1 Theoretical Background Morans autocorrelation coe64259cient often denoted as is an extension of Pear son productmoment correlation coe64259cient to a univariate series 2 5 Recall that Pearsons correlation denoted as between two variables and bot ID: 22325
Download Pdf The PPT/PDF document "Morans Autocorrelation Coecient in Compa..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Moran'sAutocorrelationCoecientinComparativeMethodsEmmanuelParadisAugust12,2020ThisdocumentclariestheuseofMoran'sautocorrelationcoecienttoquantifywhetherthedistributionofatraitamongasetofspeciesisa ectedornotbytheirphylogeneticrelationships.1TheoreticalBackgroundMoran'sautocorrelationcoecient(oftendenotedasI)isanextensionofPear-sonproduct-momentcorrelationcoecienttoaunivariateseries[2,5].RecallthatPearson'scorrelation(denotedas)betweentwovariablesxandybothoflengthnis:=nXi=1(xi�x)(yi�y) "nXi=1(xi�x)2nXi=1(yi�y)2#1=2;wherexandyarethesamplemeansofbothvariables.measureswhether,onaverage,xiandyiareassociated.Forasinglevariable,sayx,Iwillmeasurewhetherxiandxj,withi6=j,areassociated.Notethatwith,xiandxjarenotassociatedsincethepairs(xi;yi)areassumedtobeindependentofeachother.Inthestudyofspatialpatternsandprocesses,wemaylogicallyexpectthatcloseobservationsaremorelikelytobesimilarthanthosefarapart.Itisusualtoassociateaweighttoeachpair(xi;xj)whichquantiesthis[3].Initssimplestform,theseweightswilltakevalues1forcloseneighbours,and0otherwise.Wealsosetwii=0.Theseweightsaresometimesreferredtoasaneighbouringfunction.I'sformulais:I=n S0nXi=1nXj=1wij(xi�x)(xj�x) nXi=1(xi�x)2;(1)wherewijistheweightbetweenobservationiandj,andS0isthesumofallwij's:1 S0=nXi=1nXj=1wij:Quitenotsointuitively,theexpectedvalueofIunderthenullhypothesisofnoautocorrelationisnotequaltozerobutgivenbyI0=�1=(n�1).TheexpectedvarianceofI0isalsoknown,andsowecanmakeatestofthenullhypothesis.IftheobservedvalueofI(denoted^I)issignicantlygreaterthanI0,thenvaluesofxarepositivelyautocorrelated,whereasif^II0,thiswillindicatenegativeautocorrelation.Thisallowsustodesignone-ortwo-tailedtestsinthestandardway.Gittleman&Kot[4]proposedtouseMoran'sItotestfor\phylogenetice ects".Theyconsideredtwowaystocalculatetheweightsw:ˆWithphylogeneticdistancesamongspecies,e.g.,wij=1=dij,wheredijaredistancesmeasuredonatree.ˆWithtaxonomiclevelswherewij=1ifspeciesiandjbelongtothesamegroup,0otherwise.Notethatintherstsituation,therearequitealotofpossibilitiestosettheweights.Forinstance,Gittleman&Kotalsoproposed:wij=1=d ijifdijcwij=0ifdij&#x-390;c;wherecisacut-o phylogeneticdistanceabovewhichthespeciesareconsideredtohaveevolvedcompletelyindependently,and isacoecient(see[4]fordetails).Byanalogytotheuseofaspatialcorrelogramwherecoecientsarecalculatedassumingdi erentsizesofthe\neighbourhood"andthenplottedtovisualizethespatialextentofautocorrelation,theyproposedtocalculateIatdi erenttaxonomiclevels.2ImplementationinapeFromversion1.2-6,apehasfunctionsMoran.Iandcorrelogram.formulaim-plementingtheapproachdevelopedbyGittleman&Kot.Therewasanerrorinthehelppagesof?Moran.I(correctedinver.2.1)wheretheweightswerereferredtoas\distanceweights".Thishasbeenwronglyinterpretedinmybook[6,pp.139{142].Theanalysesbelowaimtocorrectthis.2.1PhylogeneticDistancesThedata,takenfrom[1],arethelog-transformedbodymassandlongevityofvespeciesofprimates:&#x-390;body-c(4.09434,3.61092,2.37024,2.02815,-1.46968)ကlongevity-c(4.74493,3.3322,3.3673,2.89037,2.30259)ကnames(body)-names(longevity)-c("Homo","Pongo","Macaca","Ateles","Galago")2 Thetreehasbranchlengthsscaledsothattherootageisone.Wereadthetreewithape,andplotit:�library(ape)�trnwk-"((((Homo:0.21,Pongo:0.21):0.28,Macaca:0.49):0.13,Ateles:0.62)"ကtrnwk[2]-":0.38,Galago:1.00);"ကtr-read.tree(text=trnwk)ကplot(tr)ကaxisPhylo() Wechoosetheweightsaswij=1=dij,wherethed'sisthedistancesmeasuredonthetree:�w-1/cophenetic(tr)ကwHomoPongoMacacaAtelesGalagoHomoInf2.38095241.02040820.80645160.5Pongo2.3809524Inf1.02040820.80645160.5Macaca1.02040821.0204082Inf0.80645160.5Ateles0.80645160.80645160.8064516Inf0.5Galago0.50000000.50000000.50000000.5000000InfOfcourse,wemustsetthediagonaltozero:ကdiag(w)-0WecannowperformtheanalysiswithMoran'sI:3 �Moran.I(body,w)$observed[1]-0.07312179$expected[1]-0.25$sd[1]0.08910814$p.value[1]0.04714628Notsurprisingly,theresultsareoppositetothosein[6]since,there,thedistances(givenbycophenetic(tr))wereusedasweights.(Notethattheargumentdisthasbeensincerenamedweight.1)Wecannowconcludeforaslighlysignicantpositivephylogeneticcorrelationamongbodymassvaluesforthesevespecies.ThenewversionofMoran.Igainstheoptionalternativewhichspeciesthealternativehypothesis("two-sided"bydefault,i.e.,H1:I6=I0).Asexpectedfromtheaboveresult,wedividetheP-valuebetwoifwedeneH1asI�I0:�Moran.I(body,w,alt="greater")$observed[1]-0.07312179$expected[1]-0.25$sd[1]0.08910814$p.value[1]0.02357314Thesameanalysiswithlongevitygives:�Moran.I(longevity,w)$observed[1]-0.1837739$expected[1]-0.25 1Theoldercodewasactuallycorrect;nevertheless,ithasbeenrewritten,andisnowmuchfaster.Thedocumentationhasbeenclaried.Thefunctioncorrelogram.phylo,whichcomputedMoran'sIforatreegivenasargumentusingthedistancesamongtaxa,hasbeenremoved.4 $sd[1]0.09114549$p.value[1]0.4674727Asforbody,theresultsarenearlymirroredcomparedto[6]whereanon-signicantnegativephylogeneticcorrelationwasfound:itisnowpositivebutstilllargelynotsignicant.2.2TaxonomicLevelsThefunctioncorrelogram.formulaprovidesaninterfacetocalculateMoran'sIforoneorseveralvariablesgivingaseriesoftaxonomiclevels.Anexampleofitsusewasprovidedin[6,pp.141{142].Thecodeofthisfunctionhasbeensimplied,andthegraphicalpresentationoftheresultshavebeenimproved.correlogram.formula'smainargumentisaformulawhichis\sliced",andMoran.Iiscalledforeachoftheseelements.Twothingshavebeenchangedfortheend-useratthislevel:1.Intheoldversion,therhsoftheformulawasgivenintheorderofthetaxonomichierarchy:e.g.,Order/SuperFamily/Family/Genus.Notre-spectingthisorderresultedinanerror.Inthenewversion,anyorderisaccepted,buttheordergivenisthenrespectedwhenplottedthecorrelo-gram.2.Variabletransformations(e.g.,log)wereallowedonthelhsoftheformula.Becauseofthesimplicationofthecode,thisisnomorepossible.Soitistheresponsibilityoftheusertoapplyanytranformationbeforetheanalysis.FollowingGittleman&Kot[4],theautocorrelationatahigherlevel(e.g.,family)iscalculatedamongspeciesbelongingtothesamecategoryandtodif-ferentcategoriesatthelevelbelow(genus).Toformalizethis,letuswritethedi erentlevelsasX1=X2=X3=:::=XnwithXnbeingthelowestone(Genusintheaboveformula):wij=1ifXki=XkjandXk+1i6=Xk+1jwij=0otherwiseknwij=1ifXki=Xkjwij=0otherwisek=nThisisthusdi erentfromtheideaofa\neighbourhood"ofdi erentsizes,butrathersimilartotheideaofpartialcorrelationwherethein uenceofthelowestlevelisremovedwhenconsideringthehighestones[4].Torepeattheanalysesonthecarnivoradataset,werstlog10-transformthevariablesmeanbodymass(SW)andthemeanfemalebodymass(FW):&#x-278;data(carnivora)&#x-278;carnivora$log10SW-log10(carnivora$SW)ကcarnivora$log10FW-log10(carnivora$FW)5 Werstconsiderasinglevariableanalysis(asin[6]):�fm1.carn-log10SW~Order/SuperFamily/Family/Genusကco1-correlogram.formula(fm1.carn,data=carnivora)ကplot(co1) Alegendnowappearsbydefault,butcanberemovedwithlegend=FALSE.Mostoftheappearanceofthegraphcanbecustomizedviatheoptionoftheplotmethod(see?plot.correlogramfordetails).ThisisthesameanalysisthantheonedisplayedonFig.6.3of[6].Whenasinglevariableisgiveninthelhsincorrelogram.formula,anobjectofclass"correlogram"isreturnedasabove.Ifseveralvariablesareanalysedsimultaneously,theobjectreturnedisofclass"correlogramList",andthecorrelogramscanbeplottedtogetherwiththeappropriateplotmethod:�fm2.carn-log10SW+log10FW~Order/SuperFamily/Family/Genusကco2-correlogram.formula(fm2.carn,data=carnivora)ကprint(plot(co2))6 Bydefault,latticeisusedtoplotthecorrelogramsonseparatepanels;usinglattice=FALSE(actuallythesecondargument,see?plot.correlogramList)makesastandardgraphsuperimposingthedi erentcorrelograms:�plot(co2,FALSE)7 Theoptionsareroughlythesamethanabove,butdonothavealwaysthesamee ectsincelatticeandbasegraphicsdonothavethesamegraphicalpa-rameters.Forinstance,legend=FALSEhasnoe ectiflattice=TRUE.3Implementationinade4Theanalysisdonewithade4in[6]su ersfromthesameerrorthantheonedonewithMoran.Isinceitwasalsodonewithadistancematrix.SoIcorrectthisbelow:�library(ade4)�gearymoran(w,data.frame(body,longevity))class:krandtestMonte-CarlotestsCall:as.krandtest(sim=matrix(res$result,ncol=nvar,byr=TRUE),obs=res$obs,alter=alter,names=test.names)Testnumber:2Permutationnumber:999Alternativehypothesis:greaterTestObsStd.ObsPvalue1body-0.062567892.15233420.0012longevity-0.229904370.34614140.414otherelements:NULL8 Theresultsarewhollyconsistentwiththosefromape,buttheestimatedcoecientsaresubstantiallydi erent.Thisisbecausethecomputationalmeth-odsarenotthesameinbothpackages.Inade4,theweightmatrixisrsttransformedasarelativefrequencymatrixwith~wij=wij=S0.Theweightsarefurthertransformedwith:pij=~wij�nXi=1~wijnXj=1~wij;withpijbeingtheelementsofthematrixdenotedasP.Moran'sIisnallycomputedwithxTPx.Inape,theweightsarerstrow-normalized:wij.nXi=1wij;theneq.1isapplied.Anotherdi erencebetweenbothpackages,thoughlessimportant,isthatinade4theweightmatrixisforcedtobesymmetricwith(W+WT)=2.Inape,thismatrixisassumedtobesymmetric,whichislikelytobethecaselikeintheexamplesabove.4OtherImplementationsPackagesphasseveralfunctions,includingmoran.test,thataremorespeci-callytargetedtotheanalysisofspatialdata.Packagespatialhasthefunctioncorrelogramthatcomputesandplotsspatialcorrelograms.AcknowledgementsIamthankfultoThibautJombartforclaricationsonMoran'sI.References[1]J.M.Cheverud,M.M.Dow,andW.Leutenegger.Thequantitativeas-sessmentofphylogeneticconstraintsincomparativeanalyses:sexualdimor-phisminbodyweightamongprimates.Evolution,39:1335{1351,1985.[2]A.D.Cli andJ.K.Ord.SpatialAutocorrelation.Pion,London,1973.[3]A.D.Cli andJ.K.Ord.Spatialandtemporalanalysis:autocorrelationinspaceandtime.InE.N.WrigleyandR.J.Bennett,editors,Quantita-tiveGeography:ABritishView,pages104{110.Routledge&KeganPaul,London,1981.[4]J.L.GittlemanandM.Kot.Adaptation:statisticsandanullmodelforestimatingphylogenetice ects.SystematicZoology,39:227{241,1990.[5]P.A.P.Moran.Notesoncontinuousstochasticphenomena.Biometrika,37:17{23,1950.[6]E.Paradis.AnalysisofPhylogeneticsandEvolutionwithR.Springer,NewYork,2006.9