Cautionary notes on the descriptive analysis of performance curves in reptiles

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1 Journl of Therml Biology 3 (26) Cutionry notes on the desriptive nlysis of performne urves in reptiles Gre gory Bulte, Griel Blouin-Demers Deprtment of Biology, University of Ottw, 3 Mrie-Curie, Ottw, Ont., Cnd KN 6N5 Reeived 2 Otoer 25; epted 29 Novemer 25 Astrt Orgnisml performne urves re importnt funtions for the study of reptilin eology nd evolution, ut their interprettion n e ffeted strongly y the hoie of nlytil pproh. We first use n exmple from the literture to demonstrte tht exluding iologilly meningful dt lters the desription of performne nd leds to non-sensil inferenes. We then use fitionl dt to show tht liner models (ANOVA) ommonly used in the desriptive nlysis of performne urves n lso e iologilly misleding or n lk iologil relevne. Our exmples demonstrte tht fitting non-liner urves to performne dt is more meningful nd voids erroneous representtion nd interprettion of these importnt iologil funtions. r 26 Elsevier Ltd. All rights reserved. Keywords: Performne urve; ANOVA; Optiml temperture; Therml retion norm; Reptiles. Introdution Therml sensitivity in reptiles is studied rodly (Huey, 982; Angillett et l., 22). The reltionship etween ody temperture (T ) nd performne (i.e., the therml retion norm) is often used to evlute nd predit the effets of environmentl temperture on reptile eology (Try nd Christin, 986; Huey, 99; Wetherhed nd Roertson, 992; Angillett et l., 22) nd to study evolutionry ptterns in therml physiology (Huey nd Bennett, 987; Huey nd Kingsolver, 989; Angillett et l., 22; Blouin-Demers et l., 23; Zhng nd Ji, 24). Approprite desription nd nlysis of the reltionship etween T nd performne is therefore prmount to interpret properly evolutionry nd eologil trends in therml eology nd evolutionry physiology. Over 25 yers go, Huey nd Stevenson (979) pulished disussion of pprohes to study therml sensitivity in etotherms. One of their gols ws to stimulte reserh in whole niml performne. In reent yers, severl Corresponding uthor. Tel.: ; fx: E-mil ddress: glouin@uottw. (G. Blouin-Demers). lortory studies hve quntified the therml retion norms of orgnisml trits in reptiles (Ji et l., 993; Witz nd Lwrene, 993; Sriner nd Wetherhed, 995; Ji et l., 996; Du et l., 2; Angillett et l., 22; Blouin- Demers et l., 23; Chen et l., 23; Elsworth et l., 23; Zhng nd Ji, 24). Despite Huey nd Stevenson s (979) omprehensive disussion, however, we elieve tht the desriptive nlysis of therml retion norms of reptiles in reent ppers often provides inomplete informtion nd, in some ses, is iologilly misleding. Therefore, our im here is to demonstrte the limittions of urrent methods nd to suggest lterntives tht irumvent those limittions. We will first omment nd re-nlyze study of therml sensitivity of ppetite in lizrds (MConnhie nd Alexnder, 24) in whih misoneption of the therml retion norm hs led to inpproprite dt nlysis. Seond, we will demonstrte tht the pproh most ommonly used to nlyze therml sensitivity in reent studies, ANOVA with T s tegoril independent vrile nd performne s ontinuous dependent vrile, n led to misrepresenttion of the iologil relity /$ - see front mtter r 26 Elsevier Ltd. All rights reserved. doi:.6/j.jtherio.25..3

2 288 ARTICLE IN PRESS G. Bulté, G. Blouin-Demers / Journl of Therml Biology 3 (26) Re-nlysis of MConnhie nd Alexnder (24) MConnhie nd Alexnder (24) mesured ppetite s the mss nd numer of melworms onsumed y Cordylus lizrds during 4-dy trils t T of 2, 22, 25, 32, nd 35 C. Fig. 3 nd of their rtile illustrte those reltionships nd show tht ppetite grdully inreses etween 2 nd 32 C, ut dereses rpidly etween 32 nd 35 C. Therefore, the reltionship etween ppetite nd T is urviliner. The uthors, however, ignored the 35 C trils in their nlyses nd fitted liner regression to the dt. Their rgument for this exlusion ws tht lizrds were showing signs of distress t 35 C nd were eting little, thus foring the rrest of the trils fter 3 dys insted of 4. The exlusion of the trils t 35 C from the nlyses onstitutes n importnt oneptul mistke leding to misrepresenttion of the iologil relity. By rejeting the 35 C trils, the uthors ignore the nturl shpe of the T - performne funtion. Performne of etotherms is n symmetri (skewed to the left) funtion of temperture defined y lower nd n upper ritil temperture (CT min nd CT mx : the tolerne rnge), n optimum temperture (T o ) nd performne redth (e.g., rnge t whih performne is X8%) (Huey, 982; Stevenson et l., 985; Angillett et l., 22). CT min nd CT mx re the miniml nd mximl T t whih performne is still possile nd T o is the T t whih performne is mximized. Therefore, the derese oserved t 35 C is norml iologil response when T pprohes CT mx nd nnot e ignored in performne nlysis if one wnts to get the est mthemtil representtion of the iologil relity. Doing otherwise nd using liner regression suggests tht ppetite inreses ontinuously with T, even t T lethl for the speies. When ll ppetite responses to T re inluded, the reltionship is lerly urviliner nd, therefore, the most pproprite desription is non-liner eqution suh s those presented y Stevenson et l. (985) nd Huey (982). We extrted the dt from Fig. 3 nd of MConnhie nd Alexnder (24) using the softwre GrphClik (version 2.4.3) nd renlyzed the dt inluding the 35 C trils. We first fitted liner regression to ompre the fit of this line to the one otined y MConnhie nd Alexnder tht exluded the 35 C trils. We then fitted the non-liner eqution provided y Stevenson et l. (985) Performne ¼ S = þ K e K2 ðt CT min Þ e K3ðT CT mx Þ, where CT mx,ct min, nd T re defined s ove nd K, K 2,ndK 3 re onstnts nd S is sling ftor. CT mx nd CT min were unville from MConnhie nd Alexnder (24), ut they ould e estimted from their dt. We estimted CT mx t 38 C euse lizrds were showing signs of distress suffiient to justify the rrest of Mss of Melworms Ingested (g) R 2 =.88 (p =.2) Body Temperture ( C) R 2 =.56 (p =.84) Fig.. Mss of melworms onsumed y Cordylus m. melnolotus s funtion of ody temperture (dt otined from MConnhie nd Alexnder (24)). The urve is the fitted non-liner performne urve. The solid line is the regression line of the originl study (exluding the 35 C dt). The dshed line is the regression line inluding the 35 C dt. When inluded in the regression, the 35 C trils mke the reltionship non-signifint. the experiment t 35 C nd CT min ws set t 2 C sed on the distriution of the dt nd euse very little food onsumption ourred t 2 C. Liner regressions inluding the 35 C trils led to different sttistil onlusions thn the originl regressions. Both reltionships eme non-signifint (numer of worms: R 2 ¼ :49, p ¼ :2 versus R 2 ¼ :9, p ¼ :; mss of worms: R 2 ¼ :56, p ¼ :84 versus R 2 ¼ :88, p ¼ :2, Fig. ) outlining tht performne is not liner funtion of temperture. On the other hnd, the non-liner urve fitted the dt well. The men devition of the predited vlues from the tul vlues ws of.85 worms for the numer of worms ingested nd of.6 g for the mss of worm ingested (Fig. ). The mjor prolem with the initil nlytil pproh is tht it implies tht performne inreses monotonilly with T. Therefore, one would onlude, sed on their reltionship, tht performne is greter t 4 C thn t 3 C, even if 3 C is the optiml temperture. Moreover, the uthors did not speify tht their strong positive reltionship ws restrited to the 2 32 C rnge. The resoning for eliminting dt from the nlysis ws tht the 35 C trils were shorter. In this se, however, the shorter durtion ws norml iologil response to inresed temperture. Hd the uthors ontinued the trils for 4 dys (s for the others T s), the lizrds would hve een unlikely to inrese their food onsumption euse of therml stress. 3. Shortflls of the ANOVA for desriing therml performne urves In the ove exmple, the nlytil prolem stemmed from misoneption of the reltionship etween T nd

3 G. Bulté, G. Blouin-Demers / Journl of Therml Biology 3 (26) performne. Although this exmple is n exeption in the therml performne literture, we elieve tht the nlytil pproh most ommonly used to desrie performne in reptiles (ANOVA) n lso provide n erroneous representtion of the therml retion norm. ANOVA is often used to nlyse the effet of T on performne in reptiles (Ji et l., 993, 996; Du et l., 2; Angillett et l., 22; Chen et l., 23; Elsworth et l., 23; Zhng nd Ji, 24). In generl, the performne of severl individuls is mesured t vrious T (usully 4 6) enompssing the rnge of T experiened y the niml (i.e., within the CT). The men performne (response vrile) t eh experimentl T (ftor) is then ompred using ftoril ANOVA nd post-ho multiple omprisons tests. If the vrition in performne within T is signifintly less thn the vrition etween T, then the men performnes etween the two (or more) T re onsidered to e sttistilly different. If there is no sttistil differene in men performne etween two or more onseutive T, then performne is onsidered to e equl etween those T. T o is thus onsidered to e ounded y onseutive nd sttistilly similr T t whih performne is mximized (Huey nd Stevenson, 979; Angillett et l., 22). A rnge inluding T o, rther thn single vlue, is thus otined. We elieve tht this pproh to desriing performne urves n e iologilly misleding. Before we use n exmple to illustrte how ANOVA n e misleding in the desription of performne urves, we need to identify two hrteristis of performne studies tht should e tken into ount in ANOVA. First, T is ontinuous vrile. In stndrd ANOVA design, however, the ftor (i.e., experimentl T ) is onsidered n unordered tegoril vrile. At the very lest, T should e onsidered n ordered tegoril vrile if ANOVA is to e used to desrie performne urves. Seond, in performne studies the sme individuls re usully mesured t eh T. Therefore, the pproprite nlysis is repeted mesures ANOVA (Potvin et l., 99; Angillett et l., 22). The mjority of reent studies we exmined, however, used stndrd ANOVA insted of repeted mesures ordered ANOVA. In ddition, ANOVA n produe results tht re not representtive of the rel retion norm. We illustrte how with fitionl dt tht we generted from tul performne urves. We generted our dt from swimming speed dt of northern wtersnkes (Nerodi sipedon) (Blouin-Demers et l., 23). We first reted popultion A in whih there ws little vrition in T o nd mximum performne ttined. From this popultion, we reted two senrios of inter-individul vrition. In popultion B, we introdued vrition in the mximum performne ttined (Y xis only) y vrying the sling ftor (S) ofstevenson et l. s (985) eqution so tht mximum performne rnged etween,75 nd,95 (undefined units of performne), ut the generl shpe of the urve remined unffeted. Consequently, individuls vry in the mximum performne they n hieve, ut they ll perform est t the sme T (Fig. 2). In popultion C, we reted vrition in T o (X xis only) y vrying K, K 2, K 3, or ll three t the sme time. Therefore, ll individuls hieved the sme mximum vlue, ut t different T (introduing vrition in T o ). In ll ses, we ssumed tht CT remined the sme ross ll individuls euse men CT re used to fit the urves in most studies. In theory, however, CT ould lso vry etween individuls nd this would produe more vrition in T o. We then ompred T o determined with ANOVA to T o otined with non-liner urve fitting. For ANOVA, we used the testing tempertures s the ftor (ordered) nd performne s the response in repeted mesures model. We used Tukey s HSD test to identify the T o rnge. Then, we fitted non-liner urve for eh individul with the eqution desried ove nd lulted the men T o. We lso determined the men rnge of the 8% performne redth, whih is inditive of the width of the performne plteu. For popultions B nd C, there ws no signifint differene etween 25 nd 35 C with the ANOVA (Fig. 2). The iologil interprettion would e tht T o is ounded y those T for oth popultions. With the urve fitting pproh, the men T o ws 3.7 C (7.3) for popultion B nd 3.9 C (7 2.) for popultion C. Those vlues re lose to one nother nd fll in the middle of the T o rnge depited y the ANOVA, ut the ANOVA does not tell us tht those two senrios represent popultions tht hve very distint ptterns of inter-individul vrition. On the other hnd, the error term ssoited with the men T o of the urve fitting pproh tells us tht T o is vrile in popultion C, ut not in popultion B. A mjor prolem with the ANOVA pproh is tht the T o rnge depends on the inter-individul vrition in performne t eh experimentl T. Consequently, the greter the inter-individul vrition in performne (suh s in popultion B), the wider the T o rnge. This eomes pprent when we ompre popultions A nd B (Fig. 2A, B) tht hve the sme T o, ut popultion B hs more individul vrition in performne. In popultion A, ll experimentl T re signifintly different from one nother. Therefore, one would interpret T o to e ner 3 C euse it is the experimentl T t whih performne is sttistilly greter thn t ny other T. For popultion B, however, the T o rnge is etween 25 nd 35 C euse performne t those T is not signifintly different sed on ANOVA. This inferene is erroneous euse performne t 25 C in popultion B is not equl to performne t 3 C (T o ), it is 2% lower. In some speies, suh s wter snkes, 5 C devition from T o dereses performne y 2% (Blouin-Demers et l., 23). Compring popultions A nd B using ANOVA led to the onlusion tht popultion A hs nrrow T o rnge nd popultion B hs wide T o rnge when their therml retion norms re identil; they only differ in the vrition in individul mximum performne. The wide

4 29 ARTICLE IN PRESS G. Bulté, G. Blouin-Demers / Journl of Therml Biology 3 (26) T o rnge in popultion B is onsequene of the vrition in individul mximum performne ttined. With suh vriility, the ANOVA will lwys led to wide T o Performne (A) (B) (C) f e d rnge. Lrge individul vriility is expeted espeilly when performne is not normlized (Huey nd Stevenson, 979) (i.e., expressed s perentge of individul mxim) s it is the se in severl reent studies (Ji et l., 993, 996; Du et l., 2; Chen et l., 23; MConnhie nd Alexnder, 24; Zhng nd Ji, 24). Normlizing y individul ontrols for ftors tht systemtilly influene solute performne (suh s size or sex) nd, thus, elimintes overll inter-individul vriility. Another prolem with the ANOVA pproh, originlly reognized y Huey nd Stevenson (979), rises when the tul T o is vrile in the popultion, suh s in popultion C. In popultion C, individul T o rnge from 29.5 to 34.2 C. ANOVA leds to the onlusion tht the popultion hs rod T o rnge when, in relity, T o is vrile (Fig. 2C). For popultion C, T o otined y urve fitting is lso misleding, ut the error term ssoited with T o is wrning out vriility nd, therefore, llows utious interprettion of the iologil signifine of T o in this popultion. Moreover, vriility is n importnt iologil metri euse it is one requirement for nturl seletion to t on trit, nd vrine in T o informs out individul vriility. 4. Conluding remrks We hve shown tht, with ANOVA, similr iologil interprettions n e otined despite different ptterns of individul vrition. It hs een rgued tht the ANOVA pproh is onservtive when the performne urve shows wide plteu euse T o must e inluded in the rnge. We elieve, however, tht wide rnge of T o does not provide muh iologil informtion. Moreover, it is often impossile from ANOVA tles nd plots to evlute wht ptterns of vrition led to wide T o rnge (i.e., vrition in T o or vrition in mximum performne). Men T o nd performne redth with their error terms re etter representtions of the therml retion norm of speies nd re essentil to ompre performne sttistilly for different groups or for different speies. For instne, Zhng nd Ji (24) ompred loomotor nd digestive performne etween three Tkydromus fter seprte ANOVA for eh speies, ut their omprisons hd to remin qulittive. Hd Zhng nd Ji (24) fitted urves to the dt for eh individul, they would hve een le to ompre men T o nd performne redth sttistilly etween the three speies Body Temperture Fig. 2. Performne urves of three hypothetil popultions. Popultion A hs little inter-individul vrition in mximum performne or optiml temperture. Popultion B hs inter-individul vrition in mximum performne, ut no vrition in optiml temperture. Popultion C hs little inter-individul vrition in mximum performne, ut hs interindividul vrition in optiml temperture. The grey lines re the individul urves nd the dshed line is the popultion men. The error rs indite one stndrd devition in performne t the ody tempertures used in the ANOVA.

5 G. Bulté, G. Blouin-Demers / Journl of Therml Biology 3 (26) If from the shpe of the urve it is ler tht T o rnge rther thn single T o vlue is more resonle for speies, one n esily lulte speified performne redth (e.g., 99% or 95%) from urve (or, lterntively, from the polygon method proposed y vn Berkum, 986). Suh redth is more esily interpretle iologilly thn rnge otined from post-ho test following ANOVA. The ANOVA pproh n e informtive out the vriility in performne t given T, whih my e of iologil interest lso. In suh se, repeted mesures design with ordered ftors ANOVA must e used. When the gol is to desrie performne urves, however, ANOVA should e voided. Huey nd Stevenson (979) stressed tht non-liner urves should e fitted to performne dt whenever possile. We reiterte their suggestion. Stevenson et l. (985) nd Huey (982) provide severl exmples of urves tht n e used to desrie therml sensitivity. In our exmple, we used logisti exponentil urve fitting pproh euse it fitted our dt well. We do not ontend tht this form of eqution is neessrily the est for ll situtions. Polynomil urve fitting or the minimum onvex polygon method (vn Berkum, 986) n lso e used to extrt desriptive sttistis from therml performne urves. The ltter polygon method is good lterntive when urve nnot e run through severl points. In ddition to voiding erroneous iologil interprettions nd loss of informtion (e.g., due to n unordered ANOVA design), the pprohes suggested here re more onsistent with the onept of therml retion norm euse performne is treted s ontinuous funtion of temperture. Finlly, if the gol of the study is eyond the generl desription of the urve, more elorte methods hve een proposed reently (Izem et l., 23; Izem nd Kingsolver, 25). Those pprohes llow quntifying the vrition in the shpe of performne urves nd prtitioning environmentl from geneti effets. Aknowledgments Funding for this study ws provided y the University of Ottw, Disovery Grnt from the Nturl Sienes nd Engineering Reserh Counil of Cnd to GBD nd grdute studies sholrship from the Fonds Que eois de Reherhe sur l Nture et les Tehnologies to GB. Referenes Angillett, M.J., Hill, T., Roson, M.A., 22. Is physiologil performne optimized y thermoregultory ehvior?: se study of the estern fene lizrd, Seloporus undultus. J. Therm. Biol. 27, Angillett, M.J., Niewirowski, P.H., Nvs, C.A., 22. The evolution of therml physiology in etotherms. J. Therm. Biol. 27, Blouin-Demers, G., Wetherhed, P.J., MCrken, H.A., 23. A test of the therml odpttion hypothesis with lk rt snkes (Elphe osolet) nd northern wter snkes (Nerodi sipedon). J. Therm. Biol. 28, Chen, X.J., Xu, X.F., Ji, X., 23. Influene of ody temperture on food ssimiltion nd loomotor performne in white-striped grss lizrds, Tkydromus wolteri (Lertide). J. Therm. Biol. 28, Du, W.G., Yn, S.J., Ji, X., 2. 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