Rarefaction and extrapolation with Hill numbers: a framework for sampling and estimation in species diversity studies

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1 Ecologcal Moogahs, 84, 04, Ó 04 by he Ecologcal ocey of Aeca Raefaco ad exaolao wh Hll ubes: a faewo fo salg ad esao seces dvesy sudes ANNE CHAO,,6 NICHOLA J. GOELLI,. C. HIEH, ELIZABEH L. ANDER, K. H. MA, ROBER K. COLWELL, 3,4 AND AARON M. ELLION 5 Isue of ascs, Naoal sg Hua Uvesy, Hs-Chu awa Deae of Bology, Uvesy of Veo, Bulgo, Veo UA 3 Deae of Ecology ad Evoluoay Bology, Uvesy of Coeccu, os, Coeccu 0669 UA 4 Uvesy of Coloado Museu of Naual Hsoy, Boulde, Coloado UA 5 Havad Uvesy, Havad Foes, 34 Noh Ma ee, Peesha, Massachuses 0366 UA Absac. Quafyg ad assessg chages bologcal dvesy ae ceal asecs of ay ecologcal sudes, ye accuae ehods of esag bologcal dvesy fo salg daa have bee elusve. Hll ubes, o he effecve ube of seces, ae ceasgly used o chaaceze he axooc, hylogeec, o fucoal dvesy of a asseblage. Howeve, ecal esaes of Hll ubes, cludg seces chess, ed o be a ceasg fuco of salg effo ad, hus, ed o cease wh sale coleeess. Iegaed cuves based o salg heoy ha soohly l aefaco eolao ad edco exaolao sadadze sales o he bass of sale sze o sale coleeess ad faclae he coaso of bodvesy daa. Hee we exeded evous aefaco ad exaolao odels fo seces chess Hll ube D, whee ¼ 0 o easues of axo dvesy cooag elave abudace.e., fo ay Hll ube D,. 0 ad ese a ufed aoach fo boh dvdual-based abudace daa ad salebased cdece daa. Usg hs ufed salg faewo, we deve boh heoecal foulas ad aalyc esaos fo sealess aefaco ad exaolao based o Hll ubes. Dealed exales ae ovded fo he fs hee Hll ubes: ¼ 0 seces chess, ¼ he exoeal of hao s eoy dex, ad ¼ he vese of so s coceao dex. We develoed a boosa ehod fo cosucg cofdece evals aoud Hll ubes, faclag he coaso of ulle asseblages of boh aefed ad exaolaed sales. he oosed esaos ae accuae fo boh aefaco ad sho-age exaolao. Fo log-age exaolao, he efoace of he esaos deeds o boh he value of ad o he exaolao age. We esed ou ehods o sulaed daa geeaed fo seces abudace odels ad o daa fo lage seces veoes. We also llusae he foulas ad esaos usg ecal daa ses fo bodvesy suveys of eeae foes sdes ad ocal as. Key wods: abudace daa; dvesy; exaolao; Hll ubes; cdece daa; eolao; edco; aefaco; sale coveage; seces chess. INRODUCION he easuee ad assesse of bologcal dvesy bodvesy s a acve eseach focus of ecology Magua 004, Magua ad McGll 0 ad a ceal obecve of ay oog ad aagee oecs Goo e al. 005; Coveo o Bologcal Dvesy [CBD], avalable ole. 7 he sles ad sll he os feuely used easue of bodvesy s he seces chess of a asseblage. eces chess feaues oely foudaoal odels of couy ecology MacAhu ad Wlso 967, Coell 978, Hubbell 00, ad s a ey ec cosevao bology May 988, Boo e al. 003 Mausc eceved Jauay 03; evsed 5 Al 03; acceed May 03. Coesodg Edo: H. Hllebad. 6 E-al: chao@sa.hu.edu.w 7 h:// 45 ad hsocal bogeogahy Wes ad Dooghue 004. I se of s uve ad uvesal aeal, howeve, seces chess s a obleac dex of bodvesy fo wo easos elaed o salg esy ad he seces abudace dsbuo. Fs, obseved seces chess s hghly sesve o sale sze he salg oble. Because os seces a asseblage ae ae, bodvesy sales ae usually colee, ad udeeced seces ae a coo oble. As a coseuece, he obseved ube of seces a well-defed bodvesy sale seces desy; sesu Goell ad Colwell 00 s ow o be a based udeesae of ue seces chess, ad s hghly sesve o he aea suveyed, he ube of dvduals coued, ad he ube of sales scoed fo seces occuece cdece; Colwell ad Coddgo 994. hus, fo a sascal

2 46 ANNE CHAO E AL. Ecologcal Moogahs Vol. 84, No. esecve, seces chess s vey dffcul o esae accuaely fo a fe sale. A secod oble wh seces chess as a easue of bodvesy s ha does o cooae ay foao abou he elave abudace of seces he abudace oble. By coug all seces eually, seces chess weghs ae seces he sae as coo oes. If wo asseblages have decal seces chess, sees uve ha ay subecve sese of dvesy should be hghe he asseblage wh oe-eual abudaces aog all he cooe seces, wheeas dvesy should be lowe he asseblage ha s doaed by he abudace of oe o a few coo seces Pelou 975. Icooag abudace o a bodvesy dex s ccal fo sudes of ay bu o all asecs of ecosyse fuco, because ae seces usually ae lle cobuo o oa easues of ecosyse fuco such as boass, oducvy, o ue eeo chwaz e al O he ohe had, ae seces soees lay ey oles ecosyse fuco e.g., o edaos; ebogh e al. 00 ad ae geeally of geae cosevao ad aagee coce ha ae coo oes May 988, Holsge ad Goleb 99; bu see Gaso ad Fulle 008. A exesve leaue addesses boh of hese ssues. Fo he salg oble, sadadzed coasos of seces chess ca be ade afe eolao wh aefaco e 979 o a coo level of abudace ades 968, Hulbe 97, beloff 97, Goell ad Colwell 00, 0, salg effo Colwell e al. 004, o sale coleeess Aloy 00, Jos 00, Chao ad Jos 0. Aleavely, bodvesy daa ca be used o esae a asyoc esao of seces chess ha s elavely deede of addoal salg effo. Mehods fo obag asyoc chess esaos clude esag he aea beeah a soohed cuve of a aaec seces abudace dsbuo Fshe e al. 943, Coolly ad Doelas 0, exedg he seces accuulao cuve by fg aaec fucos obeo ad Lloee 993, o usg oaaec asyoc chess esaos Chao 984, Colwell ad Coddgo 994 ha ae based o he feuecy of ae seces a sale. Alhough ay ecologss sll ublsh aalyses of aw seces desy daa, aefaco ad asyoc esaos based o sascal salg heoy ae becog sadad ools bodvesy aalyss Goell ad Ellso 0. Colwell e al. 0 ecely ufed he eolao ad exaolao ocedues fo seces chess. hey showed ha a sgle, sooh salg cuve wh a execao ad a ucodoal vaace, deved fo a efeece sale a colleco of dvduals [o salg us] ha would be gaheed a ycal bodvesy suvey ca be eolaed aefed o salle sale szes o exaolaed o a lage sale sze, guded by a esae of asyoc chess. hus, goous sascal coaso of seces chess ca be efoed o oly fo aefed subsales, bu also fo exaolaed chess values based o sales of abay ad eual sze. Chao ad Jos 0 develoed coveage-based aefaco ad exaolao ehodology o coae seces chess of a se of asseblages based o sales of eual coleeess eual coveage. he Colwell e al. 0 sale-sze-based aoach sadadzes based o sale effo, wheeas he Chao ad Jos 0 coveage-based aoach sadadzes based o sale coleeess, a esaed asseblage chaacesc. he sale sze- ad coveage-based egao of aefaco ad exaolao ogehe eese a ufed faewo fo esag seces chess ad fo ag sascal feeces based o hese esaes. Le he salg oble, he abudace oble has bee ecogzed fo decades he ecologcal leaue. Ecologss have oduced a lehoa of dvesy dces ha cobe seces chess ad he ooo of each seces o a sgle ec Washgo 984. hese dces ed o be hghly coelaed wh oe aohe, ae o always exessed us ha ae uve, sesble, o ha allow coasos, ad have salg ad sascal oees ha have bee ooly suded Ghe 99. Hll ubes ae a aheacally ufed faly of dvesy dces dffeg aog heselves oly by a exoe ha cooae elave abudace ad seces chess ad ovecoe ay of hese shocogs. hey wee fs used ecology by MacAhu 965, develoed by Hll 973, ad ecely eoduced o ecologss by Jos 006, 007. Hll ubes offe fve dsc advaages ove ohe dvesy dces. Fs, Hll ubes obey a uve elcao cle o doublg oey. Hll 973 oved a wea veso of he doublg oey: If wo coleely dsc asseblages.e., o seces coo have decal elave abudace dsbuos, he he Hll ube doubles f he asseblages ae cobed wh eual weghs. Chu e al. 03: Aedx B ecely oved a sog veso of he doublg oey: If wo coleely dsc asseblages have decal Hll ubes of ode elave abudace dsbuos ay be dffee, ule he wea veso, he he Hll ube of he sae ode doubles f he wo asseblages ae cobed wh eual weghs. eces chess s a Hll ube wh ¼ 0 ad obeys boh vesos of he doublg oey, bu os ohe dvesy dces do o obey eve he wea veso. A secod advaage of Hll ubes s ha hey ae all exessed us of effecve ubes of seces: he ube of eually abuda seces ha would be eeded o gve he sae value of a dvesy easue. hd, ey dvesy dces oosed he leaue, cludg he wdely used hao eoy ad he G-so dex, ca be coveed o Hll ubes

3 Febuay 04 RAREFACION/ERAPOLAION OF DIVERIY 47 by sle algebac asfoaos. Fouh, Hll ubes ca be effecvely geealzed o cooae axooc, hylogeec, ad fucoal dvesy, ad hus ovde a ufed faewo fo easug bodvesy Chao e al. 00, Goell ad Chao 03. Ffh, he coaso of ulle asseblages, hee s a dec l bewee Hll ubes ad seces coosoal slay o dffeeao aog asseblages Jos 007. hs oey ues dvesy ad slay o dffeeao. Alhough seces chess s oe of he Hll ubes, he leaue o Hll ubes ad o salg odels fo seces chess have develoed deedely. he ece leaue geeally fals o ehasze ha Hll ubes ohe ha seces chess hose wh. 0 ae also sesve o he ube of dvduals o sales colleced, alhough he ude-salg bas s ogessvely less sevee fo Hll ubes of hghe odes of. I heoy, sle aefaco cuves ca be cosuced fo ay dvesy dex by esalg Wale e al. 008, Rcoa e al. 0, alhough oly ecely has hs bee doe exlcly fo Hll ubes Goell ad Ellso 0; R. Colwell, avalable ole. 8 Asyoc esaos fo Hll ubes wh ¼ ae closely elaed o he well-ow eoy esao fo evews, see Pas 003, Chao e al. 03. Fo ¼ ad ay ege., ealy ubased esaos exs Nelse e al. 003, Goell ad Chao 03. I hs ae, we ufy he wo fudaeal faewos used fo he easuee ad esao of seces dvesy: aefaco/exaolao ad Hll ubes. ecfcally, we geealze he sale-szebased aoach of Colwell e al. 0 ad he coveage-based aoach of Chao ad Jos 0 o he ee faly of Hll ubes. We ovde asyoc esaos fo Hll ubes ad use he o l aalyc esaos fo aefaco ad exaolao fo a ecal efeece sale. o chaaceze he seces dvesy of a asseblage, we oose usg hee egaed aefaco/exaolao cuves based o he fs hee Hll ubes: seces chess, he exoeal of hao eoy whch we efe o as hao dvesy, ad he vese so coceao whch we efe o as so dvesy. he foulas ad esaos ae esed wh sulaed daa geeaed fo seces abudace odels/veoes ad aled o seveal ecal daa ses. Fally, we hghlgh he close heoecal ls bewee Hll ubes ad execed seces accuulao cuves Hulbe 97, Dauby ad Hady 0. Wh hs exaded faewo, ecologss wll be able o effecvely use Hll ubes fo a hos of obles bodvesy esao, cludg coaso of he seces dvesy of dffee asseblages e o sace, wh elable sascal feeces abou hese coasos. 8 h://ul.oclc.og/esaes WO YPE OF DAA AND MODEL o descbe odel aaees ad sale daa, we ado he oao ad eology of Colwell e al. 0 ad Goell ad Chao 03. Cosde a seces asseblage cossg of N oal dvduals, each belogg o oe of dsc seces. he oal abudace of seces s N, whee ¼,,...,, N. 0, ad N ¼ P ¼ N. Le ¼ N /N deoe he ue elave abudace of seces, so ha P ¼ ¼. We ehasze ha he uaes N,,N, N,..., N ad,,..., ae he aaees eeseg, esecvely, he ue albe uow udelyg asseblage sze, he colee seces chess of he asseblage, ad he seces absolue ad elave abudace ses. We cosde wo salg daa sucues fo efeece sales. Idvdual-based abudace daa ad odel I os bologcal suveys, a sale of dvduals s ae wh elacee fo he asseblage, ad a oal of obs seces ae obseved. If dvduals ae saled whou elacee, we eed o assue ha he asseblage sze N s uch lage ha he sale sze. Le be he ube of dvduals of he h seces ha ae obseved he sale, ¼,,..., ; we efe o as he sale seces feuecy. Le f be he ube of seces eeseed by exacly dvduals he sale, ¼ 0,,..., ; we efe o f as he abudace feuecy cous. Fo hese defos, ¼ P ¼ ¼ P f, ad obs ¼ P f. I acula, f s he ube of seces eeseed by exacly oe dvdual sgleos he sale, ad f s he ube of seces eeseed by exacly wo dvduals doubleos. he uobsevable feuecy f 0 deoes he ube of seces ese he ee asseblage, bu ae o obseved he sale. he uloal obably dsbuo s he os wdely used odel fo he obseved seces sale feueces,,..., fo gve ad,,..., :! Pð ¼ x ; ; ¼ x Þ¼ x! x! x x x : ðþ Noe ha udeeced seces,.e., ¼ 0, do o cobue o hs dsbuo. I hs odel, he deeco obably fo he h seces s sly he ue elave abudace ¼ N /N. I hs case, he sale sze s fxed. hus, he ube of dvduals eeseed by ay sgle seces s a os, whch s fxed by he salg desg. Aleavely, abudace daa ca also be colleced by salg a fxed aea o by alyg a fxed salg effo, ahe ha a fxed sale sze. Wh hs salg oocol, he sale sze s a ado vaable ad hus cao be fxed advace, lyg ha he ube of dvduals eeseed by ay sgle seces ca be lage, whou ay acula l. A cooly

4 48 used aea-based odel s he Posso oduc odel, whch assues ha dvduals of he h seces accuulae he efeece sale accodg o a Posso ocess Ross 995. hs odel ca be aced o Fshe e al. 943 ad fos he bass fo Colea e al. s 98 ado salg odel fo seces aea elaoshs. As show by Colwell e al. 0, he Posso oduc odel oduces esuls ha ae vually dsgushable fo hose based o a uloal odel. A sascal easo fo hs s ha he Posso oduc odel s closely elaed o a uloal odel; see Chao ad Chu 03 fo deals. heefoe, we cosdeed oly he uloal odel, whch ca accoodae boh dvdual-based ad aea-based abudace daa. ale-based cdece daa ad odel Whe he salg u s o a dvdual, bu a a, e, uada, lo, o ed suvey, s hese salg us, o he dvdual ogass ha ae saled adoly ad deedely. Because s o always ossble o cou dvduals wh a salg u, esao ca be based o a se of salg us whch oly he cdece esece of each seces s ecoded. he efeece sale fo such cdece daa cosss of a se of salg us. he esece o absece echcally, o-deeco of each seces wh each salg u s ecoded o fo a seces-by-salg-u cdece ax W wh ows ad colus. he value of he elee W of hs ax s f seces s ecoded he h salg u, ad 0 f s abse. he ow su of he cdece ax Y ¼ P ¼ W deoes he cdece-based feuecy of seces, ¼,,...,. Hee, Y s aalogous o he dvdual-based feuecy veco. eces ese he asseblage bu o deeced ay salg u yeld Y ¼ 0. he oal ube of seces obseved he efeece sale s obs oly seces wh Y. 0 cobue o obs. Followg Colwell e al. 0, we adoed a Beoull oduc odel, whch assues ha he h seces has s ow uue cdece obably ha s cosa fo ay adoly seleced salg u. Each elee W he cdece ax s a Beoull ado vaable sce W ¼ 0 o W ¼, wh obably ha W ¼ ad obably ha W ¼ 0. he obably dsbuo fo he cdece ax s PðW ¼ w 8; Þ ¼ Y ¼ Y Y ¼ ¼ ¼ w ð Þ w y ð Þ y : ðaþ he odel s euvale o a boal oduc odel fo he obseved ow sus Y, Y,..., Y as follows: ANNE CHAO E AL. PðY ¼ y ; ¼ ; ; :::; Þ ¼ Y ¼ y y ð Þ y : ðbþ Hee he obably of cdece occuece s he obably ha seces s deeced a salg u. I Aedx A, we descbe a oe geeal case uada salg o ee he odel ad exla how hs odel ca cooae saal aggegao. Le Q deoe he cdece feuecy cous, he ube of seces ha ae deeced exacly salg us, ¼ 0,,...,,.e., Q s he ube of seces each eeseed exacly Y ¼ es he cdece ax sale. Hee Q s aalogous o f he abudace daa. he oal ube of cdeces U ecoded he salg us s aalogous o he abudace daa. Hee U s a ado vaable ad ca be exessed as U ¼ P ¼ Q ¼ P ¼ Y, ad he ube of obseved seces s obs ¼ P ¼ Q. Hee, Q eeses he ube of uue seces hose ha ae each deeced oly oe salg u, ad Q eeses he ube of dulcae seces hose ha ae each deeced exacly wo salg us. he uobsevable zeo feuecy cou Q 0 deoes he ube of seces aog he seces ese he asseblage ha ae o deeced ay of he salg us. HILL NUMBER Abudace daa Hll 973 egaed seces chess ad seces abudaces o a class of dvesy easues lae called Hll ubes, o effecve ubes of seces, defed fo 6¼ as D ¼ ¼! =ðþ ð3aþ whch s he ube of seces he asseblage, ad he h seces has elave abudace, ¼,,...,. he aaee deees he sesvy of he easue o he elave feueces. Whe ¼ 0, he abudaces of dvdual seces do o cobue o he su E. 3a. Rahe, oly eseces ae coued, so ha 0 D s sly seces chess. Fo ¼, E. 3a s udefed, bu s l as eds o s he exoeal of he fala hao dex, efeed o hee as hao dvesy: D ¼ l D ¼ ex log!: ð3bþ! ¼ he vaable D weghs seces ooo o he feuecy. Whe ¼, E. 3a yelds so dvesy, he vese of he so coceao s as follows: D ¼ = ¼ Ecologcal Moogahs Vol. 84, No. ð3cþ

5 Febuay 04 RAREFACION/ERAPOLAION OF DIVERIY 49 whch laces oe wegh o he feueces of abuda seces ad dscous ae seces. Ivesgaos usg Hll ubes should eo, a leas, he dvesy of all seces ¼ 0, of ycal seces ¼, ad of doa seces ¼. Fo a geeal ode of, f D ¼ x, he he dvesy s euvale o ha of a dealzed asseblage wh x eually abuda seces, whch s why Hll ubes ae efeed o as effecve ubes of seces o as seces euvales. A colee chaacezao of he seces dvesy of a asseblage wh seces, ad elave abudaces,,..., s coveyed by a dvesy ofle a lo of D vs. fo ¼ 0o ¼ 3 o 4 [beyod hs chages lle]; see o he e sz 995. Alhough Hll ubes fo, 0 ca be calculaed, hey ae doaed by he feueces of ae seces ad have oo sascal salg oees. We hus esced ouselves o he case 0 houghou he ae. A exale of a dvesy ofle s show Fg. a. Hll ubes ca be egaded as he heoecal o asyoc dveses a a sale sze of fy fo whch he ue elave abudaces f,,..., g of each of seces ae ow. Whe sale sze s eleva fo dscusso, we use he oao D o deoe he asyoc Hll ubes. houghou he ae, we use D ad D echageably;.e., D ¼ D. Icdece daa As fa as we ae awae, Hll ubes have bee dscussed oly fo abudace daa ad have o evously bee defed fo sale-based cdece daa. Hee, we oose he followg Hll ubes fo sale-based cdece daa, based o he Beoull oduc odel E. a o euvalely, he boal oduc P odel E. b. Wh ehe of hese wo odels, ¼ ay be geae ha. o we fs oalze each aaee.e., dvde each by he su P ¼ o yeld he elave cdece of he h seces he asseblage. hs elave cdece s assued o be he sae fo ay adoly seleced salg u. Hll ubes of ode fo cdece daa ae defed as: 0 D ¼ ¼ 6 4 ¼ C A =ðþ 0 6¼ : ð4aþ As wh abudace daa, Hll ubes fo cdece daa also eese he heoecal o asyoc dveses whe he ube of salg us s fy. o we also use D ad D echageably;.e., D ¼ D. Hll ubes D fo abudace daa ae based o elave abudaces E. 3a, wheeas Hll ubes D fo cdece daa ae based o elave cdece he asseblage E. 4a. he aaee E. 4a deees he sesvy of D o he elave cdeces. If all he cdece obables,,..., ae decal, he Hll ubes of all odes eual he seces chess of he efeece sale. he Hll ube D fo cdece daa s eeed as he effecve ube of eually feue seces he asseblage fo whch he salg us ae daw. ha s, f D ¼ y, he he dvesy of he asseblage s he sae as ha of a dealzed asseblage wh y seces all of eual obably of cdece. E. 4a yelds seces chess fo cdece daa whe ¼ 0. As wh E. 3b, he l of D as eds o exss ad gves 0 D ¼ l D ¼ ex! ¼ ¼ log ¼ C A ð4bþ whch s eual o he hao dvesy fo cdece daa,.e., he exoeal of hao eoy based o he elave cdeces he asseblage. Whe ¼, E. 4a becoes D ¼ = ð ¼ Þ ¼ ð4cþ whch s he so dvesy fo cdece daa,.e., he vese so coceao based o elave cdeces. By aalogy o he case fo abudace daa, a lo of D vs. coleely chaacezes he seces dvesy of a asseblage wh seces ad cdece obables,,...,. Dvesy accuulao cuve I s well ow ha ecal seces chess vaes wh salg effo ad hus also vaes wh sale coleeess as easued by sale coveage; see ale-sze- ad coveage-based aefaco ad exaolao. heefoe, we ca lo he execed seces chess as a fuco of sale sze hs lo s he fala seces accuulao cuve o as a fuco of sale coveage. he asyoe of hs cuve as sale sze eds o fy s he seces chess he ee asseblage. We ow exed he coce of seces accuulao cuve o he coce of a dvesy accuulao cuve. As dscussed fo abudace daa, he Hll ube of a fxed ode defed E. 3a eeses he asyoc dvesy a a sale sze of fy. Fo o-asyoc dvesy, we defe he execed dvesy D fo a fe sale sze as he Hll ubes based o execed abudace feuecy cous fo a sale of sze, fo whch daa ae foed by aveagg aog sales of sze ae fo he ee asseblage. Maheacal foulas ad sascal esao ae deved he ex seco. ee Dscusso fo he

6 50 ANNE CHAO E AL. Ecologcal Moogahs Vol. 84, No. FIG.. a A dvesy ofle cuve, whch los Hll ubes D as a fuco of ode, 0 3. Hll ubes ae calculaed fo a Zf-Madelbo odel Magua 004 cludg 00 seces wh seces elave abudace ¼ c/, whee c s a cosa such ha P 00 ¼ ¼. he hee sold dos deoe Hll ubes fo ode ¼ 0,, ad. he dvesy ofle cuve s a oceasg fuco of. he sloe of he cuve eflecs he ueveess of seces elave abudaces. he oe ueve he dsbuo of elave abudaces, he oe seely he cuve decles. Fo coleely eve elave abudaces, he cuve s a cosa a he level of seces chess. b ale-sze-based dvesy accuulao cuve, whch los he execed dvesy D as a fuco of sze, ¼ 0,, ad. As sale sze eds o fy, each cuve aoaches D. c Coveage-based dvesy accuulao cuve, whch los he execed dvesy D as a fuco of execed coveage, ¼ 0,, ad. As sale coveage eds o uy, each cuve aoaches D. advaages of ou aoach ove he aleave aoach ha defes he o-asyoc Hll ubes as he aveage Hll ubes ove ay sales of sze ae fo he ee asseblage. Noe ha fo seces chess ad execed sale coleeess see alesze- ad coveage-based aefaco ad exaolao, hese wo aoaches gve decal foulas. Ou defo slaly ca be exeded o defe he execed dvesy D fo salg us ude he odel fo cdece daa. Based o he above defo, ou goal s o cosuc a dvesy accuulao cuve as a fuco of sale sze he ube of dvduals fo abudace daa o he ube of salg us fo cdece daa o sale coleeess. Fo exale, he odel fo abudace daa, we cosdeed he followg focal uesos: Whe a sale of fe sze daw a ado fo he ee asseblage, wha ae he heoecal foulas fo he execed dvesy of ode, D, fo hs sale? he lo D as a fuco of s he sale-sze-based dvesy accuulao cuve. As eds o fy, hese execed dveses aoach D ¼ D as gve E. 3a. A exale of a sale-sze-based dvesy accuulao cuve s gve Fg. b. Fo a sale of sze, wha s he execed sale coleeess, C, fo hs sale? he lo D as a fuco of C s he coveagebased dvesy accuulao cuve. As C eds o uy colee coveage, hese execed dveses also aoach D ¼ D. A exale of a coveage-based dvesy accuulao cuve s gve Fg. c. 3 Gve he daa fo a efeece sale of sze, wha ae he aalyc esaos fo D ad C? Raefaco eolao efes o he case,, wheeas edco exaolao efes o he case.. he egaed sale-sze- o coveage-based aefaco/exaolao salg cuve eeses he esaed dvesy accuulao cuve based o he efeece sale. 4 Whe hee ae ulle asseblages, how do we coae he dveses based o he aefaco/exaolao salg cuves? o aswe hese uesos, we deve he heoecal foulas of D fo ay fe sale sze ad he coesodg aalyc esaos, alog wh he vaaces ad cofdece evals, he ex seco. hus, sale-sze- ad coveage-based dvesy accuulao cuves ca be esaed ad coaed acoss ulle asseblages. Fo cdece daa, sla uesos ad he esao of he dvesy accuulao cuve ca be foulaed. RAREFACION AND ERAPOLAION OF ABUNDANCE DAA UING HILL NUMBER A ew esecve he exeso of he ow well-udesood aefaco ad exaolao of seces chess fo a efeshe, see Aedx B fo abudace daa, ad Aedx C fo cdece daa o he geeal case of Hll ubes s o dec, ad eues a ew esecve, based o abudace feuecy cous wh a dffee sascal faewo. We fs exed he oao f abudace feuecy cous of he efeece sale of sze oa oe geeal case. We defe he abudace feuecy cou f fo ay as he ube of seces eeseed by exacly dvduals a sale of sze. he execed value of he abudace feuecy cou f ca be exessed as follows see Aedx D fo a oof:

7 Febuay 04 RAREFACION/ERAPOLAION OF DIVERIY 5 E½f ðþš ¼ ð Þ ¼ 0; ;...; : ¼ ð5þ Noe ha E[ f 0 ] ¼ P ¼ ð Þ s he execed ube of udeeced seces a sale of sze. Fo he efeece sale of sze, he feuecy f s sly deoed as f, as we defed Abudace daa. o deve he heoecal foula of D, we fs descbe he feuecy cous execed a sale of sze. uose a ado sale of dvduals s ae fo he ee asseblage; we oba a se of abudace feuecy cous fo hs sale, ff ; ¼,..., g. Afe a fe ube of sales of sze have bee ae, he aveage of f fo each ¼,,..., eds o E[ f ], as deved E. 5. he feuecy cous execed a sale of sze ae hus fe[ f ]; ¼,..., g. Accodg o ou foulao, he execed dvesy fo a sale of sze, D, s he se of Hll ubes based o hese execed feueces. Noe ha, fo a sale sze of, he elave abudaces of seces ae sly / hee ae E[ f ] such seces, / hee ae E[ f ] such seces,..., / hee ae E[ f ] such seces. hus, Hll ubes of ode fo a sale of sze ae DðÞ ¼ " # = 3 E½ f ðþš ¼ 6¼ : ð6þ hs foula s vald fo ay sale sze, whch ca be ehe less ha he efeece sale sze o geae ha. heefoe, houghou he ae, he heoecal foulas fo aefaco ad exaolao of Hll ubes efe o E. 6. he secod colu able suazes he foulas fo he secal cases of ¼ 0,,, ad geeal fo.. Aalyc aefaco ad exaolao esaos fo Hll ubes of ode Based o a efeece sale of sze wh sale feuecy fo he h seces ad he obseved feuecy cous f ¼ f, we deve hee he aalyc esaos ˆD fo D gve E. 6. he oao ha o a dvesy, e.g., ˆD, eas a esao of ha dvesy based o he efeece sale. he obseved Hll ubes, D obs, fo he efeece sale ae sly ˆD. ha s, " # =ðþ ˆDðÞ ¼ D obs ¼ ð =Þ " # =ðþ ¼ ð =Þ f ð7þ ¼ : Ou aoach o devg aefaco foulas s based o sascal esao heoy fo he execed feuecy cous. he u vaace ubased esao fo E[ f ] s ^f ðþ ¼ ¼ f, : ð8þ ee Aedx D fo a oof. Hee, a [ 0 b f a, b. We use hs coveoal defo houghou hs ae ad he aedces. By subsuo fo E. 6, we ca oba he followg aalyc esaos of he execed dvesy of a eolaed sale of sze as follows: " # ˆDðÞ ¼ = 3 ^f ðþ, : ð9aþ ¼ E. 9a s he geeal, ealy ubased, aefaco foula fo Hll ubes of ay ode. A esao s ealy ubased f s bas eds o zeo whe he efeece sale sze s lage. he aalyc esao fo he aefaco of Hll ubes fo each of he odes ¼ 0,, ad s hus obaed by elacg E[ f ] wh ˆf he secfc foulas ovded able. he exaolao of Hll ubes of ay ode s a edco of he execed dvesy D þ fo a augeed sale of sze ¼ þ. Alhough he geeal foula E. 6 fo D also holds fo ay sale sze., ou esao ˆf E. 8 s vald oly fo,. heefoe, we cao sly elace E[ f ] by ˆf as we dd fo aefaco. Fo each, we eed o develo a dffee aoach fo exaolao by eas of a esae of seces chess fo ¼ 0, hao dvesy fo ¼, so dvesy fo ¼, ad hghe odes fo.. eces chess ¼ 0. Fo Es. 6 ad 9a, he seces chess ¼ 0 fo a sale of sze, s 0 ˆDðÞ ¼ ¼ ^f ðþ, : ð9bþ I Aedx D we show ha hs esao s decal o he adoal dvdual-based aefaco esao Hulbe 97, h ad Gassle 977. Ou ew esecve, howeve, offes a sle, aleave aoach o adoal dvdual-based aefaco of seces chess. E. 9b shows ha he adoal aefaco esao of he execed seces chess fo a sale sze of s sly he su of he esaed

8 5 ANNE CHAO E AL. Ecologcal Moogahs Vol. 84, No. heoecal foulas ad aalyc esaos fo aefaco ad exaolao of abudace-based Hll ubes of ode ¼ 0, ¼, ¼, ad ay ege ode., gve a efeece sale wh he obseved Hll ubes D obs ad esaed coveage Ĉd. ABLE. Ode/coveage heoecal foulaà fo all. 0 Ieolao esao fo, ¼ 0 0 DðÞ ¼ E½f 0 ðþš ¼ 0 E½f ðþš ^DðÞ ¼ ^f ðþ ¼ obs ¼ ¼ u vaace ubased " ¼ DðÞ ¼ex log # " 3 E½f ðþš ^DðÞ ¼ex log # 3 ^f ðþ ¼ DðÞ ¼ ¼ ¼ 3 E½f ðþš ^DðÞ ¼ ¼ ealy ubased ¼ ealy ubased 3 ^f ðþ " #. DðÞ ¼ = " # 3 E½f ðþš ^DðÞ ¼ = 3 ^f ðþ ealy ubased Coveage feuecy cous. hs dea ca be exeded easly o Hll ubes of ay odes, as we ex llusae. he exaolaed seces chess esao fo a sale of þ used hs ae s evewed Aedx B, ad he foula ogally deved by he e al. 003 s show able. hs aoach eues a esaed asyoe of seces chess. Ay oe seces chess esao ca be used. Colwell e al. 0 suggesed usg he Chao esao Chao 984 o abudace-based coveage esao ACE; Chao ad Lee 99 ad oed ha exaolao gves elable esaes oly u o aoxaely double o le he efeece sale sze. hs lao s aly a coseuece of he fac ha he asyoc esao s oly a lowe boud Chao 984. hao dvesy ¼. Fo Es. 6 ad 9a, we have he followg ealy ubased eolao esao fo he Hll ube ¼ hao dvesy: ¼ C d ðþ ¼ ¼ ð Þ " ˆDðÞ ¼ex log # ^f ðþ ¼, : ð0aþ Fo ou ew exaolao foula, we eed a esao fo he asyoe of hao dvesy. Chao e al. 03 deved he followg ealy ubased esao, ^H, of hao eoy H ¼ H ¼ P ¼ log as follows: ^Hð Þ ¼ ¼ ¼ ^C d ðþ ¼ u vaace ubased Noes: he las ow gves euaos fo sale coleeess as a fuco of sale sze. I also gves he coesodg coveage esaos fo aefed sales ad exaolaed sales fo coveage-based aefaco ad exaolao cuves. Fo he efeece sale, he obseved Hll ube of ode s ˆD ¼ D obs ¼½ P ð =Þ Š =ðþ. he coveage of he efeece sale s esaed by Ĉd ¼ f /f[ f ]/[ f þ f ]g; see E. he subseco ale-sze- ad coveage-based aefaco ad exaolao. à he feuecy cou f s defed as he ube of seces eeseed by exacly dvduals/es a sale of sze. he foula fo E[ f ] s gve E. 5 he subseco A ew esecve. A ubased esao ˆf fo E[ f ] exss fo, ad s gve E. 8 he subseco Aalyc aefaco ad exaolao esaos fo Hll ubes of ode. } Whe eds o fy, each edco eds o he esao of he asyoc dvesy: 0 ˆD ¼ obs þ ˆf 0, whee ˆf 0 s a edco fo f 0 Chao 984; see E. B.5 Aedx B. ˆD ¼ ex[^h ], whee ^H s a eoy esao develoed by Chao e al. 03; see E. 0b he subseco hao dvesy ¼. Fo a ege, ˆD ¼½ P ðþ = ðþ Š =ðþ, whee x ¼ xx.... x þ deoes he fallg facoal; see Goell ad Chao 03. # he lg ube of he secod d, w,, s defed by he coeffce he exaso x ¼ P ¼ wð; Þx. þ f ð AÞþ logðaþ ð AÞ ¼ ð0bþ

9 Febuay 04 RAREFACION/ERAPOLAION OF DIVERIY 53 ABLE. Exeded. Exaolao esao} fo a sale of sze þ 0 ^Dð þ Þ ¼ obs þ ^f 0 4 f ^f 0 þ f whee A ¼ f /[ f þ f ]. As a esul, he asyoc esao fo hao dvesy s ˆD ¼ ex[ ^H ]. he exaolaed esao fo hao dvesy of a sale of sze þ s as follows: " ˆDð þ Þ ¼ex þ ¼ log þ þ ^Hð Þ ð0cþ : Deals of he devao ae ovded Aedx E. Exesve sulaos Chao e al. 03 sugges ha he asyoc hao esao E. 0b s ealy ubased, lyg he exaolao ovded by E. 0c s vald fo a wde edco age. hs exaolao ca be safely exeded o he asyoe. so dvesy ¼. he geeal foula fo he execed so dvesy fo ay sale sze fo boh, ad. s DðÞ ¼ ¼ elable f, " ^Dð þ Þ ¼ex þ ealy ubased ¼ ¼ ^Dð þ Þ ¼ þ þ þ þ ealy ubased# 3 E½ f ðþš þ ¼! : ðaþ ee Aedx D fo oofs. A u vaace ubased esao of P ¼ s P ¼ ð Þ/[ 3 5 log # þ þ ^Hð Þ ¼ ð Þ ð Þ " # ^Dð þ Þ ¼ wð; Þð þ Þ ð Þ ð Þ = ð þ Þ ¼ ealy ubased ^C d ð þ Þ ¼ f ð Þf ð Þf þ f elable fo, ð Þ þ ] Good 953, lyg ha a esao fo he asyoc so dvesy s ˆD ¼ ˆD ¼ / P. A eolaed esao, fo E. a ca be exessed wo euvale fos: ˆDðÞ ¼ ¼ ˆf ðþ ¼ þ ¼ : ð Þ ð Þ ðbþ We ca aly E. a o a augeed sze of þ ad oba he followg exaolaed esao: ˆDð þ Þ ¼ þ þ þ þ ¼ : ð Þ ð Þ ðcþ he aefaco, exaolao, ad asyoc esaos ae all ealy ubased. hs eas ha fo ¼, he exaolao ca be safely exeded o he asyoe. Dvesy of ege ode.. Fo Hll ubes of ege ode., a ealy ubased eolao esao s gve E. 9a. A geeal exaolao esao ˆD þ s ue colcaed ad s show he las colu able see Aedx E fo devao deals. A ealy ubased esao of he ue asyoc value D ¼ D fo ay ege. s ˆD ¼½ P ðþ = ðþ Š =ðþ Goell ad Chao 03, whee x deoes he fallg facoal x x...x þ. able suazes, fo abudace daa, all heoecal foulas ad aalyc esaos fo aefaco ad exaolao of Hll ubes of ode ¼ 0,, ad ay ege ode.. he las ow of able also gves he foulas fo sale-coleeess as a fuco of sale sze; see he ex subseco. We esed ou esaos o sulaed daa geeaed fo seveal seces abudace odels ad o daa fo lage ecal daa ses Aedx F. he esuls show ha he oosed aalyc aefaco ad exaolao esaos ach efecly wh he coesodg heoecal values fo aefed ad exaolaed sales u o double he efeece sale sze. Howeve, whe he exaolaed sale sze s oe ha double he efeece sale sze, he efoace of ou edcos deeds o exaolaed age ad he ode see Dscusso. hee ae wo ds of vaace assocaed wh a eolaed o exaolaed esao. A vaace ha s codoal o he efeece sale easues oly he vaao dvesy ha would ase fo eeaedly esalg whou elacee he gve efeece sale. hs codoal vaace aoaches zeo as aoaches because he dvesy of sale sze of s fxed.e., hee s oly oe cobao of all dvduals

10 54 ANNE CHAO E AL. Ecologcal Moogahs Vol. 84, No. o all salg us. A ucodoal vaace easues he vaao dvesy ha would ase f aohe ew sale of sze wee ae fo he ee asseblage ahe ha fo he ogal efeece sale. heefoe, he ucodoal vaace does o aoach 0 whe sale sze eds o, ad all assocaed cofdece evals ae syec, whch eflecs he uceay of he ew sale. I devg a ucodoal vaace, he ube of udeeced seces us be esaed because hose udeeced seces also affec he vaao of a ew sale. I os alcaos, ucodoal vaace s oe useful because feeces ae o esced o he efeece sale. Colwell e al. 0 obaed a ucodoal aalyc vaace esao fo aefed ad exaolaed seces chess esaos. Howeve, exedg hs aalyc aoach fo vaace esaos o a geeal ode of becoes aheacally acable. heefoe, we sugges a sle, boosa ehod Aedx G, o oba ucodoal vaaces ad cofdece evals fo all aefed ad exaolaed esaos. I he oosed ocedue, we follow Colwell e al. 0 ad use he Chao fo abudace daa o Chao fo cdece daa o esae he ube of udeeced seces he efeece sale Chao 984, 987, alhough ay ohe oe esaos ca also be used. he exales Woed exales: coaso of asseblages llusae ou oosed salg cuves ad he assocaed cofdece evals based o he ucodoal vaace fo ou oosed boosa ehod. ale-sze- ad coveage-based aefaco ad exaolao I coag dveses aog ulle asseblages, sales ca be sadadzed by sale sze o by sale coleeess. Ou oosed sale-sze-based salg cuve fo Hll ubes of each secfc ode cludes he aefaco a whch los ˆD as a fuco of, whee, ; see able ad he exaolao a whch los ˆD þ as a fuco of þ fo. 0; see able ad yelds a sooh salg cuve, he wo as of whch o soohly a he o of he efeece sale, D obs. o fully cooae he effec of elave abudace o dvesy esao, we sugges log cuves fo a leas he fs hee Hll ubes ¼ 0,,. Whe hee ae ay vsble seces seces wh exeely sall elave abudace ha ae alos udeecable oal salg schees ou uo s ha he ube of udeeced seces sales o euvalely, seces chess he ee asseblage s vey had o esae; see Colwell e al. 0 ad Goell ad Chao 03 fo a evew. O he ohe had, ad coay o uo, he oo of sale coleeess ca be accuaely ad effcely esaed usg oly foao coaed he efeece sale self. ale coleeess ca be easued by sale coveage o sly coveage, a coce ogally develoed by he foude of ode coue scece, Ala ug, ad I. J. Good Good 953, 000, Good ad oul 956; accodg o Good [000], ug eve ublshed hs wo, bu gave esso o Good o ublsh. Coveage s defed as he oal elave abudaces of he obseved seces, o euvalely, he ooo of he oal ube of dvduals a asseblage ha belog o seces eeseed he sale. ug ad Good Good 953, 000, Robbs 968, Esy 983, 986 deved a sle coveage esao of he efeece sale of sze as oe us he ooo of sgleos. Robbs 968 showed ha he aveage suaed eo of hs esao /. A y eceage of coveage ca coa a fe ube of ae seces. he esaed colee of coveage s o a esae of he ube of usee seces, bu ahe esaes he ooo of he oal dvduals he asseblage ha belog o udeeced seces. Fo hs easo, exeely ae, udeeced seces do o ae a sgfca cobuo o ha ooo, eve f hee ae ay such seces. hs uvely exlas why he esao of seces chess hghly dvese asseblages s a sascally dffcul ssue, wheeas sale coveage ca be accuaely esaed. Aloy 00 ad Jos 00 deedely oosed ha sales be sadadzed o a coo level of sale coleeess as easued by sale coveage, ad develoed algohc aoaches fo coag aefed sales. Fo seces chess, Chao ad Jos 0 wee he fs o deve a aalyc ehod fo sealess coveage-based aefaco ad exaolao. Chao ad Jos 0 suggesed log aefaco ad exaolao cuves wh esec o sale coveage ahe ha wh esec o sale sze because he execed seces chess fo eual sale coveage sasfes a elcao cle o doublg oey, whch he execed seces chess fo eual sale sze does o obey. la coclusos ae vald fo he execed dvesy of ay ode ; see Aedx D fo deals. hs oey aes ossble o uafy ao coasos o ay ohe coasos bewee he agudes of he dveses of he asseblages. Fo dvdual-based abudace daa, Chao ad Jos 0 used a oe accuae sale coveage esae fo he efeece sale, as follows: ^C d ðþ ¼ f ð Þf : ðþ ð Þf þ f hey also deved a eolaed coveage esao Ĉ d fo ay aefed sale of sze, ad exaolaed coveage esao Ĉd þ fo ay augeed sale of sze þ ; see able las ow fo he foulas. he exaolaed coveage esao s elable f,. As wh sale-sze-based cuves, fo ay secfc ode, he coveage-based eolao [whch los ˆD wh esec o Ĉd] ad exaolao whch

11 Febuay 04 RAREFACION/ERAPOLAION OF DIVERIY 55 los ˆD þ wh esec o Ĉd þ o soohly a he efeece o Ĉd, D obs. he cofdece evals of execed dvesy based o he boosa ehod also o soohly. Bdgg sale-sze- ad coveage-based aoaches he sale-sze-based aoach los he esaed dvesy as a fuco of sale sze, wheeas he coesodg coveage-based aoach los he sae dvesy wh esec o sale coveage. heefoe, hese wo aoaches ca be bdged by he elaosh bewee coveage ad sale sze. Usg he coveage esaos ables ad he las ow each able, we ca cosuc a sale coleeess cuve, whch eveals sale coleeess fo a gve sale sze. Fo he ogal efeece sale, hs cuve esaes sale coleeess fo salle afed sales, as well as fo lage exaolaed sales. hs cuve also ovdes a esae of he sale sze eeded o acheve a fxed degee of coleeess. A oal sog heoy deved by Rasusse ad a 979 secfes ha salg sos whe sale coveage eaches a edeeed value. he sale coleeess cuve hus ovdes foao abou whehe we should coue o so salg. If ulle asseblages ae o be saled ad coaed, Chao ad Jos 0 suggesed ha ecologss should sale each asseblage o he sae degee of coleeess. uch eually colee sales fo dffee asseblages ca be coaed decly, whou ay eed fo aefaco o exaolao. ee Woed exales fo llusao. RAREFACION AND ERAPOLAION OF INCIDENCE DAA UING HILL NUMBER Fo cdece daa, aallel devaos o hose fo abudace daa yeld euaos fo he heoecal execed dveses fo ay sale sze he secod colu able. Hee, sale sze fo cdece daa eas ube of salg us. he aalyc esaos fo aefed sales ad aalyc esaos fo exaolaed sales ae also gve able. he asyoc esao fo each ode of Hll ubes ¼ 0,, s ovded he foooes of able. Full devao deals alog wh a elcao cle aea Aedx H; hee we hghlgh he followg dffeeces fo he odels ad esaos fo abudace daa. Fs, fo abudace daa, ou devao was based o a odel whch he seces feuecy follows a boal dsbuo chaacezed by ad he ue elave abudace. I coas, fo cdece daa, we assue he seces cdece-based feuecy Y follows a boal dsbuo chaacezed by he oal ube of salg us ad cdece obably. Wh abudace daa, P ¼ ¼, bu wh cdece daa, P ¼ ca exceed. ecod, he oal ube of dvduals a efeece sale of abudace daa s fxed by desg. I coas, oal ube of cdeces a efeece sale of salg us s a ado vaable U, wh execao EU ¼ P ¼. heefoe, P ¼ ca be accuaely esaed by U/. Fo abudace daa, he ube of dvduals ay aefed o exaolaed sale sze s fxed, wheeas fo cdece daa, he ube of cdeces ay salg us s ado wh execao EU ¼ P ¼, whch s esaed by Uˆ ¼ U/. hd, fo abudace daa, he ay devaos of ou esaos ae based o feuecy cous f, he ube of seces eeseed by exacly dvduals o obseved es a sale of sze. he coesodg cdece feuecy cou s Q, he ube of seces ecoded exacly salg us a sale of salg us. he sale coleeess cuve as a fuco of abudace, develoed by Chao ad Jos 0, s evewed Aedx B, ad he foulas aea he las ow of able. I Aedx C we deve, fo he fs e, he coesodg sale coleeess cuve as a fuco of salg us fo cdece daa. Wh such a cuve, ecologss ca obecvely uafy he sale coleeess fo ay colee abudace o cdece daa ses. hese cuves hel deee a sale sze eeded a desgg a suvey. All foulas ae suazed he las ow of able. Based o he foulas able, fo each ode of he Hll ubes D fo cdece daa, we ca oba a egaed aefaco/exaolao salg cuve wh cofdece evals. ascal feece heoy les ha he oosed eolaed esao fo dvesy s ubased fo ¼ 0 ad ealy ubased fo ¼ ad. We suo hese clas wh sulao ess Aedx F. As wh he abudace daa, he efoace of ou exaolaed esaos deeds o he ode of Hll ubes ad he edco age of exaolao. ulao ess ovde soe geeal usage gudeles see Dscusso. I ables ad, we suaze he oees ad efoace of each dex based o heoy ad aalyses of ecal ad sulaed daa ses. WORKED EAMPLE: COMPARION OF AEMBLAGE Exale : Abudace daa coag sde seces dvesy wo eaes ace e al. 0 ovded seces abudace daa fo sales of sdes fo fou exeeal foes caoy-aulao eaes a he Havad Foes. he eaes wee esablshed o sudy he log-e coseueces of loss of he doa foes ee, ease heloc suga caadess, caused by a oave sec, he heloc woolly adelgd Adelges sugae; Ellso e al. 00. Daa fo wo eaes ae used hee fo llusao: he heloc-gdled eae, whch ba ad cabu of heloc ees wee cu

12 56 ANNE CHAO E AL. Ecologcal Moogahs Vol. 84, No. he heoecal foulas ad aalyc esaos fo aefaco ad exaolao of Hll ubes based o cdece daa fo ¼ 0, ¼, ¼, ad ay ege ode., gve a efeece sale wh he obseved Hll ubes D obs ad esaed coveage Ĉsale. ABLE. Ode/coveage heoecal foula fo all. 0à Ieolao esao, Y ¼ 0 DðÞ ¼ E½Q 0 ðþš ¼ 0 E½Q ðþš ^DðÞ ¼obs ¼ Y u vaace ubased ¼ " DðÞ ¼ex log # 3 E½Q ðþš U U ¼ " ^DðÞ ¼ex ^U log ^U # 3 ^Q ðþ ¼ ealy ubased ¼ DðÞ ¼ 3 E½Q ðþš ^DðÞ ¼ 3 ^Q ðþ ealy ubased ¼ U ¼ ^U. " # " DðÞ ¼ = 3 E½Q ðþš ^DðÞ # = ¼ 3 ^U ^Q ðþ ealy ubased U ¼ ¼ Coveage C sale ðþ ¼ ð Þ ¼ ¼ Y Y ^C sale ðþ ¼ ealy ubased U Y Noes: he las ow gves euaos fo sale coleeess as a fuco of sale sze, ad he coesodg coveage esaos fo aefed sales ad exaolaed sales. ee Aedx C fo ¼ 0 ad Aedx H fo. 0 fo oao ad all devao deals. Fo he efeece sale, he obseved Hll ube of ode s D obs ¼½ P ¼ ð=uþ Q Š =ðþ. he coveage of he efeece sale s esaed by Ĉsale ¼ Q /Uf[ Q ]/[ Q þ Q ]g. U ¼ P Y. 0 Y ¼ P ¼ Q deoes he oal ube of cdeces sales. à Fo ay sale sze of, Q s defed as he ube of seces deeced exacly salg us. U s defed as he execed oal ube of cdeces salg us: U ¼ P ¼ E½Q ðþš ¼ P ¼. A ubased esao ˆQ fo E[Q ] exss fo, ad s gve E. H.5 Aedx H. A ubased esao fo U s Uˆ ¼ P ¼ ^Q ðþ¼u/. } Whe eds o fy, each edco eds o he esao of he asyoc dvesy: 0 ˆD ¼ obs þ ˆQ 0, whee ˆQ 0 s a edco fo Q 0 Chao 987; see E. C.5 Aedx C. ˆD ¼ ex[^h sale ], whee ^H sale s a eoy esao fo cdece daa; see E. H.7 Aedx H. Fo a ege, ˆD ¼½ P Y Y ðþ =ðu ðþ ÞŠ =ðþ, whee x ¼ xx.... x þ. ee able fo he defo of he lg ube of he secod d: w,. ad he ees lef lace o de o c ee oaly by adelgd fesao; ad he heloc-logged eae, whch heloc ees wee cu ad eoved fo he los Ellso e al. 00. he abudace feuecy daa fo he wo eaes sued ove wo los e eae ae abulaed able 3, ad he a abudace dsbuos ae show Fg.. We used he daa fo hese wo eaes o llusae he cosuco of wo yes sale-sze- ad coveage-based of aefaco ad exaolao cuves of Hll ubes. he cosuced salg cuves wee he used o coae sde seces dveses bewee he wo eaes. he efeece sale sze ube of dvdual sdes fo he gdled eae was 68, ad he obseved seces chess, hao dvesy, ad so dvesy.e., Hll ubes fo ¼ 0,, fo hs efeece sale sze wee, esecvely, 6,.06, ad 7.84 sold os Fg. 3a ad b. he sale sze fo he logged eae was 5, ad he coesodg obseved Hll ubes fo ¼ 0,, wee 37, 4.4, ad 6.76, esecvely. hus, udgg fo he usadadzed aw daa he efeece sales, he logged eae aeas o have hghe obseved seces chess ad hao dvesy, bu lowe so dvesy ha he gdled eae. e : Coae sale-sze-based salg cuves u o a base sale sze Fg. 3. We fs cosuced, fo each of he wo eaes, he egaed sale-szebased aefaco ad exaolao cuves fo Hll

13 Febuay 04 RAREFACION/ERAPOLAION OF DIVERIY 57 ABLE. Exeded. Exaolao esao fo þ salg us} 0 ^Dð þ Þ ¼obs þ ^Q ^Q 0 þ Q elable f, " ^Dð þ Þ ¼ex þ ealy ubased ¼ ubes of ¼ 0,,. I Fg. 3a, we show hese salesze-based cuves wh 95% cofdece evals based o a boosa ehod. We exaolaed u o double he efeece sale sze.e., u o sze 336 fo he gdled eae ad sze 504 fo he logged eae. I each lo, exce fo al, sall sale szes, oe of he cofdece evals fo he hee cuves esec, ad he a ode of dvesy s seces chess. hao dvesy. so dvesy. Fo ay fxed sale sze o coleeess he coaso age, f he 95% cofdece evals do o ovela, he sgfca dffeeces a a level of 5% aog he execed dveses whehe eolaed o exaolaed ae guaaeed. Howeve, aally ovelag evals do o guaaee osgfcace chee ad Gelea 00. he cuve fo seces chess ¼ 0 ceases seely wh sale sze boh eaes, bu he cuves fo hao ad so dvesy ¼ ad ¼ level off beyod he efeece sale, llusag ha hghe ode Hll ubes ae ceasgly doaed by he feueces of he oe coo seces ad ae, heefoe, less sesve o salg effecs. o coae dveses bewee he gdled ad logged eaes, we show Fg. 3b, fo each fxed value of ¼ 0,, ad, he sale-sze-based aefaco ad exaolao of hese wo los wh 95% cofdece evals u o a base sale sze. We sugges he base sale sze o be double he salles efeece sale Q! 3 5 Y U log Y # þ U þ ^H sale ð Þ ^Dð þ Þ ¼ þ 3 þ þ U= þ Y. 0 ealy ubased " ^Dð wð; Þð þ Þ ðþ þ Þ ¼ ðu=þ ð þ Þ ealy ubased ^C sale ð þ Þ ¼ Q U ¼ elable fo, ð ÞQ ð ÞQ þ Q Y ðy Þ U ð =Þ Y ðþ Y ðþ þ # = sze o he axu efeece sale sze, whcheve s lage he easo fo ou suggeso wll becoe cleae he secod exale. ee Box fo syseac ses o deee a base sale sze. I hs exale, he base sale sze s 336 double he salle efeece sale sze. he esaed Hll ubes ca he be coaed acoss asseblages fo ay sale sze less ha he base sze. I a adoal aefaco, he daa fo he logged eae would be aefed o a sale sze of 68 dvduals o ach he abudace he gdled eae. Fo hs aefed sale, he Hll ubes of ¼ 0,, ae esaed o be 3.7, 3.83, ad 6.68, esecvely. he oosed egaed salg cuve allows elable coasos fo ay sale sze u o a abudace of 336. Acoss hs age of abudace, Fg. 3b eveals ha he logged eae s oe dvese fo all bu he salles sale szes fo seces chess ¼ 0 ad hao dvesy ¼, alhough he cofdece evals ovela. I coas, fo so dvesy ¼, he gdled eae s oe dvese, alhough aga he wo cofdece evals ovela. e : Cosuc a sale coleeess cuve o l sale-sze- ad coveage-based salg cuves Fg. 4. Based o E., he coveage fo he gdled eae s esaed as 93% fo he efeece sale of sze 68 dvduals, ad he coveage fo he logged eae s 94% fo he efeece sale of 5 dvduals. I s foave o exae how he sale coleeess vaes wh sale sze see he foulas he las ow able. I Fg. 4, we lo he sale coleeess cuve as a fuco of sale sze fo each of he wo eaes, u o double he efeece sale sze. Fo ay sale sze less ha 68, he cuve shows ha he sale coleeess fo he gdled eae s esaed o be hghe ha ha logged eae, alhough he cofdece evals ovela. Whe sale sze s lage ha 68, he esaes of sale coveages de seces abudace feuecy cous wo caoy aulao eaes Ellso e al. 00, ace e al. 0. ABLE 3. Gdled Logged f f Noe: he daa clude as of, f whee f efes o he ube of seces eeseed by exacly dvduals. Fo he gdled eae, obs ¼ 6 seces, ¼ 68 dvduals; fo he logged eae, obs ¼ 37 seces, ¼ 5 dvduals.

14 58 ANNE CHAO E AL. Ecologcal Moogahs Vol. 84, No. FIG.. Ra abudace dsbuos fo sde daa fo he gdled ad logged eaes of ease heloc suga caadess a a sudy fo he Havad Foes, Peesha, Massachuses, UA. I he gdled eae, ba ad cabu of heloc ees wee cu ad he ees lef lace o de o c ee oaly by adelgd fesao, ad he logged eae, heloc ees wee cu ad eoved fo he los. he oooal abudace o he y-axs o a logahc scale s calculaed as he ooo of he axu abudace. fo he wo eaes dffe lle. If we aly a adoal aefaco aoach o sadadze sale coveage, a sale sze of ;68 dvduals he logged eae gves a sale coveage of 93%. hus, he dvesy odeg of he wo eaes fo 93% of he asseblage dvduals s he sae as ha fo a sadadzed sale of 68 dvduals. he sale coleeess cuve fgue ovdes a bdge bewee sale-sze- ad coveage-based salg cuves, as wll be exlaed he ex se. e 3: Coae coveage-based salg cuves u o a base coveage Fg. 5. Fo he sale coleeess cuve Fg. 4, whe sale sze he gdled eae s doubled fo 68 o 336 dvduals, he sale coveage s ceased fo 93% o 96%. I he logged eae, whe sale sze s doubled fo 5 o 504 dvduals, he coveage s ceased fo 94% o 97%. I Fg. 5a, we ese, fo each eae, he coesodg coveage-based aefaco ad exaolao cuves wh 95% cofdece evals fo dvesy of ¼ 0,, whe he coveage s exaolaed o he value fo a doublg of each efeece sale sze. I Fg. 5b, we coae he coveage-based dveses of he wo eaes fo ¼ 0 lef ael, ¼ ddle ael, ad ¼ gh ael u o he coveage of 96%. hs s ou base coveage he lowes coveage fo doubled efeece sale szes o he axu coveage fo efeece sales, whcheve s lage. ee Box fo suggesos o he choce of base coveage. Because he cease coveage fo he exaolao s sall, ad he esaed dvesy fo ¼ ad hadly chage beyod he efeece sales, he exaolao as Fg. 5b ae ealy vsble fo hese wo odes of. ce he wo cofdece bads do o esec fo seces chess ¼ 0 f coveage exceeds 50% Fg. 5b, lef ael, seces chess he logged eae s sgfcaly hghe ha he gdled eae fo ay sadadzed sale coveage bewee 50% ad 96%. Fo hao dvesy ¼, he logged eae s oe dvese, bu he cofdece bads ovela. Fo so dvesy ¼, whe coveage s less ha 70%, boh eaes have alos he sae dvesy, bu whe coveage s geae ha 70%, he so dvesy fo he gdled eae s slghly hghe. Coag Fgs. 3b ad 5b, we see ha he salesze- ad coveage-based cuves fo ¼ 0 ad ¼ exhb cosse dvesy odegs bewee he wo eaes. Howeve, fo ¼, he sale-sze-based cuves do o esec Fg. 3b, bu he coveage-based cuves have wo cossg os Fg. 5b. ee Dscusso fo oe coasos of he wo yes of cuves. Exale : Icdece daa coag seces dvesy of ocal as aog fve ses We used he ocal a seces daa colleced by Logo ad Colwell 0 fo fve elevaos o he Bava asec, a 30- couous gade of we foes o Cosa Rca s Alac sloe. he fve ses ae, esecvely, a elevaos of 50, 500, 070, 500, ad 000. eces esece o absece was ecoded each salg u, whch cossed of all woe as exaced fo a - foes floo lo. ee Logo ad Colwell 0 fo salg ad daa deals. A sale-by-seces cdece ax was oduced fo each of he fve ses. he cdece feuecy cous ae gve Colwell e al. 0: able 6. he los fo a feuecy dsbuos of he fve ses ae show by Logo ad Colwell 0: Fg. 3. A egaed aefaco ad exaolao cuve fo seces chess was eseed by Colwell e al. 0: Fg. 4b. hey cocluded ha seces chess aog he fve ses was sgfcaly dffee oe of he cofdece evals esec, exce fo vey sall szes, ad ha chess has he odeg: e : Coae sale-sze-based salg cuves u o a base sale sze Fg. 6. Fo each of he fve ses, Fg. 6a shows he obseved Hll ubes ad salesze-based aefaco ad exaolao los wh 95% cofdece evals fo hee salg cuves Hll ubes of ¼ 0,, u o double he efeece sale sze. o coae dvesy aog he fve elevaos, we fs deeed he base sale sze. he efeece sale szes ube of salg us fo each elevao 50, 500, 070, 500, ad 000 ae, esecvely, 599, 30, 50, 00, ad 00. he base sale sze would be 599 whch s lage ha 3 50 ¼ 300, double he salles efeece sale; see Box fo he choce of hs base sale sze. A advaage of hs

15 Febuay 04 RAREFACION/ERAPOLAION OF DIVERIY 59 FIG. 3. a ale-sze-based aefaco sold les ad exaolao dashed les, u o double he efeece sale sze of sde seces dvesy based o he Hll ubes ¼ 0,, fo he heloc gdled eae ad he logged eae. he 95% cofdece evals gay-shaded egos wee obaed by a boosa ehod based o 00 elcaos. Refeece sales ae deoed by sold dos. he ubes aeheses ae he sale sze ad he obseved Hll ubes fo each efeece sale. b Coaso of sale-sze-based aefaco sold les ad exaolao dashed cuves, u o he base sale sze of 336 dvduals.e., double he salle efeece sale sze of sde seces dvesy fo Hll ubes of ode ¼ 0 lef ael, ¼ ddle ael, ad ¼ gh ael. Refeece sales each eae ae deoed by sold dos. he ubes aeheses ae he sale sze ad he obseved Hll ubes fo each efeece sale. choce of base sale sze s ha o daa ae excluded fo ou aalyss. Howeve, a dawbac s ha he exaolao age fo soe sales could exceed he doubled efeece sale szes. Fo hao ad so dveses, he edco bases ae al beyod he double efeece sale szes, bu fo seces chess such cases we should be cauous abou he edco bas. ee Dscusso fo suggesos o exaolao age. Nex fo each secfc ode of, we lo he salesze-based eolao ad exaolao cuves wh 95% cofdece bads fo hese fve elevaos ogehe he sae fgue, as llusaed Fg. 6b. Exaolaos ae exeded o he base sale sze of 599 fo all ses. Ou lo of ¼ 0 coesods o Fg. 4b Colwell e al. 0. We hee exed he aoach o clude cuves fo ¼ ad ¼, ad also clude coveagebased los. Fo he hee odes of, dvesy of he ses s cossely odeed as Fg. 6b. All cofdece evals ae oovelag exce fo vey sall szes, lyg he dvesy of ay ode ¼ 0,, s sgfcaly dffee aog he fve elevaos fo ay fxed sale sze u o 599 salg us. e : Cosuc a sale coleeess cuve o l sale-sze- ad coveage-based salg cuves Fg. 7. he sale coveages fo he fve ses 50, 500, 070, 500, ad 000 wee esaed as 99.8%, 97.60%, 98.39%, 98.89%, ad 99.64%, esecvely, dcag ha salg s ealy colee fo all ses. A suay of coveage esaos fo cdece daa aea able. ee Aedx C fo esao deals. Fo ay fxed sale sze,300 salg us, he sale coveage fo he wo lowe elevaos 50 ad 500 s sgfcaly lowe ha coveage a hghe elevaos 070, 500, 000. Whe sale sze s geae ha 300, he ae esss, bu he 95% cofdece bads beg o ovela.

16 60 ANNE CHAO E AL. Ecologcal Moogahs Vol. 84, No. BO. yseac ses o deee base sale sze fo he sale-sze-based aefaco ad exaolao, ad base coveage fo he coveage-based aefaco/exaolao. Exale s used o llusae each se. he efeece sale sze fo he h sale s deoed by, ¼,,...,, ad he coesodg sale coveage esae s deoed by C. Fo abudace daa, sale sze efes o he ube of dvduals; fo cdece daa, sale sze efes o he ube of salg us. e 0. e he axu exaolaed ao, eual o he ao of he exaolaed sale sze ad he efeece sale sze. Fo ag feeces abou seces chess ¼ 0, we sugges he axu exaolaed sze should be double he efeece sale sze, ha s, ¼. Fo feeces fo dvesy of, ca be ay osve ube,.e., s sascally safe o exaolae o he asyoe. a ale-sze-based aefaco/exaolao e. Coue he axu efeece sale sze, a ¼ axf,,..., g. I Exale, a ¼ axf599, 30, 50, 00, 00g¼599. e. Coue he u es efeece sale szes, b ¼ f,,..., g. I Exale fo ¼, b ¼ f98, 460, 300, 400, 400g¼300. e 3. he suggesed base sale sze s he axu of a ad b, base ¼ axf a, b g. I Exale fo ¼, base ¼ axf599, 300g¼599. b Coveage-based aefaco/exaolao e. Coue he axu coveage of efeece sale szes, C a ¼ axfc, C,..., C g. I Exale, C a ¼ axf0.998, 0.976, , , g¼ e. Coue he u coveage of es efeece sale szes, C b ¼ fc, C,..., C g. I Exale fo ¼, C b ¼ f0.9968, , , , g¼ e 3. he suggesed base coveage s he axu of C a ad C b, C base ¼ axfc a, C b g. I Exale fo ¼, C base ¼ axf0.9964, g¼ e 3: Coae coveage-based salg cuves u o a base coveage Fg. 8. Fg. 8a shows, fo each lo, he coesodg coveage-based aefaco ad exaolao cuves fo Hll ubes of ¼ 0,, whe he coveage s exaolaed o he value fo a doublg of each efeece sale sze. Fo he sale coleeess cuve Fg. 7, whe he sale sze each se s doubled, he sale coveage ceases vey slghly fo all ses. hee s lle chage a dvesy fo ¼ ad. hus, he exaolaed oos of he cuves Fg. 8a ae ealy vsble, as we also oed Fg. 5. Whe all sale szes ae doubled, he u value of he coveage values of hese doubled sale szes aog he fve ses s 99.08% fo 500 elevao. Howeve, s less ha he coveage 99.64% of he efeece sale fo 000 elevao. I ode o use all daa, we selec ou base coveage o be 99.64% Box. Fg. 8b coaes coveage-based aefaco ad exaolao cuves u o he base coveage of 99.64%. All hee coveage-based dveses show he sae odeg by elevao as he sale-sze-based coaso. Noe of he cofdece evals ovela exce a vey sall coveage values, lyg sgfca dffeeces a dvesy aog he fve elevaoal asecs a coaable coveage. FIG. 4. Plo of sale coveage fo aefed sales sold le ad exaolaed sales dashed le as a fuco of sale sze fo sde sales fo he heloc gdled ad logged eaes. he 95% cofdece evals wee obaed by a boosa ehod based o 00 elcaos. Refeece sales ae deoed by sold dos. Each of he wo cuves was exaolaed u o double s efeece sale sze. he ubes aeheses ae he sale sze ad he esaed sale coveage fo each efeece sale.