It is made available under a CC-BY-NC-ND 4.0 International ... · Consortium, Chun Jimmie Ye, Alkes...

1

Transethnicgeneticcorrelationestimatesfromsummarystatisticssupportwidespread

non-additiveeffects

BrielinC.Brown,AsianGeneticEpidemiologyNetwork-Type2Diabetes(AGEN-T2D)Consortium,ChunJimmieYe,AlkesL.Price,NoahZaitlen

Abstract

Theincreasingnumberofgeneticassociationstudiesconductedinmultiple

populationsprovidesunprecedentedopportunitytostudyhowthegeneticarchitectureof

complexphenotypesvariesbetweenpopulations,aproblemimportantforbothmedicaland

populationgenetics.Herewedevelopamethodforestimatingthetransethnicgenetic

correlation;thecorrelationofcausalvarianteffectsizesatSNPscommoninpopulations.

Unlikesomepriorapproaches,wetakeadvantageoftheentirespectrumofSNP

associationsandutilizeonlysummary-levelGWASdata,therebyavoidingthe

computationalcostsandprivacyconcernsassociatedwithgenotype-levelinformationwhile

remainingscalabletohundredsofthousandsofindividualsandmillionsofSNPs.Weapply

ourmethodtogeneexpression,rheumatoidarthritis,andtype-twodiabetesdataand

overwhelminglyfindthatthegeneticcorrelationissignificantlylessthan1.Wearguethat

thisisevidenceforthepresenceofnon-additiveordifferentialtaggingeffectsthatmodify

themarginaleffectsizesatSNPscommoninbothpopulations.Ourmethodisimplemented

inapythonpackagecalledpopcorn.

Introduction

Manycomplexhumanphenotypesvarydramaticallyintheirdistributionsbetween

populationsduetoacombinationofgeneticandenvironmentaldifferences.Forexample,

northernEuropeansareonaveragetallerthansouthernEuropeans1andAfricanAmericans

haveanincreasedrateofhypertensionrelativetoEuropeanAmericans2.Thegenetic

contributiontopopulationphenotypicdifferentiationisdrivenbydifferencesincausal

allelefrequencies,effectsizes,andgeneticarchitectures.Understandingtherootcausesof

phenotypicdifferencesworldwidehasprofoundimplicationsforbiomedicalandclinical

practiceindiversepopulations,thetransferabilityofepidemiologicalresults,aidingmulti-

ethnicdiseasemapping3,4,assessingthecontributionofnon-additiveandrarevariant

effects,andmodelingthegeneticarchitectureofcomplextraits.Inthisworkweconsidera

centralquestionintheglobalstudyofphenotype:dogeneticvariantshavethesame

phenotypiceffectsindifferentpopulations?

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2

WhilethevastmajorityofGWAShavebeenconductedinEuropeanpopulations5,

thegrowingnumberofnon-Europeanandmulti-ethnicstudies4,6,7provideanopportunity

tostudygeneticeffectdistributionsacrosspopulations.Forexample,onerecentstudyused

mixed-modelbasedmethodstoshowthatthegenome-widegeneticcorrelationof

schizophreniabetweenEuropeanandAfricanAmericansisnonzero8.Whilepowerful,

computationalcostsandprivacyconcernslimittheutilityofgenotype-basedmethods.In

thiswork,wemaketwosignificantcontributionstostudiesoftransethnicgenetic

correlation.First,weexpandthedefinitionofgeneticcorrelationtobetteraccountfora

transethniccontext.Second,wedevelopanapproachtoestimatinggeneticcorrelation

acrosspopulationsthatusesonlysummarylevelGWASdata.Similartootherrecent

summarystatisticsbasedheritabilitymethods9–11,ourapproachavoidsprivacyconcerns

andisscalabletohundredsofthousandsofindividualsandmillionsofmarkers.Unlike

traditionalapproachesthatfocusonthesimilarityofGWASresults12–16weutilizetheentire

spectrumofGWASassociationswhileaccountingforLDinordertoavoidfiltering

correlatedSNPs.

Inasinglepopulation,thegeneticcorrelationoftwophenotypesisdefinedasthe

correlationcoefficientofSNPeffectsizes17,18.Inmultiplepopulations,differencesinallele

frequencymotivatemultiplepossibledefinitionsofgeneticcorrelation.Hereweconsider

boththecorrelationofalleleeffectsizesaswellasthecorrelationofallelicimpact,which

takesintoaccountthefrequencyofthevariantinthepopulation.Forexample,avariant

mayhaveamuchhighereffectsizebutmuchlowerfrequencyinonepopulation.Therefore,

wedefinethetransethnicgeneticeffectcorrelation(ρge)asthecorrelationcoefficientofthe

per-alleleSNPeffectsizes,andthetransethnicgeneticimpactcorrelation(ρgi)asthe

correlationcoefficientofthepopulation-specificallelevariancenormalizedSNPeffectsizes.

Whileotherdefinitionsofthegeneticcorrelationarepossible(seediscussion),these

quantitiescapturetwoimportantquestionsaboutthestudyofdiseaseinmultiple

populations:towhatextentdothesamemutationsinmultiplepopulationsdifferintheir

phenotypiceffectsandtowhatextentarethesedifferencesmitigatedorexacerbatedby

differencesinallelefrequency?

Toestimategeneticcorrelation,wetakeaBayesianapproachwhereinweassume

genotypesaredrawnrandomlyfromtheirpopulationandeffectssizeshaveanormalprior

(theinfinitesimalmodel19).Whileunlikelytorepresentreality,thismodelhasbeenused

successfullyinpractice8,20–23.Theinfinitesimalassumptionyieldsamultivariatenormal



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distributionontheobservedteststatistics(Z-scores),whichisafunctionoftheheritability

andgeneticcorrelation.RatherthanpruningSNPsinLD10,24,25,thisallowsustoexplicitly

modeltheresultinginflationofZ-scores.Wethenmaximizeanapproximateweighted

likelihoodfunctiontofindtheheritabilityandgeneticcorrelation.Thismethodis

implementedinapythonpackagecalledpopcorn.Thoughderivedforquantitative

phenotypes,popcornextendseasilytobinaryphenotypesundertheliabilitythreshold

model.Weshowviaextensivesimulationthatpopcornproducesunbiasedestimatesofthe

geneticcorrelationandthepopulationspecificheritabilities,withastandarderrorthat

decreasesasthenumberofSNPsandindividualsinthestudiesincreases.Furthermore,we

showthatourapproachisrobusttoviolationsoftheinfinitesimalassumption.

WeapplypopcorntoEuropeanandYorubangeneexpressiondata26aswellasGWAS

summarystatisticsfromEuropeanandEastAsianrheumatoidarthritisandtype-two

diabetescohorts,27,28.OuranalysisofgEUVADISshowsthatoursummarystatisticbased

estimatorisconcordantwiththemixedmodelbasedestimator.Wefindthatthemean

transethnicgeneticcorrelationacrossallgenesislow(ρge=0.320(0.009)),butincreases

substantiallywhenthegeneishighlyheritableinbothpopulations(ρge=0.772(0.017)).We

findthegeneticeffectcorrelationinRAandT2Dtobe0.463(0.058)and0.621(0.088),

respectively.

Acrossallphenotypesconsidered,weoverwhelminglyfindthatthetransethnic

geneticcorrelationissignificantlylessthanone.Therearemanyphenomenathat

contributetothis,including:untypedandunimputedrarevariantscorrelatedwith

observedSNPs;gene-geneinteractionsordominanceeffects,gene-environment

interactions,includingepigeneticeffects,thataredifferentialbetweenpopulations;and

differencesinsub-phenotypecomposition.Ourresultsthereforeshowthatthese

phenomenasignificantlyaltertheeffectsizesofSNPscommontobothpopulations.

Methods

ConsiderGWASofaphenotypeconductedintwodifferentpopulations.Assumewe

haveN1individualsgenotypedonMSNPsinstudyoneandN2individualsgenotypedonthe

sameSNPsstudytwo.LetX1,X2bethematricesofmean-centeredgenotypesinstudyone

andstudytwo,respectively,andletY1,Y2benormalizedphenotypesoftheindividualsin

studyoneandtwo,respectively.Letf1,f2bevectorsoftheallelefrequenciesoftheMSNPs

commontobothpopulations.AssumingHardy-Weinbergequilibrium,theallelevariances



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areσ12=2f1(1–f1),σ22=2f2(1–f2).Letβ1,β2bethe(unobserved)per-alleleeffectsizesfor

eachSNPinstudiesoneandtwo,respectively.Theheritabilityinstudyoneisthenh12=Σi

σ1i2β1i2(andlikewiseforstudytwo).Theobjectiveofthisworkistoestimatetransethnic

geneticcorrelationfromsummarystatisticsofcommonvariants (and

likewiseforstudytwo)andestimatesofpopulationLDmatrices(Σ1andΣ2)fromexternal

referencepanels.

Therearemultiplepossibledefinitionsoftransethnicgeneticcorrelationandinthis

workweconsiderthegeneticeffectcorrelationρge=Cor(β1,β2),previouslydefinedbyLeeet

al17andimplementedinGCTA,whileadditionallydefiningthegeneticimpactcorrelationρgi

=Cor(σ1β1,σ2β2).Intuitively,thegeneticeffectcorrelationmeasurestheextenttowhichthe

samevarianthasthesamephenotypicchange,whilethegeneticimpactcorrelationgives

moreweighttocommonallelesthanrareonesseparatelyineachpopulation.Forexample,

iftheeffectsizesarethesameineachpopulationρge=1butρgi<1becauseofallele

frequencydifferencesbetweenthepopulations.Inthiscaseρgi<ρgehowevertheopposite

canalsobetrue.Ifrarealleleshavelargereffectsizesandtherearemanyallelescommonin

studyonebutrareinstudytwo,thenρgiwillbegreaterthanρge.Inthiscase,thedifferences

ineffectsizesaremitigatedbycorrespondingdifferencesinallelefrequency.

Weassumethegenotypesaredrawnrandomlyfromeachpopulationandthat

phenotypesaregeneratedbythelinearmodelY1=X1β1+ε1(likewiseforphenotypetwo).

Wheneffectsizesβareassumedinverselyproportionaltoallelefrequency,asiscommonly

done21,23,weshow(Appendix)thatunderthelinearinfinitesimalgeneticarchitecture,the

jointdistributionoftheZ-scoresfromeachstudyisasymptoticallymultivariatenormalwith

mean andvariance:

(1)

However,wheneffectssizesareassumedindependentofallelefrequencyweshow:

(2)

Z1 =(X1/�1)>Y1p

N1

�!0

Var(Z) =

"⌃1 +

N1+1M h2

1⌃21 ⇢gi

ph21h

22

pN1N2

M ⌃1⌃2

⇢gip

h21h

22

pN1N2

M ⌃2⌃1 ⌃2 +N2+1M h2

2⌃22

#

Var(Z) =

2

4⌃1 +

N1+1k�2

1k1h21⌃1�2

1⌃1 ⇢geph21h

22

pN1N2p

k�21k1k�2

2k1

⌃1

p�21�

22⌃2

⇢gep

h21h

22

pN1N2p

k�21k1k�2

2k1

⌃2

p�22�

21⌃1 ⌃2 +

N2+1k�2

2k1h22⌃2�2

2⌃2

3

5



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Giventheseequationsforvariance,thequantitiesρgiorρgeandh12,h22canbe

estimatedbymaximizingthemultivariatenormallikelihood

,whereCiseitheroftheabovecovariance

matrices(1)and(2).BecauseΣ1andΣ2areestimatedfromfiniteexternalreferencepanels,

maximumlikelihoodestimationoftheabovemultivariatenormalleadstooverfitting.We

implementedtwooptimizationstoavoidthisproblem.First,wemaximizeanapproximate

weightedlikelihoodthatutilizesonlythediagonalelementsofeachblockofVar(Z).This

allowsustoaccountfortheLD-inducedinflationoftestsstatisticsbutdiscardscovariance

informationbetweenpairsofZ-scoresandthereforeleadstoovercountingZ-scoresofSNPs

inhighLD.Tocompensateforthis,wedownweightZ-scoresofSNPsinproportiontheirLD.

Second,ratherthancomputethefullproductsΣ12,Σ22andΣ1Σ2overallMSNPsinthe

genome,wechooseawindowsizeWandapproximatetheproductby

.TheseoptimizationsaresimilartothoseemployedbyLDscore

regression21.Thefulldetailsofthederivationandoptimizationareprovidedinthe

appendix.

Results

SimulatedGenotypesandSimulatedPhenotypes

Inordertoverifythatpopcornyieldsanunbiasedestimateoftheheritabilities(h12,

h22)andgeneticcorrelations(ρge,ρgi),weappliedpopcorntosummarystatisticsfrom

simulatedGWAS.Wesimulated50,000European-like(EUR)and50,000EastAsian-like

(EAS)individualsat248,953SNPsfromchromosomes1-3withallelefrequencyabove1%

inbothEuropeanandEastAsianHapMap3populationswithHapGen229.HapGen2

implementsahaplotyperecombinationwithmutationmodelthatresultsinexcesslocal

relatednessamongthesimulatedindividuals.Toaccountforthislocalstructure,we

randomlyremovedoneindividualfromeachpairwithglobalIBDcoefficientπ>0.15,

leaving20,085simulatedEURand20,006simulatedEAS.Additionally,weuseda5MB

windowinouranalysistoaccommodatelongerrangeLD.Fromthesesimulatedindividuals,

1000perpopulationwerechosenuniformlyatrandomtoserveasanexternalreference

panelforestimatingΣ1andΣ2.

l(⇢g{i,e}, h21, h

22|Z,⌃,�) / �ln(|C|)� Z>C�1Z

(⌃a⌃b)ii =w=i+WX

w=i�W

raiwrbiw



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Ineachsimulationeffectsizesweredrawnfroma“spikeandslab”model,where

withprobabilitypand with

probability1-p.ρgiwasanalyticallycomputedfromthesimulatedeffectsizesandallele

frequenciesinthesimulatedreferencegenotypes.Quantitativephenotypesweregenerated

underalinearmodelwithi.i.d.noiseandnormalizedtohavemean0andvariance1,while

binaryphenotypesweregeneratedunderaliabilitythresholdmodelwhereindividualsare

labeledcaseswhentheirliabilityexceedsathreshold ,withKthe

populationdiseaseprevalence30.

Wevariedh12,h22,ρge,andρgi,aswellasthenumberofindividualsineachstudy(N1,

N2),thenumberofSNPs(M),thepopulationprevalenceK,andproportionofcausalvariants

(p)inthesimulatedGWASandgeneratedsummarystatisticsforeachstudy.Theresults

showninFigure1andFigureS1demonstratethattheestimatorsarenearlyunbiasedand

thatstandarderrorthatdecreaseswith(roughly)thesquarerootofthenumberofSNPs

andindividuals.Furthermore,byvaryingtheproportionofcausalvariantspweshowthat

ourestimatorisrobusttoviolationsoftheinfinitesimalassumption(FigureS2).Finally,we

showinTableS1thatourestimatesoftheheritabilityofliabilityincasecontrolstudiesare

nearlyunbiased.

Simulationswithnonstandarddiseasemodels

Ourapproach,aswellasgenotype-basedmethodssuchasGCTAmakeassumptions

aboutthegeneticarchitectureofcomplextraits.Previousworkhasshownthatviolationsof

theseassumptionscanleadtobiasinheritabilityestimation31,thereforewesoughtto

quantifytheextentthatthisbiasmayeffectourestimates.Wesimulatedphenotypesunder

sixdifferentdiseasemodels.Independent:effectsizeindependentofallelefrequency.

Inverse:effectsizeinverselyproportionaltoallelefrequency.Rare:onlySNPswithallele

frequencyunder10%affectthetrait.Common:onlySNPswithallelefrequencybetween

40%and50%affectthetrait.Difference:effectsizeproportionaltodifferenceinallele

frequency.Adversarial:differencemodelwithsignofbetasettoincreasethephenotypein

thepopulationwherethealleleismostcommon.Additionalgeneticarchitecturesare

possible,includingoneswhereeffectsizesarenotadirectfunctionofMAF32.

Wesimulatedphenotypesusinggenotypeswithallelefrequencyabove1%or5%

andcomparedthetrueandestimatedgeneticimpactandeffectcorrelationamongall

�1i,�2i ⇠ N✓0,

h21 ⇢ge

ph21h

222

⇢gep

h21h

22 h2

2

�◆

�1i,�2i = (0, 0)

⌧ = ��1(1�K)



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models(Table1).WefindthatwhenonlySNPswithfrequencyabove5%inboth

populationsareused,thedifferenceinρgeandρgiisminimalexceptinthemostadversarial

cases.Evenintheadversarialmodel,thetruedifferenceisonly7%.Thoughunlikelyto

representreality,thefournonstandarddiseasemodelsresultinsubstantialbiasinour

estimators.WhenSNPswithallelefrequencyabove1%inbothpopulationsareincluded,

thedifferencesaremorepronounced.Thisisbecausethenormalizingconstant1/σrapidly

increasesastheSNPbecomesmorerare.Indeed,asSNPsbecomemorerarehavingan

accuratediseasemodelbecomesincreasinglyimportantandthereforeweproceedwitha

5%MAFcutoffinouranalysisofrealdataandusethenotationhc2torefertotheheritability

ofSNPswithallelefrequencyabove5%inbothpopulations(thecommon-SNPheritability).

Note,however,thatoneoftheadvantagesofmaximumlikelihoodestimationingeneralis

thatthelikelihoodcanbereformulatedtomimicthediseasemodelofinterest.

ValidationofPopcornusinggeneexpressioninGEUVADIS

Tofurthervalidateourapproach,wecomparedthecommon-SNPheritability(hc2)

andgeneticcorrelationestimatesofpopcorntoGCTAinthegEUVADISdatasetforwhich

rawgenotypesarepubliclyavailable.gEUVADISconsistsofRNA-seqdatafor464

lymphoblastoidcellline(LCL)samplesfromfivepopulationsinthe1000genomesproject.

Ofthese,375areofEuropeanancestry(CEU,FIN,GBR,TSI)and89areofAfricanancestry

(YRI).RawRNA-sequencingreadsobtainedfromtheEuropeanNucleotideArchivewere

alignedtothetranscriptomeusingUCSCannotationsmatchinghg19coordinates.RSEMwas

usedtoestimatetheabundancesofeachannotatedisoformandtotalgeneabundanceis

calculatedasthesumofallisoformabundancesnormalizedtoonemilliontotalcountsor

transcriptspermillion(TPM).ForeQTLmapping,CaucasianandYorubansampleswere

analyzedseparately.Foreachpopulation,TPMsweremediannormalizedtoaccountfor

differencesinsequencingdepthineachsampleandstandardizedtomean0andvariance1.

Ofthe29763totalgenes,9350withTPM>2inbothpopulationswerechosenforthis

analysis.

Foreachgeneweconductedacis-eQTLassociationstudyatallSNPswithin1

megabaseofthegenebodywithallelefrequencyabove5%inbothpopulationsusing30

principalcomponentsascovariates.WefoundthatGCTAandpopcornagreeontheglobal

distributionofheritability(FigureS3)andthatGCTA’sestimatesofgeneticcorrelationhave

asimilardistributiontopopcorn’sgeneticeffect(GE)andgeneticimpact(GI)correlation



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estimates(Figure2).WhilethenumberofSNPsandindividualsincludedineachgene

analysisaretoosmalltoobtainaccuratepointestimatesofthegeneticcorrelationonaper-

genebasis(N=464,M=4279.5),thelargenumberofgenesenablesaccurateestimationof

theglobalmeanheritabilityandgeneticcorrelation.

Common-SNPheritabilityandgeneticcorrelationofgeneexpressioningEUVADIS

Wefindthattheaveragecis-hc2oftheexpressionofthegenesweanalyzedwas

0.093(0.002)inEURand0.088(0.002)inYRI.Ourestimatesarehigherthanpreviously

reportedaveragecis-heritabilityestimatesof0.055inwholebloodand0.057inadipose33,

whichcouldariseforseveralreasons.First,weremove68%ofthetranscriptsthatare

lowlyexpressedineithertheYRIorEURdata.Second,estimatesfromRNA-seqanalysisof

celllinesmightnotbedirectlycomparabletomicroarraydatafromtissue.

Theaveragegeneticeffectcorrelationwas0.320(0.010)whiletheaveragegenetic

impactcorrelationwas0.313(0.010).Notably,thegeneticcorrelationincreasesasthecis-

hc2ofexpressioninbothpopulationsincreases(Figure3).Inparticular,whenthecis-hc2of

thegeneisatleast0.2inbothpopulationsthegeneticeffectcorrelationwas0.772(0.017)

whilethegeneticimpactcorrelationwas0.753(0.018).

Inordertoverifythattherewerenosmall-samplesizeorconditioningbiasesinour

analysis,weanalyzedthegeneticcorrelationofsimulatedphenotypesoverthegEUVADIS

genotypes.Wesampledpairsofheritabilitiesfromtheestimatedexpressionheritability

distributionandsimulatedpairsofphenotypestohavethegivenheritabilityandagenetic

effectcorrelationof0.0overrandomlychosen4000baseregionsfromchromosome1ofthe

gEUVADISgenotypes.Withoutconditioning,theaverageestimatedgeneticeffect

correlationwas-0.002(0.003),indicatingthattheestimatorremainedunbiased.

Furthermore,theaverageestimatedgeneticeffectcorrelationwasnotsignificantlydifferent

from0.0conditionalontheestimatesofheritabilitybeingaboveacertainthreshold(Figure

S4).Notehoweverthatthesmallsamplesizeincreasesthevarianceoftheestimator

substantially.Becausethegeneticcorrelationisboundedbetween-1and1,thismayinduce

biaswhenthetruevalueisclosetotheboundaryandthesamplesizeissmall.

Wefindthatwhiletheaveragegeneticcorrelationislow,thegeneticcorrelation

increaseswiththecis-hc2ofthegene,indicatingthatascis-geneticregulationofgene

expressionincreasesitdoessosimilarlyinbothYRIandEURpopulations.Thismayhelp

interprettherecentobservationthatwhiletheglobalgeneticcorrelationofgeneexpression



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acrosstissuesislow33,cis-eQTL’stendtoreplicateacrosstissues34.Asthepresenceofacis-

eQTLindicatessubstantialcis-geneticregulation,ananalysisofeQTLreplicationacross

tissuesisimplicitlyconditioningontheheritabilityofgeneexpressionbeinghighand

thereforemayindicatemuchhighergeneticcorrelationthanaverage.

SummarystatisticsofRAandT2D

Finally,wesoughttoexaminethetransethnicρgiandρgeinRAandT2Dcohortsfor

whichrawgenotypesarenotavailable.WeobtainedsummarystatisticsofGWASfor

rheumatoidarthritisandtype-2diabetesconductedinEuropeanandEastAsian

populations.WeremovedtheMHCregion(chromosome6,25–35Mb)fromtheRA

summarystatistics.Weestimatedthecommon-SNPheritabilityandgeneticcorrelation

using2539629SNPsgenotypedorimputedinbothRAstudiesand1054079SNPs

genotypedorimputedinbothT2Dstudieswithallelefrequencyabove5%in1000genomes

EURandEASpopulations.Thehc2andgeneticcorrelationestimatesarepresentedinTable

2.OurRAhc2estimatesof0.177(0.015)and0.221(0.026)forEURandEAS,respectively,

arelowerthanapreviouslyreportedmixed-modelbasedheritabilityestimatesof0.32

(0.037)inEuropeans35.Similarly,ourT2Dhc2estimatesof0.242(0.013)and0.105(0.021)

forEURandEAS,respectively,arelowerthanapreviouslyreportedmixed-modelbased

estimateof0.51(0.065)inEuropeans35.Westressthatthisdiscrepancyislikelyduetothe

differencebetweencommon-SNPheritabilityhc2andtotalnarrow-senseheritabilityhg2.

Furthermore,estimatesoftheheritabilityofT2Dfromfamilystudiescanvary

significantly36,37.

WefindthegeneticeffectcorrelationinRAandT2Dtobe0.463(0.058)and0.621

(0.088),respectively,whilethegeneticimpactcorrelationisnotsignificantlydifferentat

0.455(0.056)and0.606(0.083).Thetransethnicgeneticimpactandeffectcorrelationfor

bothphenotypesaresignificantlydifferentfromboth1and0(Table2),showingthatwhile

thereiscleargeneticoverlapbetweenthephenotypes,theperalleleeffectssizesdiffer

significantlybetweenthetwopopulations.Thisisevidenceforthepresenceofdifferential

taggingofuntypedandunimputed,possiblyrarevariantsaswellasgene-gene,gene-

environmentorothernon-additiveinteractions,themarginaleffectssizesofwhichare

modifiedbychangesinallelefrequencybetweenthetwopopulations.Whilewithin-locus

(dominance)interactionsmayalsoplayarole38,themagnitudeofthiseffecthasbeen

debated39.



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Discussion

Wehavedevelopedthetransethnicgeneticeffectandgeneticimpactcorrelationand

providedanestimatorforthesequantitiesbasedonlyonsummary-levelGWASinformation

andsuitablereferencepanels.Wehaveappliedourestimatortoseveralphenotypes:

rheumatoidarthritis,type-2diabetesandgeneexpression.WhilethegEUVADISdataset

lacksthepowerrequiredtomakeinferencesaboutthegeneticcorrelationofsingleorsmall

subsetsofgenes,wecanmakeinferencesabouttheglobalstructureofgeneticcorrelationof

geneexpression.Wefindthattheglobalmeangeneticcorrelationislow,butthatit

increasessubstantiallywhentheheritabilityishighinbothpopulations.Inallphenotypes

analyzed,thegeneticcorrelationissignificantlydifferentfromboth0and1.Ourresults

showthatglobaldifferencesinSNPeffectsizeofcomplextraitscanbelarge.Incontrast,

effectssizesofgeneexpressionappeartobemoreconservedwherethereisstronggenetic

regulation.

Itisnotpossibletodrawconclusionsaboutpolygenicselectionfromestimatesof

transethnicgeneticcorrelation.Theeffectssizesmaybeidentical(!!" = 1)whilepolygenicselectionactstochangeonlytheallelefrequencies.Similarly,theeffectssizesmaybe

different(!!" < 1)withoutselection.DifferencesineffectssizesatcommonSNPscanresultfrommanyphenomena.Themarginaleffectsofageneinteractioncanbealtered

substantiallyviadifferencesinallelefrequency,whichmayariseduetodrift.Variantsthat

arerareinbothpopulationsbutdifferentiallytaggedmayalsocontribute,thoughweexpect

thiseffecttobesmall.Anothercontributionlikelycomesfromvariantsthatarerarein

populationonebutcommoninpopulationtwo(andviceversa),whichwillbefilteredinour

analysisbutmaybedifferentiallytagged.

Whilethegeneticcorrelationofmultiplephenotypesinonepopulationhasa

relativelystraightforwarddefinition,extendingthistomultiplepopulationsmotivates

multiplepossibleextensions.Inthisworkwehaveprovidedestimatorsforthecorrelation

ofgeneticeffectandgeneticimpactbutotherquantitiesrelatedtothesharedgeneticsof

complextraitsbetweenpopulationsincludethecorrelationofvarianceexplained

andproportionofsharedcausalvariantsbetweenthetwo

populations.Interestingly,whileourgoalwastoconstructanestimatorthatdeterminedthe

extentofgeneticsharingindependentofallelefrequency,weobservethatthecorrelationof

geneticeffectandgeneticimpactaresimilar.Furthermore,oursimulationsshowthatunder

⇢ge = Cor(�21�

21 ,�

22�

22)



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arandomeffectsmodelutilizingonlySNPswithallelefrequencyabove5%inboth

populationsthetruegeneticeffectandgeneticimpactcorrelationaresimilar.Weconclude

thatatvariantscommoninbothpopulations,differencesineffectsizeandnotallele

frequencyaredrivingthetransethnicphenotypicdifferencesinthesetraits.

Estimatesofthetransethnicgeneticcorrelationareimportantforseveralreasons.

Asdetailedinthiswork,theyprovideinsightintotheextentthatnon-additivegenetic

variationaffectsaphenotype,furtheringourunderstandingofthegeneticarchitectureof

complextraits.Furthermore,theymayhelpinformbestpracticesfortransethnicmeta-

analysis,potentiallyofferingimprovementsovercurrentmethodsthatutilizefsttocluster

populationsforanalysis4.Finally,thetransethnicgeneticcorrelationconstrainsthelimitof

outofsamplephenotypepredictivepower.Ifthemaximumwithinpopulationcorrelationof

predictedphenotypePtotruephenotypeYis ,thenthemaximumoutof

populationcorrelationis (Appendix).

Ourapproachtoestimatinggeneticcorrelationhastwomajoradvantagesover

mixed-modelbasedapproaches.First,utilizingsummarystatisticsallowsapplicationofthe

methodwithoutdata-sharingandprivacyconcernsthatcomewithrawgenotypes.Second,

ourapproachislinearinthenumberofSNPsavoidingthecomputationalbottleneck

requiredtoestimatethegeneticrelationshipmatrix.Conceptually,ourapproachisvery

similartothattakenbyLDscoreregression.Indeed,thediagonaloftheLDmatrixproduct

inonepopulationareexactlytheLD-scores( ).Onecouldignoreourlikelihood-

basedapproachanddefinecross-populationscores inordertoexploitthe

linearrelationship (asimilarapproachcanbetakenfor

thegeneticeffectcorrelation).SinceLD-scoreregressionhasbeensuccessfullyusedto

computethegeneticcorrelationoftwophenotypesinasinglepopulation,thisderivation

canbeviewedasanextensionofLD-scoreregressiontoonephenotypeintwodifferent

populations.Themaindifferenceinourapproachischoosingmaximumlikelihoodrather

thanregressioninordertofitthemodel.Acomparisonofourmethodtotheldscsoftware

showstheyperformsimilarlyasheritabilityestimators(FigureS5).

Ofcourse,ourmethodisnotwithoutdrawbacks.First,itrequiresalargesample

sizeandlargenumberofSNPstoachievestandarderrorslowenoughtogenerateaccurate

estimates.UntilrecentlylargesampleGWAShavebeenrareinnon-Europeanpopulations,

⇢max

Y P

=q

h21

⇢max

Y P

= ⇢ge

qh21

⌃21ii = li

ci =X

m

r1imr2im

E[Z1iZ2i] =

pN1N2

M⇢gi

qh21h

22ci



http://dx.doi.org/10.1101/036657


12

thoughtheyarebecomingmorecommon.Similarly,referencepanelqualitymaysufferin

non-Europeanpopulationsandthismayimpactdownstreamanalysis40.Second,itislimited

toanalyzingrelativelycommonSNPs,bothbecausehavinganaccuratediseasemodelis

importantfortheanalysisofrarevariantsandbecauseeffectsizeandcorrelationcoefficient

estimateshaveahighstandarderroratrareSNPs21.Third,ouranalysisiscurrentlylimited

toSNPsthatarepresentinbothpopulations.Indeeditiscurrentlyunclearhowbestto

handlepopulation-specificvariantsinthisframework.Finally,admixedpopulationsinduce

verylong-rangeLDthatisnotaccountedforinourapproachandwearethereforelimited

toun-admixedpopulations21.

Ouranalysisleavesopenseveralavenuesforfuturework.Weapproximately

maximizethelikelihoodofan!×!multivariatenormaldistributionviaamethodthatutilizesonlythediagonalelementsofeachblock,discardingcovarianceinformation

betweenZ-scores.Abetterapproximationmaylowerthestandarderroroftheestimator,

facilitatingananalysisofthegeneticcorrelationoffunctionalcategories,pathwaysand

geneticregions.Wewouldalsoliketoextendouranalysistoincludepopulationspecific

variantsaswellasvariantsatfrequenciesbetween1-5%orlowerthan1%.Oursimulations

indicatethathavinganaccuratediseasemodelisimportantfordeterminingthedifference

betweenthegeneticeffectandgeneticimpactcorrelationwhenrarevariantsareincluded.

Maximumlikelihoodapproachesarewellsuitedtodifferentgeneticarchitectures,for

exampleonecouldexplicitlymodeltherelationshipbetweenallelefrequencyandeffectsize

,whereα=-1correspondstotheinverseassumptionandα=0corresponds

totheindependenceassumption.However,amorethoroughapproachwouldincorporate

additionalsourcesofinformationsuchastheeffectofselection.

2f(1� f) / �↵



http://dx.doi.org/10.1101/036657


Appendix

Consider two GWAS of a phenotype conducted in di↵erent populations populations. Assume we have N1

individuals genotyped or imputed to M SNPs in study one and N2 individuals genotyped or imputed to

M SNPs in study two. Let X1, X2 and Y1, Y2 be the matrices of mean-centered genotypes and phenotypes

of the individuals in study one and two, respectively, with f1, f2 the allele frequencies of the M SNPs

common to both populations. Assuming Hardy-Weinberg equilibrium, the allele variances are �21 = 2f1(1�

f1),�22 = 2f2(1� f2). Let �1,�2 be the (unobserved) per-allele e↵ect sizes for each SNP in studies one and

two, respectively. Define the genetic impact correlation ⇢gi

= Cor(p�21�1,

p�22�2) and the genetic e↵ect

correlation ⇢ge

= Cor(�1,�2). We present a maximum likelihood framework for estimating the heritability of

the phenotype in study 1 and it’s standard error, the heritability of the phenotype in study 2 and it’s standard

error, and the genetic e↵ect and impact correlation of the phenotype between the studies and it’s standard

error given only the summary statistics Z1, Z2 and reference genotypes G1, G2 representing the populations

in the studies. We assume that genotypes are drawn randomly from populations with expected correlation

matrices ⌃1 (and similarly for study two), and that every SNP is causal with a normally distributed e↵ects

size (though this assumption is not necessary in practice, see Figure S1).

Genetic impact correlation

Let X 01 = X1p

�

21

(and similarly for study 2) be normalized genotype matrices. We consider the standard linear

model for generation of the phenotypes, where

Y1 = X 01�1 + ✏1

Y2 = X 02�2 + ✏2

For convenience of notation let h2ix

= ⇢gi

ph21h

22. We assume the SNP e↵ects follow the infinitesimal model,

where every SNP has an e↵ect size drawn from the normal distribution, and that the residuals are independent

for each individual and normally distributed:

0

@ �1

�2

1

A ⇠ N

0

@

2

4 0

0

3

5 ,1

M

2

4 h21IM h2

ix

IM

h2ix

IM

h22IM

3

5

1

A (1)

0

@ ✏1

✏2

1

A ⇠ N

0

@

2

4 0

0

3

5 ,

2

4 (1� h21)IM 0

0 (1� h22)IM

3

5

1

A (2)

1



http://dx.doi.org/10.1101/036657


where h21, h

22 are the heritability of the disease in study one and two, respectively, and ⇢

gi

is the genetic

impact correlation.

Using the above model, we compute the distribution of the observed Z scores as a function of the reference

panel correlations and the model parameters (h21, h

22, ⇢gi). Given a distribution for Z and an observation of

Z we can then choose parameters which give the highest probability of observing Z. First, we compute the

distribution of Z. It is well known that the Z-scores of a linear regression are normally distributed given

� when the sample size is large enough. Since P(Z) / P(Z|�)P(�) and the product of normal distributions

is normal, we only need to compute the unconditional mean and variance of Z to know its distribution.

Specifically, let Z = [Z>1 , Z>

2 ]>, then it’s mean is

E[Z] = E

2

4X

0>1 Y1pN1

X

0>2 Y2pN2

3

5 =

2

41pN1

(E[X 0>1 X 0

1]E[�1] + E[X 0>1 ]E[✏1])

1pN2

(E[X 0>2 X 0

2]E[�2] + E[X 0>2 ]E[✏2])

3

5 = 0

The within-population variance is:

Cov[Z1i, Z1j ] = E[Z1iZ1j ] = EX,�,✏

[E[Z1iZ1j |X,�, ✏]]

=1

N1EX,�,✏

[X 0>1i (X

01�1 + ✏1)(X

01�1 + ✏1)

>X 01j ]

=1

N1EX,�

[X 0>1i X

01�1�

>1 X 0>

1 X2j ] +1

N1EX,✏

[X 0>1i ✏1✏

>1 X

01j ]

=h21

MN1EX

[X 0>1i X

01X

0>1 X 0

1j ] +1� h2

1

N1EX

[X 0>1i X

01j ]

=h21

MN1(N1Mr1ij +N1

MX

m=1

r1imr1jm +N21

MX

m=1

r1imr1jm) +1� h2

1

N1r1ij

= r1ij +N1 + 1

Mh21⌃1(i)⌃

(j)1

where rpij

= ⌃pij

is the correlation coe�cient of SNP i and j in population p. Similarly, the between-

population variance is:

Cov[Z1i, Z2j ] =1p

N1N2EX,�

[X 0>1i X

01�1�

>2 X 0>

2 X 02j ] +

1pN1N2

EX,✏

[X 0>1i ✏1✏

>2 X

02j ]

=h2ix

MpN1N2

EX

[X 0>1i X

01X

0>2 X 0

2j ]

=h2ix

MpN1N2

(N1N2

MX

m=1

r1imr2jm)

=

pN1N2

Mh2ix

⌃1(i)⌃(j)2

where ⌃(i) denotes the i’th row of ⌃ and ⌃(j) denotes the j’th column. The covariance of the Z-scores is

2



http://dx.doi.org/10.1101/036657


thus

C = Var(Z) =

2

4 ⌃1 +N1+1M

h21⌃

21 h2

gx

pN1N2

M

⌃1⌃2

h2gx

pN1N2

M

⌃2⌃1 ⌃2 +N2+1M

h22⌃

22

3

5 (3)

and Z ⇠ N (0, C).

Genetic e↵ect correlation

Let hex

= ⇢ge

ph21h

22. We modify the procedure above to use mean-centered instead of normalized genotype

matrices and model the distribution of the e↵ect sizes as

0

@ �1

�2

1

A ⇠ N

0

B@

2

4 0

0

3

5 ,

2

64h

21

k�21k1

IM

h

expk�2

1k1k�22k1

IM

h

expk�2

1k1k�22k1

IM

h

22

k�22k1

IM

3

75

1

CA (4)

Notice that a linear model with e↵ects sizes acting on un-normalized genotypes is the same as a linear model

with e↵ect sizes acting on normalized genotypes under the substitution �1,2 !q

�21,2�1,2. Therefore the

covariance of Z-scores on the per allele scale can be immediately inferred from the prior derivation

C = Var(Z) =

2

64⌃1 +

N1+1k�2

1k1h21⌃1�2

1⌃1 h2gx

pN1N2p

k�21k1k�2

2k1

⌃1

p�21�

22⌃2

h2gx

pN1N2p

k�21k1k�2

2k1

⌃2

p�22�

21⌃1 ⌃2 +

N2+1k�2

2k1h22⌃2�2

2⌃2

3

75

Approximate maximum likelihood estimation

Let C =

2

4 C11 C12

C21 C22

3

5 be either of the above covariance matrices written in block form. We approximately

optimize the above likelihood as follows: first we find h21 and h2

2 by maximizing the likelihood corresponding

to C11 and C22, then we find ⇢gi

or ⇢ge

by maximizing the likelihood corresponding to C12:

l(h21|Z1,⌃,�) ⇡ �

MX

i=1

w11i

✓ln(C11ii) +

Z21i

C11ii

◆

l(h22|Z2,⌃,�) ⇡ �

MX

i=1

w22i

✓ln(C22ii) +

Z22i

C22ii

◆

l(⇢g{i,e}|Z, h2

1, h22,⌃,�) ⇡ �

MX

i=1

w12i

✓ln(C12ii) +

Z1iZ2i

C12ii

◆

Because we are discarding between-SNP covariance information (Cov(Z1i, Z1j)), highly correlated SNPs will

be overcounted in our approximate likelihood. As a simple example, notice that two SNPs in perfect LD

will each contribute identical terms to the approximate likelihood, and therefore should be downweighted

by a factor of 1/2. The extent to which SNP i is over-counted is exactly the i’th entry in it’s corresponding

3



http://dx.doi.org/10.1101/036657


LD-matrix product. Therefore we let wgi

jki

= 1/ (⌃j

⌃k

)ii

and wge

jki

= 1/⇣⌃

j

q�2j

�2k

⌃k

⌘

ii

to reduce the

variance in our estimates of the parameters h21, h

22, ⇢gi and ⇢

ge

.

Furthermore, rather than compute the full products ⌃21, ⌃

22 and ⌃1⌃2 over all M SNPs in the genome, we

choose a window size W and approximate the product by (⌃a

⌃b

)ii

=P

w=i+W

w=i�W

raiw

rbiw

. Though maximum

likelihood estimation admits a straightforward estimate of the standard error via the fisher information, we

found these estimates to be inaccurate in practice. Instead, we use block jackknife with block size equal to

min(100, M

200 ) SNPs to ensure that blocks are large enough to remove residual correlations.

Out of population prediction of phenotypic values

Consider using the results of a GWAS with perfect power in population 2 to predict the phenotypic values of

a set of individuals from population 1. This defines the upper limit of the correlation of true and predicted

phenotypic values. Let the true values of the e↵ects sizes in population 2 be �2. Let the true phenotypes

in population 1 be Y = X1�1 + ✏1 while the predicted phenotypes are P = X1�2. We are interested in the

correlaiton of the predicted and true phenotypes ⇢MAX

Y P

= Cor(Y, P ). Notice that, given X, the true and

predicted phenotype of each individual is an a�ne transformation of a multivariate normal random variable

(�) 2

4 Yi

Pi

3

5 =

2

4 X(i) 0M

0M

X(i)

3

5

2

4 �1

�2

3

5+

2

4 ✏i

0

3

5

and therefore (Yi

, Pi

) for individual i is multivariate normal with expected covariance matrix

EX

[Cov(Yi

, Pi

)] = EX

2

4 X(i) 0M

0M

X(i)

3

5

2

641

k�21k1

IM

h

expk�2

1k1k�22k1

IM

h

expk�2

1k1k�22k1

IM

h

22

k�22k1

IM

3

75

2

4 X(i) 0M

0M

X(i)

3

5>

= EX

2

64

Pm

X

2im

k�21k1

h

ex

Pm

X

2imp

k�21k1k�2

2k1

h

ex

Pm

X

2imp

k�21k1k�2

2k1

h

22

Pm

X

2im

k�22k1

3

75

=

2

4 1 hex

qk�2

1k1

k�22k1

hex

qk�2

1k1

k�22k1

h22k�2

1k1

k�22k1

3

5

and therefore the expected correlation E[Cor(Yi

, Pi

)] is h

exph

22

qk�2

1k1

k�22k1

k�22k1

k�21k1

= ⇢ge

ph21. The expected popula-

tion correlation tends to the sample correlation as the number of samples increases, therefore

⇢MAX

Y P

= Cor(Y, P ) ! ⇢ge

qh21 (5)

as N ! 1

4



http://dx.doi.org/10.1101/036657


13

DescriptionofSupplementaldata

Supplementaldataincludefivefiguresandonetable.

Acknowledgements

TheauthorswouldliketoacknowledgeLiorPachterandHilaryFinucaneforinsightful

discussionabouttheproblem.BCBissupportedbytheNSFGRFP.ALPissupportedbyNIH

grantR01HG006399.NZissupportedbyNIHgrantK25HL121295.

WebResources

Popcornisavailableathttps://github.com/brielin/popcorn

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FigureTitlesandLegends



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Figure1:Standarderrorofourestimateofgeneticeffectandimpactcorrelationasa

functionofthenumberofSNPsandindividuals.Phenotypesweresimulatedwithh12=0.5,

h22=0.5andρgi,e=0.5andgenotypesweresimulatedwithHapGen2tomatchEuropean(EUR)

andEastAsian(EAS)populations.

Figure2:DistributionofgeneticcorrelationcomparisonbetweenpopcornandGCTA.

Distributionwascomputedusingagaussiankdeonthesetofgeneticcorrelationestimates.

Figure3:Geneticcorrelationasafunctionofheritabilityforgeneexpression.Themeanand

standarderrorofthegeneticcorrelationofthesetofgeneswithh12andh22exceeding

thresholdhineachanalysis(y-axis)isplottedagainsth(x-axis).

Tables

Table1:Trueandestimatedvaluesofthegeneticimpactandeffectcorrelationinsimulated

EUR-likeandEAS-likegenotypes.Resultsaretheaverageof100simulationswith

phenotypeheritabilityof0.5ineachpopulation.

Table2:HeritabilityandgeneticcorrelationofRAandT2DbetweenEURandEAS

populations.EURRAdatacontained8,875casesand29,367controlsforastudyprevalence

of0.23.EASRAdatacontained4,873casesand17,642controlsforastudyprevalenceof

0.22.RAdiseaseprevalencewasassumedtobe0.5%inbothpopulations7.T2DEURdata

contained12171casesand56862controlsforastudyprevalenceof0.18.T2DEASdata

contained6952casesand11865controlsforastudyprevalenceof0.37.T2DEUR

prevalencewasassumedtobe8%27whileT2DEASprevalencewasassumedtobe9%41.

RA T2D

MAF>0.01 MAF>0.05

Model Independent 0.500 0.478 0.490 0.459 0.500 0.489 0.490 0.469

Inverse 0.438 0.500 0.520 0.493 0.479 0.500 0.503 0.489

Rare 0.500 0.482 NA NA 0.500 0.498 0.916 0.731

Common 0.500 0.500 0.308 0.286 0.500 0.500 0.298 0.277

Difference 0.500 0.419 0.428 0.414 0.500 0.461 0.479 0.468

Adversarial 0.695 0.588 0.730 0.676 0.698 0.648 0.739 0.698

⇢ge ⇢gi ⇢ge ⇢gi ⇢ge ⇢gi ⇢ge ⇢gi



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hEUR2obs 0.265(0.022) 0.146(0.008)

hEAS2obs 0.320(0.038) 0.093(0.020)

hEUR2lia 0.177(0.015) 0.242(0.013)

hEAS2lia 0.221(0.026) 0.105(0.021)

ρge 0.463(0.058) 0.621(0.088)

pρge>0 1.37e-15 1.70e-12

pρge<1 2.53e-20 1.66e-05

ρgi 0.455(0.056) 0.606(0.083)

pρgi>0 8.16e-16 2.85e-13

pρgi<1 4.87e-22 2.06e-06



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