad which w rfr to as. This was modlld usig a lft- ad righttrucatd ormal distriutio. Th stimat of is show i Fig. 4a. Whil this distriutio is ot sigificatl diffrt from uiform at th 5% lvl a o-uiform modl for is rfral o th asis of I 953.3 vs 955.7. Th fit from th 0 aalss is show i Fig. 4. I this cas audac is stimatd to 467 lats usig q whil usig q 0 it is stimatd to 758 lats Tal. Ths stimats ar slightl highr tha thos otaid wh assumig uiform ad caus a highr roortio of lats ar stimatd to at gratr distacs whr dtctio roailit is lowr s Fig. 4. I this aml truth is ukow. To furthr assss th mthods a oulatio of kow si was stalishd ad aalss of a surv coductd o th oulatio ar giv i W di ad W Fig.. 0
4. Discussio I ma circumstacs quadrat or lot samlig is likl to rov adquat for lat survs. W liv howvr that th mor coml mthods of this ar ar ottiall usful i th followig circumstacs. First if idividual lats ar asil ovrlookd quadrat samlig ma rquir aistakig sarchig o hads ad ks svrl limitig th amout of groud that ca covrd. Th mthods of this ar i which dtctio o th trasct li is allowd to ucrtai ma rov a mor cost-ffctiv otio. I th cas of cowslis aov w stimat that aroimatl 30% of th cowslis at distac ro wr missd i th thick vgtatio rst i Ju dsit vr carful sarchig alog th trascts th osrvrs. Scod if lat distriutio i th surv rgio is vr clumd or if lats ar sarsl distriutd quadrat samlig ma giv oor rcisio. For similar rsourcs it is ossil to covr a sustatiall highr roortio of th surv rgio with li trasct samlig tha with quadrat samlig caus it is ot cssar to dtct all lats withi th covrd stris. This rovids ttr satial covrag of surv ffort ad hc highr rcisio. ottial rolms du to isufficit lis could with a markdl o-uiform distriutio of lats ar avoidd usig our mthods. Th solutio to th rolm of o-uiformit advocatd Mlvill ad Wlsh 00 to locat all lats withi a sust of stris so that th dtctio fuctio ma stimatd without assumig uiformit sic w kow for ach lat whthr or ot it was dtctd from th trasct li is usatisfactor for two rasos: sustatial rsourcs ma rquird to sur comlt umratio withi th sust ad wh th umr of stris i th sust is vr small ithr o or two stris i th cass rstd Mlvill ad Wlsh th rsultig stimatd dtctio fuctio ma urrstativ of th dtctio fuctio for th rmaiig stris du to variatio i vgtatio hight ad colour light or othr factors. Our dsig avoids oth rolms comlt umratio is ot rquird awhr ad th
itrsctio squars ar sufficitl umrous ad ar sstmaticall srad through th surv rgio so that rrstativss is assurd. If rformac of th mthod is i dout a stratg suggstd th associat ditor is to slct a radom saml of itrsctio squars ad attmt comlt umratio of ths. Rlativ to comlt umratio of whol stris this has th advatag that itrsctio squars ar small so that a rasoal umr of squars ca survd this mas ad caliratio will th mor roust to variatio i dtctailit across th surv rgio. ot that ifrc rlatd to th couts of lats i stris is dsig-asd ad th samlig fram is th st of stris of half-width w i ach dirctio that full sa th surv rgio. Th fiit oulatio corrctio is th tak as o mius th roortio of th surv rgio that is ffctivl covrd that roortio is th ara of samld stris with half-width tak as th ffctiv stri half-width w 0 g d dividd th si of th surv rgio. If th grid of stris is fid th a uiform distriutio of octs with rsct to distac from th li withi 0 w is ot assurd. Radomiatio of th grid locatio surs that this assumtio is mt i th ss that th avrag distriutio of octs distacs from th li ovr th ifiit of ossil radomiatios is uiform Fwstr t al. 005 i r.. Howvr for a sigl raliatio of th dsig sciall if th umr of stris samld is small or if th octs satial attr is markdl aggrgatd it ma rov ficial to modl a ouiformit osrvd for that raliatio. I most studis w would ct that th addd variac from modllig o-uiformit would outwigh th rductio i ias rsultig i a icrasd ma squar rror. Th roosd ootstra mthod was assssd for th cas of covtioal distac samlig Sctio.3 simulatio W di. Th simulatios wr asd o th Flcfaulds stud for which th fiit oulatio corrctio was sufficitl larg to dd. With this corrctio th mthod was foud to work wll for uiforml distriutd oulatios as
would ctd for clustrd oulatios ad for oulatios hiitig a liar trd i dsit through th rgio W Tal s W Fig. for a sigl raliatio of ach t of oulatio. Wh ootstra rsamls wr costraid so that ach rsaml has th corrct umr of dg squars of ach t th rformac of th ootstra was rlativl oor which might ctd giv that for aml ach corr uit was of a uiqu t i ths simulatios ad so ach aard actl oc i ach rsaml. stimator ˆ had highr variac tha ˆ i all simulatios ct wh clustrig was trm ad i that cas th ootstra stimat of th variac showd uward ias so that th imrovd rcisio was ot rflctd i th stimatd variac W Tal. Hc w rcommd us of stimator ˆ. Sulmtar matrials W di W Tal ad W Figs ad rfrcd i Sctios..3.6 3 ad 4 ar availal udr th ar Iformatio lik at th Biomtrics wsit htt://www.tis.org/iomtrics. KOWDGMTS TM s rsarch is sosord FT ad FS III Quadro omuitário d oio SFRH/BD/08/00. STB ackowldgs th suort of th vrhulm Trust. RFRS Borchrs D.. Bucklad S.T. Godhart.W. lark.d. ad Hdl S.. 998. Horvit- Thomso stimators for doul-latform li trasct survs. Biomtrics 54-37. Borchrs D.. Bucklad S.T. ad Zucchii W. 00. stimatig imal udac: losd oulatios. Srigr Vrlag odo. 3
Borchrs D.. ad Burham K.. 004. Gral formulatio for distac samlig. I dvacd Distac Samlig 6-30. S.T. Bucklad D.R. drso K.. Burham J.. aak D.. Borchrs ad. Thomas ds. Oford Uivrsit rss Oford. Borchrs D.. aak J.. Southwll. ad ato.g.m. 006. ccommodatig umodld htrogit i doul-osrvr distac samlig survs. Biomtrics 6 37-378. Bucklad S.T. 984. Mot arlo cofidc itrvals. Biomtrics 40 8-87. Bucklad S.T. drso D.R. Burham K.. aak J.. Borchrs D.. ad Thomas. 00. Itroductio to Distac Samlig. Oford Uivrsit rss Oford. Bucklad S.T. drso D.R. Burham K.. aak J.. Borchrs D.. ad Thomas. ditors. 004. dvacd Distac Samlig. Oford Uivrsit rss Oford. Bucklad S.T. Magurra.. Gr R.. ad Fwstr R.M. 005. Moitorig chag i iodivrsit through comosit idics. hil. Tras. R. Soc. od. B 360 43-54. Bucklad S.T. Summrs R.W. Borchrs D.. ad Thomas. 006. oit trasct samlig with tras or lurs. Joural of lid colog 43 377-384. Fwstr R.M. Bucklad S.T. Burham K.. Borchrs D.. aak J.. ad Thomas. i r.. stimatig th coutr rat variac i distac samlig. Fwstr R.M. aak J.. ad Bucklad S.T. 005. i trasct samlig i small ad larg rgios. Biomtrics 6 856-859. aak D.. Borchrs ad. Thomas ds. Oford Uivrsit rss Oford. aak J.. 999. Distac samlig with iddt osrvrs: rducig ias from htrogit wakig th coditioal iddc assumtio. I Mari Mammal Surv ad ssssmt Mthods 37-48. G.W. Garr S.. mstru J.. aak B.F.J. Mal.. McDoald ad D.G. Rortso ds. Balkma Rottrdam. aak J.. ad Borchrs D.. 004. Mthods for icomlt dtctio at distac ro. I dvacd Distac Samlig 09-89. S.T. Bucklad D.R. drso K.. Burham J.. aak D.. Borchrs ad. Thomas ds. Oford Uivrsit rss Oford. 4
ukacs.m. Frakli.B. ad drso D.R. 004. assiv aroachs to dtctio i distac samlig. I dvacd Distac Samlig 60-80. S.T. Bucklad D.R. drso K.. Burham J.. aak D.. Borchrs ad. Thomas ds. Oford Uivrsit rss Oford. Marqus F.F.. ad Bucklad S.T. 003. Icororatig covariats ito stadard li trasct aalss. Biomtrics 59 94-935. Mlvill G.J. ad Wlsh.H. 00. i trasct samlig i small rgios. Biomtrics 57 30-37. Stridrg S. Bucklad S.T. ad Thomas. 004. Dsig of distac samlig survs ad Gograhic Iformatio Sstms. I dvacd Distac Samlig 90-8. S.T. Bucklad D.R. drso K.. Burham J.. aak D.. Borchrs ad. Thomas ds. Oford Uivrsit rss Oford. Thomas. aak J.. Stridrg S. Marqus F.F.. Bucklad S.T. Borchrs D.. drso D.R. Burham K.. Hdl S.. ollard J.H. Bisho J.R.B. ad Marqus T.. 005. Distac 5.0. Rsarch Uit for Wildlif oulatio ssssmt Uivrsit of St. drws UK. htt://www.ruwa.st-ad.ac.uk/distac/ 5
Tal. stimats of dsit lats/ha of cowslis at Flcfaulds Madow 95% cofidc limits i rackts. Trascts udac stimat 95% c.i. ovtioal /S lis 980 466 059 distac /W lis 3 670 849 samlig Both ˆ 048 689 593 Both ˆ 873 568 34 ovtioal /S lis 978 468 045 distac /W lis 4 680 857 samlig with Both ˆ 05 696 59 covariats Both ˆ 877 574 34 llowig for Both ˆ 3 507 05 56 0 Both ˆ 4 58 99 90 llowig for Both ˆ 3 758 9 70 o-uiform Both ˆ 4 467 03 78 ad 0 6
Figur catios Fig.. Dsig of th Flcfaulds surv which comriss two sstmatic grids of stris o with stris ruig aroimatl /S ad th othr with stris ruig /W. Th stris ar of width w ad th trasct lis ru dow th middl of ach stri. ach grid is radoml surimosd o th surv rgio. Fig.. Schmatic largmt of a crossovr of rdicular trascts. Dtctios mad from th /W li ol ar idicatd + from th /S li ol * ad from oth lis o. Withi rgio distacs from th /W li ar rcordd. Withi rgio B distacs from th /S li ar rcordd. Withi th itrsctio squar oth ad ar rcordd th suscrit idicats dtctd from th /W li ol idicats dtctd from th /S li ol ad dtctd from oth lis. Th rag for ach of ad is [ 0 w ] that is w rcord asolut distac from ach li ad w trucat osrvatios at distac w. Fig. 3. Two amls of a squar grid dashd lis surimosd ovr a crossd dsig so that covrd rgios withi grid squars ar sstmaticall sacd through th surv rgio. oaramtric ootstra rsamlig is imlmtd rsamlig squars togthr with th associatd data from th grid. I th lft-had schm th covrd rgios rsamld this rocdur ar crosss whil i th right-had schm th ar s. Fig. 4a Histogram of th comid distacs from th /S lis of lats dtctd from th /W lis ad distacs from th /W lis of lats dtctd from th /S lis for cowsli data. Th dottd li is th stimat of assumd to th sam for oth latforms scald to hav th sam ara as th histogram. lot of scald stimatd dtctio roailit dashd li stimatd dottd li ad stimatd roailit dsit fuctio for osrvd distacs solid li for cowsli data. Th histogram shows umr of dtctios distac from th li from which th wr dtctd. ll fuctios hav scald to hav th sam ara as th histogram. 7
Fig.. Marsh Fc 8
Fig.. w * B w + * + o 9
Fig. 3. 30
Fig. 4. a distac for all lis omid is Frquc 0 5 0 5 0 Frquc 0 0 0 30 40 50 0.0 0.5.0.5 Distac 0.0 0.5.0.5 Distac 3