Sakamoto, M., Venditti, C., & Benton, M. J. (2017). 'Residual diversityestimates' do not correct for sampling bias in palaeodiversity data.Methods in Ecology and Evolution, 8(4), 453–459.https://doi.org/10.1111/2041-210X.12666

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1

‘Residualdiversityestimates’donotcorrectforsamplingbiasin1

palaeodiversitydata2

3

SHORTTITLE:Donotuseresidualsmethod4

5

WORDCOUNT:3,6356

7

ManabuSakamoto1,ChrisVenditti1andMichaelJ.Benton28

9

1SchoolofBiologicalSciences,UniversityofReading,Reading,RG66AJ,UK10

2SchoolofEarthSciences,UniversityofBristol,Bristol,BS81RJ,UK11

12

EMAIL:m.sakamoto@reading.ac.uk13

14

15

2

ABSTRACT16

1. Itiswidelyacceptedthatthefossilrecordsuffersfromvarioussampling17

biases–diversitysignalsthroughtimemaypartlyorlargelyreflectthe18

rockrecord–andmanymethodshavebeendevisedtodealwiththis19

problem.Onewidelyusedmethod,the‘residualdiversity’method,uses20

residualsfromamodelledrelationshipbetweenpalaeodiversityand21

sampling(sampling-drivendiversitymodel)as‘corrected’diversity22

estimates,buttheunorthodoxwayinwhichtheseresidualsaregenerated23

presentsseriousstatisticalproblems;theresponseandpredictor24

variablesaredecoupledthroughindependentsorting,renderingthenew25

bivariaterelationshipmeaningless.26

2. Here,weusesimplesimulationstodemonstratethedetrimental27

consequencesofindependentsorting,throughassessingerrorratesand28

biasesinregressionmodelcoefficients.29

3. Regressionmodelsbasedonindependentlysorteddataresultin30

unacceptablyhighratesofincorrectandsystematically,directionally31

biasedestimates,whenthetrueparametervaluesareknown.Thelarge32

numberofrecentpapersthatusedthemethodarelikelytohave33

producedmisleadingresultsandtheirimplicationsshouldbereassessed.34

4. Wenotethatthe‘residuals’approachbasedonthesampling-driven35

diversitymodelcannotbeusedto‘correct’forsamplingbias,andinstead36

advocatetheuseofphylogeneticmultipleregressionmodelsthatcan37

includevariousconfoundingfactors,includingsamplingbias,while38

simultaneouslyaccountingforstatisticalnon-independenceowingto39

sharedancestry.Evolutionarydynamicssuchasspeciationareinherently40

3

aphylogeneticprocess,andonlyanexplicitlyphylogeneticapproachwill41

correctlymodelthisprocess.42

KEYWORDS43

Palaeodiversity;residuals;modeling;samplingbias;fossilrecord;independent44

sorting 45

4

INTRODUCTION46

IthasbeenwellknownsincethetimeofDarwinthatthefossilrecordislargely47

incomplete(Darwin1859),promptinggenerationsofmacroevolutionary48

researcherstotakeacautiousapproachwheninterpretingpatternsof49

palaeodiversitythroughtime(Raup1972;Raup1976;Raup1991;Prothero50

1999;Smith&McGowan2007;Alroy2010b).Therehavebeenmanyattemptsto51

accountforthissamplingbias(Raup1972;Raup1976;Smith&McGowan2007;52

Alroy2010b),butoneapproachinparticular,oftenreferredtoasthe‘residual53

diversity’method,devisedbySmithandMcGowan(2007)(andmodifiedby54

Lloyd(2012)),hasbeenwidelyused(citationcount~215toAug2016;Google-55

Scholar).56

57

Usingregressionresidualsasdata‘corrected’forconfoundingfactorsisawidely58

usedmethodinbiology,socialsciences,economics(King1986;Freckleton59

2002),andeveninpalaeodiversitystudies(Raup1976).However,Smithand60

McGowan’s(2007)approachdiffersfromtheseclassicalresidualsapproachesin61

onekeyway:the‘residuals’aregeneratednotasregressionresiduals(ε=y-ŷ)62

fromasimpleregressionofdiversity(y)onaproxyofsampling(x),butfrom“a63

modelinwhichrockareaatoutcropwasaperfectpredictorofsampleddiversity”64

(Smith&McGowan2007),herereferredtoasthesampling-drivendiversity65

model(SDDM).TheSDDMisconstructedasaregressionmodelbetweenysorted66

fromlowtohighvalues(y’)andxsortedfromlowtohighvalues(x’),wherethe67

relationshipbetweenthesetwoindependentlysortedvariablesy’andx’is68

assumedtorepresenttheSDDgeneratingprocess–thoughthereisnoreasonto69

assumeassuch.‘Residuals’areobtainedasthedifferencebetweentheSDDM70

5

predictionsŷ’andtheobservedvaluesy,whicharethentreatedasthe‘residual71

diversityestimates’(figure1).72

73

However,independentlysortingyandxasoutlinedabovedecouplesapaired,74

bivariatedataset,andisobviouslyproblematicinstatistics.Modelfittingon75

decoupleddata(e.g.y’andx’)willleadtospuriouspredictionsand‘residuals’as76

theestimatedregressioncoefficientswillbebasedonaforced(false)linear77

relationship(figure1b).However,owingtocontinuedwideuseoftheSDDMasa78

preferredmethodforidentifyingsupposedly‘true’palaeodiversitysignals(as79

recentlyas(Grossnickle&Newham2016)),itappearsthatthisbasicstatistical80

conceptissomehowoverlooked.Whileithasbeensuggestedthattheuseof81

formationcountsto‘correct’palaeodiversitytimeseriesdataisunlikelytobe82

meaningfulbecauseofsubstantialredundancyofthetwometrics(Bentonetal.,83

2011;Benton2015),andarecentstudyhasscrutinizedtheperformanceof84

SDDMresidualsinaccuratelypredictingtruesimulatedbiodiversitysignals85

(Brocklehurst2015),theperformanceoftheSDDMitselfhasneverbeen86

assessed.Here,wedemonstratethedetrimentaleffectsofdecouplingdatain87

regressionmodellingusingsimplesimulations.88

89

90

MATERIALANDMETHODS91

Wefirstgeneratedrandomdeviates,x,samplingfromanormaldistribution(μ=92

0,σ=1),atasamplesizen=100(seeSIforothersamplesizesn=30and1000).93

Wethencalculatedyusingalinearrelationshipintheformofy=a+bx+e,94

whereaistheintercept,bistheslopeandeisGaussiannoise.Forsimplicity,we95

6

fixeda=0.4andb=0.6,whilevaryinge(μe=0,σe=0.05,0.1,0.25,0.5)–other96

valuesofaandbshouldreturnsimilarifnotidenticalresults(though,b=197

wouldbemeaningless).FollowingSmithandMcGowan(2007),wesortedyandx98

independentlyofeachothertogeneratey’andx’,andfittedanordinaryleast99

squares(OLS)regressionmodeltoy’onx’(SDDM).Forcomparison,wefittedan100

OLSregressionmodeltoyonxintheiroriginalpairedbivariaterelationship(the101

standardregressionmodel,SRM),theperformanceofwhichservesasa102

benchmark.103

104

TotestSmithandMcGowan’s(2007)assertionthattheSDDMisindeed“amodel105

inwhichrockareaatoutcropwasaperfectpredictorofsampleddiversity”,we106

evaluatedwhethertheestimatedregressioncoefficientsαandβsignificantly107

differedfromthetrueregressionparameters,aandb,usingat-test.Werepeated108

theprocedureover5000simulationsandcalculatedthepercentageoftimesthe109

estimatedcoefficientsdifferedsignificantlyfromthetrueparameters.Wewould110

expectabout5%ofthesimulationstoresultinregressioncoefficients111

significantlydifferentfromthetrueparametersbychancealone;anything112

substantiallyabovethisthresholdwouldindicatethatthemodelhas113

unacceptablyhighTypeIerrorratesorfalselyrejectingatruenullhypothesis,114

whereournullhypothesisisthattheSDDMcancorrectlyestimatethe‘true’115

modelparameters.116

117

Inaddition,wetestedforbiasintheestimatedregressionslopes,i.e.whetherthe118

estimatessystematicallydeviatedfromthesimulationparameterb=0.6.The119

meanofthe5000slopeswassubjectedtoat-testagainstafixedvalueof0.6.If120

7

deviationswererandom,thenwewouldnotexpecttofindanysignificant121

differencesbetweenthemeanslopeandthetheoreticalvalue,withallslopes122

randomlydistributedaroundit.123

124

125

RESULTS126

SRMcoefficientsweresignificantlydifferentfromthetruemodelparametersin127

only~5%ofthe5000iterationsacrossσe(figure2a;table1;SI),within128

acceptablelevelsofrandomlydetectingastatisticalsignificance.Variationin129

regressionlinesacross5000iterationsaredistributedrandomlyaboutthe130

simulatedline(figure3a),withnosignificantdifferencebetweenthemean131

regressionslopeandthesimulationparameterb=0.6(table2;SI).Incontrast,132

SDDMcoefficientsweresignificantlydifferentfromthetrueparameters(figure133

2b)ataratemuchhigherthantheconventionallyaccepted5%(table1;SI).The134

meanslopeoftheregressionmodelssignificantlydifferedfromthesimulation135

parameterb,inasystematicallyanddirectionalmanner(figure3b;table2;SI)–136

SDDMregressioncoefficientsarenotonlyincorrectbutgrosslymisleading.This137

systematicbiasincreaseswithincreasednoiseinthedata(table2)–themore138

noisethereisinthedata,themorepositivetherelationshipbetweeny’andx’139

becomes.140

141

142

DISCUSSION143

Byestablishing“amodelinwhichrockareaatoutcropwasaperfectpredictorof144

sampleddiversity”,SmithandMcGowan(2007)attemptedtocreateasampling-145

8

drivendiversitymodel.However,theirSDDMisnotbasedonanyhypothesized146

orempiricalrelationshipbetweendiversityandsampling,orformulatedfrom147

firstprinciples.Thisisincontrasttootherwell-formulatedbiologicalmodels148

suchasvariousscalingmodelswheretheparameterofinterest(i.e.scaling149

coefficientortheslopeofthebivariaterelationship)isfoundedonfirst-principle150

theories,e.g.the2/3ruleforthescalingofareawithmass.Rather,theSDDMis151

basedontheassumptionthaty’andx’(yandxsortedindependentlyofeach152

other)formtheexpectedtheoreticalbivariaterelationshipbetweenyandx,153

whichthisstudyshowstobeincorrect(figures2,3),asonewouldexpectsince154

thereisnoreasontoassumesuchathing.155

156

Afurtherandperhapsmoreseriousproblemwithusingaforcedpairingofy’and157

x’isthateachdatapoint(pairofy’iandx’i)doesnotrepresentanaturalpairing158

andhasnomeaning;thenewpairingisactuallyyiandxj,wheretheithandjth159

ordersareindependentofeachother.Forinstance,usingthemarinegeneric160

diversityandrockareadataofSmithandMcGowan(2007)(figure4),thelowest161

marinegenericdiversityisintheCambrian,TommotianStage(529–521million162

yearsago[Ma];genuscount=309),whilethesmallestmarinerockoutcroparea163

(afterremoving0valueddata(Smith&McGowan2007))isfromtheEarly164

Permian,Asselian/SakmarianStage(299–290Ma;rockarea=1).Similarly,the165

highestdiversityisrecordedforthePliocene(5.3–2.58Ma;genuscount=3911)166

whilethelargestrockareaisfoundintheCenomanian(100–94Ma;rockarea=167

373).Thesetwoextremepointsalonedemonstratethatthepaireddiversityand168

rockareavaluesaremillionsofyearsapart,andareindependentofeachother169

(figure4).170

9

171

Thismaybeobvious,butindependentlysortingyandxhasseriousstatistical172

consequences.Forinstance,inSmithandMcGowan’s(2007)data,log10marine173

genericdiversityhasnosignificantrelationshipwithlog10rockareaintheir174

originalpairedbivariatedata(figure4;r2=0.0398;p=0.0979),butoncesorted,175

hasasignificantlystrongpositiverelationshipwithlog10rockareasorted176

independentlyoflog10diversity(figure4;r2=0.903;p<0.001).Thisgeneral177

patternistrueinatleasttwomoredatasets(Bensonetal.2010;Benson&178

Upchurch2013)(figuresS1andS2).Theindependentsortingprocedurehas179

forcedastrongbutfalselinearrelationshipbetweentwovariablesthat180

otherwisedonotshowanysignificant(orifsignificant,averyweak)181

relationship.Infact,tworandomlygenerateddeviates(e.g.sampledfroma182

normaldistribution)thathavenorelationshipwitheachother(figure5a),once183

sortedindependentlyfromlowesttohighestwillinevitablyhaveasignificant184

andstrongrelationship(r2=~1;figure5b).Perhapsmoredetrimental,isthefact185

thattheindependentlysortedbivariaterelationshipwillalwaysbestrongly186

positive–asimulatednegativerelationshipbetweenxandy(figure5c)willhave187

astrongandpositiverelationshiponcetheyaresortedindependently(figure188

5d).189

190

Insomeclades(namelyMesozoicdinosaurs),diversitymeasurescanhavevery191

stronglypositiverelationshipswithsomesamplingmetrics,suchasgeological192

formationcounts(β=0.868;r2=0.85;p<0.001(Barrett,McGowan&Page193

2009))orfossilcollectioncounts(β=0.865;r2=0.79;p<0.001(Butleretal.194

2011)),whichwouldjustifycorrectingforsuchconfoundingfactors,ifthe195

10

samplingmetricswereindeednon-redundantwithdiversity(Bentonetal.2011;196

Bentonetal.2013;Benton2015).However,eveninsuchcases,itdoesnot197

changethefactthatthemodelledrelationshipobtainedfromtheSDDMwillstill198

besystematicallybiased(figure3),andalternativemethodsshouldbe199

considered.200

201

Itisproblematictostipulatethatthisforcedrelationshipisthe‘true’relationship202

betweensampledpalaeodiversityandtherockrecord.Oursimulationsshow203

thatregressionmodelsfittedonindependentlysorteddatahaveunacceptably204

highTypeIerrorrateswhenthedatagenerationprocessesareknown,meaning205

thatSmithandMcGowan’s(2007)approachisnotstatisticallyviable.In206

particular,thattheslopesareincorrectlyestimatedatveryhighrates(~100%207

whenσe=0.5)hassevereconsequencesinthatSDDMpredictionsare208

systematicallybiased(figures2b,3b),leadingtoerroneous‘residuals’.209

Inferencesmadefromsuchproblematic‘residuals’(e.g.Smith&McGowan2007;210

Barrett,McGowan&Page2009;Bensonetal.2010;Butleretal.2011;Benson&211

Upchurch2013)willinevitablybemisleading(Brocklehurst2015),lackingany212

biologicalorgeologicalmeaning.213

214

Givenoursimulations,westronglyrecommendagainstusingtheSDDM215

approachinmodellingtherelationshipbetweenpalaeodiversityandrockrecord216

data;thestandardregressionusingunsorteddataisasensibleoption.However,217

usingtheresidualsofaregressionmodelasdataforsubsequentanalyseshas218

alsolongbeenknowntointroducebiasedstatisticalestimates(King1986;219

Freckleton2002).Successiveseriesofmodellingremovesvarianceanddegrees220

11

offreedomfromsubsequentmodelparameterestimation,sothefinalmodels221

andstatisticalanalysesdonotaccountfortheremovederrorsappropriately222

(King1986).Instead,onecandirectlymodeltheconfoundingeffectsalongwith223

effectsofinterest(e.g.environment,climate,etc)throughmultipleregressions224

(OLS,GLMsorgeneralizedleastsquares[GLS]).Inthecontextofpalaeodiversity225

studies,onecanfitamultipleregressionmodelusingsomediversitymetricas226

theresponsevariableandsamplingproxyasaconfoundingcovariate,alongside227

additionalpredictorvariablessuchassealevel,temperature,etc.Theresulting228

modelcoefficientsfortheenvironmentalpredictorswouldbetheeffectsof229

interestafteraccountingfortheundesiredeffectsofrockavailability.Since230

diversitymeasuresarefrequentlytakenascounts,itisadvisabletousemodels231

thatappropriatelyaccountforerrorsincountdata,suchasthePoissonor232

negativebinomialmodels(O'Hara&Kotze2010).Whetherornottoincludetime233

seriesterms(e.g.autoregressive[AR]terms)dependsonthelevelofserial234

autocorrelationinthetimeseriesdataandonsamplesize;palaeontologicaltime235

seriestendtobeshort,with30timebinsorfewerbeingfairlytypical(Mesozoic236

dinosaursonlyspanamaximumof26geologicalstages(Butleretal.2011;237

Benson&Mannion2012)),inwhichcasecomplexmodelsfacetherisksofover-238

parameterisation.ModelselectionproceduresusingtheAkaikeInformation239

Criterion(Akaike1973)orsimilarindicescanhelpmakethisdecision(Burnham240

&Anderson2002).However,wedonotlightlyadvocatetheuseoftimeseries241

modelling,especiallyifthedependentvariable,sampleddiversity,isintheform242

ofcounts,inwhichcaseappropriatetimeseriesmethodsareseverelyunder-243

developed(butseegeneralisedlinearautoregressivemovingaverage[GLARMA]244

models(Dunsmuir&Scott2015)orPoissonexponentiallyweightedmoving245

12

average[PEWMA]models(Brandtetal.2000)),butmoreimportantlysince246

therearemoreappropriatealternativemethods,i.e.phylogeneticapproaches(?247

Silvestroetal.2015;Sakamoto,Benton&Venditti2016).248

249

Fundamentally,macroevolutionarystudiesaimtoincreaseourunderstandingof250

evolutionaryprocesses(speciationandextinctionthroughtime),ratherthanthe251

resultingpatternsorphenomena(sampleddiversity,e.g.richness).Thus,we252

shouldseektocharacterizetheprocessusingbiologicallymeaningfuland253

interpretablemodelsinsteadofdescribingthepatterns.Further,simply254

exploringerrorinthefossilrecordinitselfseemsratherfruitlessbecause255

uncertaintydependsonthequestionsbeingposed;palaeontologicalstudiesof256

macroevolutionshouldbenodifferentthanotherstatisticalapproachesinthe257

naturalsciencesinthatuncertaintyisassessedwhileexploringthephenomena258

ofinterest(Benton2015).Explicitlyphylogeneticapproaches(e.g.Lloydetal.259

2008;Didier,Royer-Carenzi&Laurin2012;Stadler2013;Stadleretal.2013;260

Sakamoto,Benton&Venditti2016)offerthebestandmostappropriatemeansto261

tacklequestionsofevolutionaryprocesses.Especiallywhenextrinsiccausal262

mechanismsforchangesinbiodiversityaretestedusingregressionmodels,263

ignoringphylogenyisinseriousviolationofstatisticalindependence264

(Felsenstein1985;Harvey&Pagel1991).Itisalsoworthnotingthat265

subsamplingapproaches(e.g.Alroy’sSQS(Alroy2010a;Alroy2010b;Alroy266

2010c))aregainingwidepopularityasmodernmethodstoaccountforsampling267

bias,theyarenotwithoutproblems(Hannisdaletal.2016),andcertainlydonot268

takesharedancestrydescribedbyphylogenyintoaccount,thusalsosuffering269

statisticalnon-independence(Felsenstein1985;Harvey&Pagel1991),andcan270

13

frequentlyresultinincorrectinterpretationofthedata.Forinstance,while271

recentstudiesusingbinnedtimeseriesapproaches(includingSDDMandSQS)272

haveledtomixedconclusionsregardingthelong-termdemiseofdinosaurs273

beforetheirfinalextinctionattheCretaceous-Paleogene(K-Pg)boundary66274

millionyearsago(Ma)(Barrett,McGowan&Page2009;Lloyd2012;Brusatteet275

al.2015),anexplicitlyphylogeneticBayesiananalysishasstronglysuggested276

thatdinosaurswereindeedinalong-termdeclinetensofmillionsofyearsprior277

totheK-Pgmassextinctionevent,inwhichspeciationratewasexceededby278

extinctionrateanddinosaurswereincreasinglyincapableofreplacingextinct279

taxawithnewones(Sakamoto,Benton&Venditti2016).Suchevolutionary280

dynamicscannotbeidentifiedusingtime-binned(tabulated)data.Phylogenetic281

mixedmodellingapproaches(Hadfield2010)furtherallowtheincorporationof282

confoundingvariablessuchassamplingbutalsoenvironmentaleffects283

(Sakamoto,Benton&Venditti2016).Therefore,inordertoadvanceour284

understandingoftheevolutionarydynamicsofbiodiversity,speciationand285

extinctionthroughtime(ortheunderlyingprocessgeneratingtheobserved286

patternsinsampleddiversity,e.g.taxonrichness),whileaccountingforsampling287

andphylogeneticnon-independence,itisimperativethatwehaveanabundance288

oflarge-scalecomprehensivephylogenetictreesoffossil(andextant)taxa.289

290

291

ACKNOWLEDGEMENTS292

WethankJoBaker,CiaraO’DonovanandHenryFerguson-Gowfordiscussion293

andinsightfulcomments.WealsothankNeilBrocklehurstandMichelLaurinfor294

14

reviewingthismanuscriptandprovidinghelpfulcommentary.Wehaveno295

conflictsofinterest.296

297

298

FUNDING299

MSandCVarefundedbyLeverhulmeTrustResearchProjectGrantRPG-2013-300

185(awardedtoCV).MJBisfundedbyNaturalEnvironmentResearchCouncil301

StandardGrantNE/I027630/1.302

303

304

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Silvestro,D.,Antonelli,A.,Salamin,N.&Quental,T.B.(2015)Theroleofclade399competitioninthediversificationofNorthAmericancanids.Proceedings400ofTheNationalAcademyofSciences,USA,112,8684-8689.401

Smith,A.B.&McGowan,A.J.(2007)Theshapeofthephanerozoicmarine402palaeodiversitycurve:Howmuchcanbepredictedfromthesedimentary403rockrecordofwesternEurope?Palaeontology,50,765-774.404

Stadler,T.(2013)Recoveringspeciationandextinctiondynamicsbasedon405phylogenies.JournalOfEvolutionaryBiology,26,1203-1219.406

Stadler,T.,Kuhnert,D.,Bonhoeffer,S.&Drummond,A.J.(2013)Birth-death407skylineplotrevealstemporalchangesofepidemicspreadinHIVand408hepatitisCvirus(HCV).ProceedingsofTheNationalAcademyofSciences,409USA,110,228-233.410

411

17

TABLES412Table1.TypeIerrorrates(%)forSRM(StandardRegressionModel)andSDDM413(Sampling-DrivenDiversityModel)estimates(interceptαandslopeβ)across414residualerror(σe).415416

σeSRM SDDM

α β α β0.05 5.34 4.90 26.1 28.50.10 4.84 4.92 40.2 48.40.25 4.82 4.78 57.3 91.30.50 5.48 5.14 68.7 100.0

417

18

Table2.t-testresultsbetweenmeanregressionslopesof5000iterationsandthe418theoreticalslopeb=0.6,forSRM(StandardRegressionModel)andSDDM419(Sampling-DrivenDiversityModel)acrossresidualerror(σe).420421

σeSRM SDDM

mean-slope t-value p-value mean-slope t-value p-value0.05 0.6 1.230 0.220 0.602 20.9 00.10 0.6 -1.790 0.073 0.607 46.0 00.25 0.6 -0.042 0.967 0.646 131.0 00.50 0.6 0.685 0.493 0.775 244.0 0

422

19

FIGURES423

424Figure1.Procedureforgenerating‘residuals’fromasampling-drivendiversity425

model.(a)Apaired,bivariatedatasetx(samplingproxy)andy(sampled426

diversity)wassimulatedsothatxisrandomlydrawnfromanormaldistribution427

(μ=0,σ=1)andyiscalculatedasy=a+bx+ewherea=0.4,b=0.6andeis428

noise(μ=0,σ=0.5).ThethickblacklineistheexpectedrelationshipY=a+bx.429

Verticallinesrepresentthetrueresidualsordeviationsinyfromthethickline.430

(b)FollowingSmithandMcGowan(2007)xandyaresortedfromlowtohigh431

valuesindependentofeachother(x’andy’respectively),andanordinaryleast432

squares(OLS)regressionmodel(pinkline)isfittedtoy’onx’.Despitethepink433

linesupposedlyrepresentingthedatageneratingprocess,itisclearthatitisnot434

agoodestimatorofthetrueknowngeneratingprocess,thethickline.(c)The435

OLSmodelfrom(b)isusedasthesampling-drivendiversitymodel(SDDM)or436

theexpectedrelationshipbetweenyandx,fromwhich‘residuals’arecomputed437

asthedeviationsinyfromthepinkline(verticalpinkdottedlines).Itis438

immediatelyclearthatthereisasubstantialdifferencebetweenthetrue439

residuals(a)andtheSDDM‘residuals’(c).440

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Figure2.Regressionmodellingonadecoupledbivariatedatasetfailstoestimate443

thesimulationslopeparameter.(a)Abivariatedataset(yandx)wasgenerated444

soastofollowatheoreticalrelationship(thickline)withintercepta=0.4,slope445

b=0.6andnoise(e[μe=0,σe=0.5]).Thebest-fitregressionline(blue)isnot446

significantlydifferentfromthetheoreticalline(dashed95%confidenceintervals447

encompassthethickline;seetable1forTypeIerrorratesover5000448

simulations),withyandxformingamoderatelystrongrelationship(r2=0.526)449

appropriateforthedegreeofemodelled.Regressionmodelresiduals(vertical450

lines)shownostructure,asexpected.(b)Thebivariatedatain(a)weresorted451

independentlyofeachother(y’andx’),towhicharegressionmodelwasfitted.452

Thebest-fitsampling-drivendiversitymodel(SDDM)regressionline(pink)453

deviatesstronglyfromthetheoreticalrelationship(dashed95%confidence454

intervalsdonotencompassthethickline;table1),andy’andx’formavery455

strong(butfalse)linearrelationship(r2=0.973).Regressionresiduals(vertical456

lines)showclearstructure.Onepairofmodelcomparisonoutof5000457

simulationsisshown.458

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Figure3.SDDMregressionpredictionsaresystematicallybiased.(a)Standard461

regressionlines(blue)for5000simulateddatasetsatσe=0.5deviaterandomly462

aroundthetheoreticalrelationship(thickline)withthemeanslopeshowingno463

significantdifferencefromthetheoreticalslopeb=0.6(table2).(b)SDDM464

regressionlinesondecoupleddatasets(pink)deviatesystematicallyawayfrom465

thetheoreticalrelationship(thickline),withasignificantdifferencebetweenthe466

meanregressionslopeandthetheoreticalslope(table2).467

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Figure4.Thedifferencebetweentheoriginalpaired,bivariaterelationship(a)470

andtheforced,falserelationship(b)shownusingthedatafromSmithand471

McGowan(2007).Log-transformedmarinegenericdiversityhasanon-472

significantandweakrelationshipwithlog-transformedrockarea(β=0.105;r2=473

0.0398;p=0.0979;a).However,oncediversityandrockareaaresorted474

independentlyofeachotherfollowingSmithandMcGowan(2007),thenthe475

relationshipbecomessignificantandstrong(β=0.499;r2=0.903;p<0.001;b).476

Pointsarecolouredaccordingtotheirgeologicalagewithcoolercoloursonthe477

olderandwarmercoloursontheyoungerendsofthetimescale.Filledand478

outlinecoloursin(b)correspondtotheagesoftherockrecordanddiversity479

respectively,anddemonstratevisuallythemismatchbetweeny’andx’.Dashed480

linesareconfidenceintervals,whiledottedlinesarepredictionintervals.481

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Figure5.Independentlysortinganytwovariablesresultsinaforcedpositive484

relationship.(a)Tworandomlygeneratedvariablesyandxshownosignificant485

relationshipsacross1000simulations,withtheslopesoftheregressionlines486

(blue)distributedaroundtheexpectedslopeofzero.(b)Whenregression487

modelsarefittedonindependentlysorteddatasets(y’andx’),estimatedslopes488

aresignificantlydifferentfromtheexpectedvalueofzero,andresultinastrong489

positiverelationship(r2=~1;insetpink)despitetheunrelatednatureofthe490

originaldatasets(r2=~0;insetblue).(c)Abivariatedataset(yandx)was491

generatedsoastofollowatheoreticalrelationship(thickline)withintercepta=492

0.4,slopeb=-0.6andnoise(e[μe=0,σe=0.5]).Standardregressionlines(blue)493

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deviaterandomlyaroundthetheoreticalrelationshipwiththemeanslope494

showingnosignificantdifferencefromthetheoreticalslopeb=-0.6.(d)However495

oncesortedindependently,regressionlines(pink)deviatesystematicallyaway496

fromthetheoreticalrelationship,withallestimatedslopesbeingpositive.Thus497

SDDMslopeestimatesaresystematicallyanddirectionallybiased.498

499