AmJHumGenet306186431978MultifactorialInheritancewithCulturalTransmi - PDF document

AmJHumGenet30:618-643,1978MultifactorialInheritancewithCulturalTransmissionandAssortativeMating.I.DescriptionandBasicPropertiesoftheUnitaryModelsJOHNRICE,'C.ROBERTCiONINGER,ANDTHEODOREREICHThefamilialtransmissionofcomplexdevelopmentaltraitsoftenappearstobeinfluencedbythevariationofmanydiscretegeneticandenvironmentalfactors[1].Thetransmissionofsuchtraitsmaybedescribedandquantifiedbymultifactorialmodelsinwhichmanyfactorsofsmalleffectplayarole.DifferentformulationsofpolygenicormultifactorialmodelsofcontinuousphenotypesorthresholdcharactershavebeendescribedbyseveralauthorssincetheseminalworkofFisher[2]andarereviewedelsewhere[3,4].Althoughsomeformulationsallowforadditiveenvironmentalcontributions,differencesbetweenpolygenicandculturalinheritancehavereceivedlittleattention.Environmentaleffectsmaysometimessimulategenetictransmission[5],butotherpatternsmayalsooccur[6,7].Cavalli-SforzaandFeldman[6]notedthatculturalinheritancemaybeeffecteddirectlybysociallearningandmodelingorbyparentsteachingtheiroffspringcertaincustomsandpreferencesaboutdiet,environment,andotheractivities.Theyemphasizetheplasticityofgenotypesinresponsetotheenvironmentanddealexplicitlywiththeeffectofparentalphenotypesonthedevelopmentofthesamephenotypeintheoffspring.Cavalli-SforzaandFeldmanrecognizethatothergeneticandenvironmentalfactorsbesidesthephenotypeoftheparentmayinfluencethedevelopmentofachild,buttheirmodelonlydealswithdirectphenotype-to-phenotypeeffects.Incontrast,Morton[8]andRaoetal.[9,10]describelinearmodelsoffamilialresemblancewhichareprimarilyconcernedwithfamilymembersrearedcontem-poraneously.Theirmodelsallowforenvironmentaleffectscommontochildrenrearedtogether("commonenvironment")whichmayincreasephenotypicresemblance.Theiroriginalmodel[9]alsoallowsforacorrelationbetweenmidparent-genotypeandcommonenvironm

entbutdoesnotallowfordirectculturaltransmissionfromparenttooffspring.Theirlatermodel[10]allowsforsometypesofnongenetictransmissionbetweengenerationsatequilibrium,butthepropertiesofsuchmodelshavenotbeensystematicallydescribed.ReceivedApril28,1978;revisedAugust8,1978.ThisworkwassupportedinpartbyU.S.PublicHealthServicegrantsAA-00209,AA-03539,MH-14677,MH-25430,andResearchScientistDevelopmentAwardMH-00048(C.R.C.).IAllauthors:DepartmentofPsychiatry,WashingtonUniversity,SchoolofMedicineandtheJewishHospitalofSt.Louis,216SouthKingshighway,St.Louis,Missouri63110.©1978bytheAmericanSocietyofHumanGenetics.Allrightsreserved.618 MULTIFACTORIALINHERITANCETofurtherourunderstandingofthetransmissionofcomplexdevelopmentaltraitslikeheight,weight,bloodpressure,diabetes,hypercholesterolemia,intelligence,andpsychiatricphenotypes,morecomprehensivemodelsareneededinwhichtheeffectofbothpolygenicandculturalinheritancearespecifiedinpedigreesextendedoverseveralgenerations.Inaddition,theeffectsofnonrandommatingmustbetakenintoaccountsincesuchtraitsoftenshowstrongassortativemating[II-13].In1918,Fisherpresentedanextensivetreatmentofassortativematingforpolygenictraits.AnannotatedaccountofFisher'spaperhasbeenpresentedbyMoranandSmith[14],andmanyofhisresultshavebeenderivedusingamoreelementarymethodbyCrowandFelsenstein[15].Thesetreatmentsareonlyvalidforpolygenictraitswithoutculturaltransmission.Therefore,theextensionofthemultifactorialmodeltoincludeculturaltransmissionrequiresageneralizationofFisher'streatmentofassortativemating.Inthispaperwewilldescribeageneralmodelofmultifactorialinheritancewithculturaltransmissionandassortativemating.Forclaritywewilldistinguishamongthepolygenicmodelinwhichthereisnoculturaltransmission,thecultukurlmodelinwhichthereisnogenetictransmission,andothermultifu(to/ialmodelsinwhichboth

geneticandculturaltransmissionarepossible.Weallowforthresholdeffects,commonenvironment,maternalandpaternaleffects,andassortativematingwithoutselection.Wefollowthedistinctionsamongassortativemating,selectivemating,andinbreedingdescribedbyLewontinetal.[16].TheeffectsofselectivematingonthepolygenicmodelhavebeenconsideredbyWilson[17].andWilson'scritiqueofFisher'smodelofassortativematinghasbeencriticizedelsewhere[18].InourmodelandthatofFisher,genefrequenciesdonotchangewithassortativematingbydefinition.Inaddition,weassumethattheoverallfrequenciesofculturalfactorsrelevanttothedevelopmentofatraitdonotchange,ratheronlythedistributionoftheseeventsbetweenandwithinfamiliesischanged.Apositivephenotypiccorrelationbetweenmatesinducesapositivecorrelationbetweenthetransmissibleculturalfactorsofthematesaswellasbetweenthegenesrelevanttothetrait.Thisleadstoanincreaseinthevarianceofthetraitinthepopulationwhichdependsontheextenttowhichculturalfactorsareinherited.Weareconcernedwithboththeculturalandthegeneticfactorswhichinfluencethetransmissionofatraitfromparenttooffspringsinceaprioriassumptionsabouttheirrelativecontributionsmayobscureratherthanenhanceourunderstanding.Itisnecessarytoprovideamodelwhichallowsforbothtypesoftransmissionsimultane-ously,ratherthanassumeforconveniencethatculturalfactorsarenegligible.However,formanytraitsthereisinsufficientdatatoestimatebothheritabilityandtheeffectofculturalinheritance.Accordingly,weintroducetheunitirvmodelwhichdoesnottrytofurtherpartitionthetransmissiblevariance,butwhichdoesallowforhypothesistestingandforapproximateestimationofthecombinedimportanceofgeneticandculturalfactors.DESCRIPTIONOFTHEMODELSThefollowingassumptionsdefinethegeneralmultifactorialmodelandaremoregeneralthanmanypriorformulationsofthismodel:(1)AquantitativecharacterPmay619 bepartitioned

asP=A+B+E,whereAandBdenotealltheeffectsofgeneticandculturalfactorstransmittedfromparenttooffspring,respectively,andEdenotesallothereffectswhichareusuallyrandomenvironmentalinfluences,withthecovariance(A,E)andcovariance(B,E)equalto0.(2)Thetransmissiblefactorsactadditivelyandeachhassmalleffectrelativetothetotalphenotypicvariance.(3)TherearemanytransmissiblefactorssothatPmaybeassumedtobenormallydistributed.Weassume,further,thatthejointdistributionofA,B,andEisatrivariatenormal.(4)InordertoallowforbothbiologicalinheritancethroughDNAandculturalinheritance,noaprioriassumptionsaremadeabouttransmissionprobabilitiesorheritability.InordertoallowforassortativematingbasedondirectphenotypicpreferencesasinFisher'sfirstmodel([14],page50),weintroducetwoadditionalassumptions:(5)Assortativematingisrecurrent(i.e.,occurseverygeneration)andisbaseddirectlyonphenotypicpreferences.(6)Thephenotypiccorrelationbetweenmatesisofconstantmagnitudethroughgenerations.Thecausalsystemoftransmissionfromparenttooffspringforthegeneralmultifactorialmodelwillbeexplicatedindetailinthesecondpaperinthisseries119].Briefly,thepathcoefficientfromtheadditivegeneticvalue(A)ofaparenttotheadditivegeneticvalue(A)ofhisoffspringis1/2sincetheparentcontributesprecisely1/2ofhisgenes.Thepathcoefficientfromthetransmissibleculturalvalue(B)ofaparentto(B)ofhisoffspringisdenotedby13andwillbediscussedindetailbelow.TheprimarycorrelationmbetweentheparentalphenotypesinducescorrelationsbetweentheAandBoftheparents.Inaddition,evenifAandBwithinthesameindividualareuncorrelatedinarandommatingpopulation,anonzerocorrelationwispresentatequilibriumifassortativematingoccurs.Thisformulationofthemultifactorialmodelisdescribedbyfourparameters:h2=V4IVp,b2=VBIVP,(3,andm,whereVdenotesthevarianceofthesubscriptedvariable.Otherparametersmaybeintroducedtodesc

ribephenomenasuchasacommonenvironmentofrearingoraprimarygene-culturecovariance.Thecorrelationsbetweenrelativesintermsoftheparametersofthemodelcanbedeterminedbytheuseofpathanalysis[20-22]toprovideasystemofnonlinearequationsforparameterestimation.Toassesstheeffectofassortativematingonarandommatingpopulation,itisfirstnecessarytocomputethechangesinvarianceofAandBwhichresultfromtheassortativemating.Theparametersatequilibriumcanthenberelatedtotherandommatingparameters,andtheequilibriumcorrelationstotherandommatingones.Itisthenpossibletopartitiontheequilibriumvariancetodeterminetheproportionattributabletoassortativemating.Wewilltreatthreesubmodelsofthegeneralmodelinwhatfollows,anddefertheanalysisofthefullmodeltoacompanionpaper.Thefirstsubmodelistheculturalmodel,whichresultsifVA=0.ThesecondistheusualpolygenicmodelwhichresultsifVB=0.Thethirdistermedthepseudopolygenicmodelandresultswhen(3=1/2inthegeneralmodel.Eachofthethreemodelsischaracterizedasunitarysincetransmissionisdeterminedbyasinglevariable(viz.B,A,orT=A+B).Eachofthesemodelsisinturnaspecialcaseoftheunitarvmodelwhichisdefinedbelow.Inadditiontotheirownusefulness,thedevelopmentofthesemodelsisrequisiteforthegeneraltreatment.RICEETAL.620 MULTIFACTORIALINHERITANCEDESCRIPTIONOFTHECULTURALMODELIntheculturalmodel,weassumethatV.,=0,sothatP=B+Ewithcov(B,E)=0.Inarandommatingpopulation,thepathcoefficient/3isequaltothecorrelationcoeffi-cientbetweentheBofaparentandtheBofhisoffspring.Onceassortativematingoccurs,thevarianceisincreasedinsuccessivegenerationsuntilequilibriumisreached.Inthemultipleregressionequationofanoffspringonhisparents,itisassumedthatthepartialregressioncoefficients'isunaffectedbyassortativemating:BP='BB(.-1)I+3'B(F)n,+fI'',",(1)whereMdenotesthemother,Fdenotesthefather,F'nistheresidualtermcorrespond-ingtofluctuationaboutmi

dparentculturalvalue,andf'nisthepartialregressioncoeffi-cientforF'nItisshownlaterthatinanequilibriumpopulationtheregressioncoefficient3'andthepathcoefficient/3areequal.Weuseaprime(')heretoem-phasizethatthesearetheconcrete(unstandardized)variablesrelatedbyregressioncoefficientsratherthanpathcoefficients,althoughweassume,forconvenience,thatallvariableshavezeromean.ThecorrespondingpathequationisgivenbyB11+1=/3nB(A)1+/311B(F)n+f,,F,,(1.1)where3=,Bandf,,-SiF'n(TB'Rn+1C'+Considerthepathdiagramdisplayedinfigure1wherendenotesthegenerationnumber,inthephenotypiccorrelationbetweenmates,andrBBnthecorrelationbetweentheculturaltransmissiblefactorsofmatesingenerationn.Forsimplicity,uncorrelatedresidualtermsarenotshownsincetheydonotcontributetofamilialresemblance.Althoughweassumethatinisrecurrentandofconstantmagnitudethroughgenera-tions,I'BB,dependsonnbecausetheratioofthevarianceoftransmissiblefactorsVB,,tothetotalphenotypicvarianceVp,,changeswithrecurrentassortativematinguntilequilibriumisreached.Specifically,fromfigure1,thecorrelationbetweenthetransmissiblefactorsis/'BB,=nmb,,2.Assumptions5and6maybechangedifthecorrelationbetweenmatesisnotbaseddirectlyonthephenotype.Fisher'streatmentoftherelationshipbetweenin(Fisher'spi)andrBB(thenongeneticanaloguetoFisher'sA)isvalidevenforculturaltransmission.Onlythecaseofdirectphenotypicpreferencewillbepresentedhere,butotherpatternsareeasilyelaborated[19].ChangesinVaricanceDuetoAssortativeMatingBytakingthecovarianceoftherighthandsideofequation(1)withitself,andrecallingthatCOV(B',,+I,B',,+,)=VB',,+,,wehaveVB#I+1=/312VB'(1)n,+312VB1(F)n+2/3'2rB'B',,B'(.1J)n,CB'(F)II+f',VFn(2)Considerfirstarandommatingpopulationatequilibriumdenotedhereasgenerationo,withVB',,+,=VB'(11X,,=VAF,},,=VB'OandwithCOV(B(,.1,,B(,,,,)=0.Equation(2)reducesto621 mPnPnbnjjbnfltrBBnPan=

/2UBS+,g'-l~Pn+1Bn+1bn+lFIG.1.-Parentaldeterminationofanoffspring'sphenotypefortheculturalmodel.Thegenerationnumberoftheparentsandtheoffspringaredenotedbynandn+1,respectively.Thedouble-headedarrowindicatesthecorrelationinducedbytheprimarycorrelationm.VB-,=23"''Vil,+f'',VF',(2.1)sothatf'I(r,,=(1-2j3,2)1.(2.2)Nowsupposethatassortativematingoccursandaffectsonlythevarianceoftransmissiblefactors(assumption7).andspecificallythatVf,,,,=V,,,andf,,'=f,'foralln,sothatfn'oF,=f0o(F'o.Fromequation(1)and(2.2),thechangeinthevarianceoftransmissiblefactorsfromrecurrentassortativematingisseentofollowtherecursionrelationVB'+I=2/132(1+IrfnB',B)VB',n+(1-2/312)VB'.(3)Underrecurrentassortativematingforamultifactorialtrait,equilibriumisdefinedbythegenerationswhereVB,,,=VB,,+,=VB,.Quantitiesnotindexedbygenerationsareassumedtobeatequilibriumunderassortativemating,andquantitiessubscriptedbyorefertotheirrandommatingvalues.Atequilibrium,equation(3)yieldsI-2/312V-=I-2/312(1+I'B')VBR¼,.(3.1)Whentheassortativematingpopulationisatequilibrium,thepathcoefficient/3fromtheBofaparenttotheBofhisoffspringis/3=/3'(JTB,)/(C(BR)=/3'.Intherandommatingpopulation,/3,,=/3'(Bo)l((/rB,,)=3'.andwehavetheremarkablefactthatthepathcoefficient/3isthesameunderbothequilibriumconditionsandequalstheRICEETAL.622 MULTIFACTORIALINHERITANCEPIN'P4mBROBPPFIG.2.-Pathdiagramdepictingparent-offspringandsibling-siblingcorrelationsatequilibriumfortheculturalmodel.partialregressioncoefficient13'.Thisisanalogoustothefactthatthepathcoefficientbetweenthegenicvaluesofparentandoffspringis1/2inanequilibriumpopulation,whetheritismatingrandomlyorassortatively([22],p.268).Since/3'=/3andrBfB?=rmB,therewillbenoconfusioninreturningtotheconventionofrepresentingbothstandardizedandunstandardizedquantitiesbythesamesymbolwithouttheuseofcumbersomeprimes.Thech

angeinvarianceoftransmissiblefactors,AVB,,dependsonthesignofthecorrelationbetweenmates,increasingifthecorrelationispositiveanddecreasingifthecorrelationisnegative.Thismaybeseenfromequation(3.1)or(3.2),notingthat(1+rBn)isalwayspositive:AVBVB132(rB)(3.2)Giventhatassortativematingonlychangesthevarianceoftransmissiblefactors(assumption7),andspecificallythatVE=VE,=V*:",thephenotypicvarianceatequilibriumisseentodependon/3,rmn,andVUp,:VP=Vi+VE=VUP+AVB=VP,)+2[32rBBVB)(4)1-2t32(I+rBB)RelationshipbetweenParametersunderRandomandAssortativeMatingThepathdiagraminfigure2depictstherelationshipbetweenmates,parent-offspringpairs,andsiblingsatequilibrium.Residualtermsareomittedsincetheydonot623 contributetofamilialresemblance.Bypathanalysis,itcanbeshownthatatequilibriumunderassortativemating,B=lfl),(5)wherebV=VB=/3j-(1+in)b2,(6)where-,),,=parent-offspringcorrelation,andr,()=2/32(I+I-BB)b2,(7)wherer,,=correlationoftwosibseachrearedbythesameparents.Underrandommating,in=0andequations(6)and(7)reduceto'/)(*=f3/1,2(6.1)and,,*'2/32/,).(7.1)Equation(6)mayberearrangedtogiveausefulexpressionfortheproportionofthetotalvarianceduetoculturalfactors:=J)()(8)13(1+in)Fromequations(5)and(8),thecorrelationbetweentransmissiblefactorsofmatesisI'B=MIO+m)(9)Further,fromstraightforwardapplicationsofequations(3)through(9),therandommatingparametersr,(,*,red*,andVP,aregivenby1*-(1+m)r,)((-2/32)-2/3mr,2(10)I.])-(I+M)2(1-2/32)-2nij0r,,,,*=2/3r,,,*,and(11)I[1-+2in72'_IVP1.(12)Theseequationsmayinturnbesolvedfortheassortativematingparametersatequilibrium,yieldingI2-I-2/32)l-22-8nrp,*(/3-rp(*)]}r=(1+rn)4r(/3-.0(13)r-2=3r[I+mr)]1and(14)1O+mn[3(+rn)JVP(I~~-2/32)(1I+rn)21V).(15)[(I(1+-n)2-2mr,,,Ifr),,,*=/3,theexpressionsgiveninequations(13)to(15)simplify,andtheRICEETAL.624 MULTIFACTORIALINHERITANCE1.0r*0=0.6P0.8r~~~~~~~~0r~.4

0.r*.=0.20.6rpoO.1\b20.40.200.20.40.6FIG.3.-Relationshipbetween,8andb02inarandommatingpopulationfortheindicatedparent-offspringcorrelations.parametersassumetheirmaximalvalue:rp,(=3(1+m),(13.1)rQ(=2,((1+m),and(14.1)VP=~I-232(l82V.(15.1)Fromequation(6.1),itcanbeseenthatb,=Iwhenrp,*=/3,sothatrBBn=mforalln.Thus,thelargestincreasesinparametervalueswilloccurwhen,/,m,andb02arehigh.Propertiesof/Therelationshipbetween[3andb2isillustratedintermsofrandommatingparametersinfigure3andintermsofassortativematingparametersinfigure4.Thecorrelationmisfixedat.4,andtherandommatingvaluesofifareshownforvariousvaluesofrp,)inapopulationatequilibrium.Equation(10)isfirstusedtocalculater,0*from/3,rp,andm=.4,andthenb2isfoundbydividingthisvalueofrp,0*by/3.Figures3and4demonstratethatfortheculturalmodelthephenotypicparent-offspring625 1.0rp:O0.7Po'0.60.8ro=.rpo=0.40.b2X0.4-0.200.20.40.6FIG.4.-Relationshipbetweenf3andb02inapopulationwithm=0.4fortheindicatedparent-offspringcorrelations.correlationmaybeapoorindicatoroftheimportanceoftransmissiblefactors.If/3issmall,thenrp,,maybesmallwithb2large,andif/8islarge,thenrp(,maybelargewithb2small.Indeed,theeffectivenessofaparentasateacherormodelforsociallearningisacrucialconsiderationintheassessmentoffamilialfactors.Although/3isnotfixedat1/2asisthepathcoefficientforthepolygenicmodel,otherconsiderationsrestricttherangeof3andthecorrelationbetweenparentsandoffspring.Inorderforthevalueoftheparent-offspringcorrelationinequation(13)tobereal,thetermundertheradicalmustnotbenegative;thatis,(1-2,32)[1-2,2-8mrO*(,3-rp,,*)].Therefore,solvingfortherootsof/,anupperboundfor1,31isgivenby1/31-2mIrp,,*+[1/2+4mIrpo*I(1+m)].(16)Atediouscomputationshowsthatequation(16)isalsoequivalenttotheconstraintsthatIrp,,,--2(1+m)rp,,*16or626RICEETAL.b22b2(16.2) MULTIFACTORIALINHERITANCETherequirem

entthatthephenotypicandculturalvariancereachafiniteequilibriumpointalsoplacesconstraintsonthemagnitudeof1.Rearrangingequation(3.1),weobservethatVBo1-2/32(1+mb2)(17)NotingthattheratioofVB,toVBbecomessmallandthatbapproaches1asVBincreases,weobtaintheboundthat11(18)Intherandommatingpopulationwemusthave1131l/V27sothatvaluesof1/81greaterthanthosegivenbyeitherequations(16)or(18)orl/\,I2arenotpossible.As/3approachesthislatterbound,thephenotypicvarianceapproachesinfinity.Thus,attheupperboundwith/3=l/V2(1+m)andb2=1,theparent-offspringcorrelation,obtainedfromequation(13.1)wouldbe[(I+mn)/2]i,avaluemuchlargerthanthoseusuallyobservedinpractice.Also,ifselectionpressurepreventedtoolargeanincreaseinthephenotypicvariance,itcouldreduce/3insucceedinggenerations.Withculturalinheritance/3mightvaryfromgenerationtogenerationaswellasfromfamilytofamily.Accordingly,iffamilieswithhigh/3wereselectedagainst,/8woulddecrease.TheApproachtoEquilibriumTheapproachtoequilibriumunderrecurrentassortativematingdependsonthemagnitudesof/3andmandisusuallyquiterapidforbothculturalandbiologicalinheritance.Equation(3)canbeusedtodeterminethenumberofgenerationsrequiredtoreachequilibriumasshowninfigure5.Infigure5theparametersmandrp,arefixedat.4and.5,respectively,andequilibriumvarianceswerechosentobe1.0.Thephenotypicvariancethroughsucceedinggenerationsisplottedforvariousvaluesof/3withn=0denotingtherandommatinggeneration.Evenwith/31/2,equilibriumiscloselyapproachedwithinafewgenerations.Accordingly,assumption6(thatthephenotypiccorrelationbetweenmatesisconstantthroughgenerations)isnotparticu-larlyrestrictive.MaternalandPaternalEffectsInthepresenceofculturalinheritance,/3neednotbethesameforeachparent.Inmanycontemporaryculturesthemothertendstohaveagreaterresponsibilityinchildrearingsothatthecorrelationbetweentransmissiblefacto

rsmaybegreaterbetweenmotherandchildthanbetweenfatherandchild.Insometraits,suchasbirthweight,uterineenvironmentmayplayamediatingroleinthetransmissionofthetraitandseparatecorrelationsbetweenmotherandchild(rMo)andbetweenfatherandchild(rbo)areneeded.Accordingly,let/Mand/Fdenotethemother-childandfather-childpathcoefficientbetweentransmissiblefactors,respectively.Equation(1)thenbecomesBn+1=/3MB(M)n+/3B(F)ni+(1-/M2-F2)2(BoFoF.,(19)sothat627 RICEETAL.8075012345678910nFIG.5.-Approachtoequilibriumthrough10generationsexpressedintermsofpercentageofthefinalequilibriumvalueofthepopulationvariance.misfixedat.4,rp0isfixedat.5,and(3variesfrom.45forthetopcurveto.55forthebottomone.VB1-(PM2+13F2+2f3MJ3FrBB)(20)andVPVP0+(P8M2+PAF2)0OBVBo-(11-(PM2+13F2+2PMJ3FrBB)(21)Intherandommatingpopulation,thefather-offspringcorrelationrFp*isequalto/3Fbo2,andthemother-offspringcorrelationisf3Mbo2.Otherformulaemaybeeasilymodifiedbysubstituting(PM+PF)for2,(PM2+PF2)for2/82,and(PM2+OF2+2PMPFrBB)for2/32(1+rBB).Additionalextensionsinwhich/dependsonboththesexoftheparentandthesexofthechild,assuggestedbysomeforjuveniledelinquency[23],areeasilyelaborated.THEPOLYGENICMODELThespecialcaseofthegeneralmultifactorialmodelinwhichalltransmissiblefactorsareadditivegenesistermedthepolygenicmodel.Inthiscase,theabovederivationsarevalidwith/8fixedat1/2andBreplacedbyA.628 MULTIFACTORIALINHERITANCEInthissetting,theregressionofanoffspringonhisparentswhichisgiveninequation(1)becomesA11+1=V12An+112A4+2(orA°/0S)S,(22)whereSisthesegregationfrommid-parentgenotype.Similarly,setting/3=1/2inequations(3.1)and(4),thevariancesunderassortativematingatequilibriumareVA=1i(23)andVP=VPO+1rAVA,,(24)Theseexpressionsagreewithequations(4.8.11)and(4.9.6)ofCrowandKimura[24]withalargenumberofeffectivegenes.Theirderivationdependsheavilyupontheassumptionofp

olygenicinheritance,whereasintheprecedingderivation,theonlygeneticassumptionrequiredisthatthepathcoefficientfromAofaparenttoAofhisoffspringis1/2.THEPSEUDOPOLYGENICMODELNowconsiderthespecialcaseofthegeneralmodelwith,3=1/2.Equations(1)and(22)maybecombinedtoyieldQn+I=1/2Qn+1/2Qn+½/2ro00F'(25)whereF'hasvariance1,andQiisgivenbyQ,=Ai+B,.TheearlierargumentsmaybeagainappliedtoshowVGIV/'Q,and(26)1-rrQQVp=Vp,,+1rQQO(27)whereq2=VQ,and(28)rQQ=mq2(29)InthegeneralcasewecoulddefineT0+±=An+1+Bn+1(30)=1/2(M)n+/2A(F)n+13B(,)n,,+/3B(F)n+f'F.(30.1)butfor/3+1/2,theanalogueofequation(25)doesnothold,sothatgeneticandculturalfactorsmustbeconsideredseparately[19].Thecasewhen,3=1/2istermedthepseudopolygenicmodelsinceallcorrelationsbetweenrelativesinintactfamilies(exceptthecorrelationbetweenmonozygotictwins)maybeexpressedintermsofmandq2.Asaresult,b2andh2cannotbeestimatedseparatelyunlessseparationdataare629 present.Sincethepseudopolygenicmodelmimicsgeneticinheritanceinintactfamilies,theaprioriassumptionofapolygenicmodelisuntestable.Techniques,suchasthosefortheresolutionofheterogeneitydescribedbyReichetal.[25-27],whicharevalidforthepseudopolygenicmodel,arethereforedesirable.THEUNITARYMODELToanalyzeeachoftheabovemodelsinauniformway,weintroduceanewmodel,theunitarymodel,whereweassumeP=T+E,andthattheregressionofanoffspringonhisparentsisgivenbyTn+1=TT(M)n+TT(F)n+\|1-22(ofTo/orF.)F(31)sothatVT(1-2r2)VT(and(32)12r2(I+rTT)vPv~~eI-272(I1+r7T)'where2=VTand(34)VPrTT=mt2.(35)Submodelsoftheunitarymodelinclude:(1)T=3,T=B(theculturalmodel);(2)T=1/2,T=A(thepolygenicmodel);and(3)T=1/2,T=A+B(thepseudopolygenicmodel).Wehave:rpo=Tr(1+m)t2,(36)roo=2T2(1+rTT)t2(37)r*(1+m)rpo(1-2r2)-2rmrpo2(38)(1+m)2(1-2r2)-22mrpo2r00*=2rrpo*,(39)VPO=[]-I+M)2(I-2T2)V,(40)1-2r2-{(-2T2)[122T-8mrpo*(T-ro*)]})4PO~~~~~4m(r-rpo*)-2rr0[I+mr]and(42)

VP[(I-2r2)(I+m)21V2m.(43)[I(12-2)(I+M)2-2mrp2J630RICEETAL. MULTIFACTORIALINHERITANCEEXTENSIONSOFTHEUNITARYMODELCommonEnvironmentItfollowsfromequations(36)and(37)thatifrisequalto1/2(asinthepolygenicorpseudopolygenicmodel)orislessthan1/2(asmayoccurwithculturalinheritance),thenfullsiblingpairscannotbemoresimilarthanparent-offspringpairs.However,r0oisoftenobservedtobegreaterthanr,,.Fromthepolygenicmodeltheincreasedsimilarityoffullsibsisoftenattributedtotheeffectsofdominancedeviation[24]sincethisincreasesr0,oandnotrpO.However,Morton[8]describesthenotionofdominancedeviationforpolygenesas"farfetched"andcitesdatashowingthatdominancedecreaseswithsmallgeneeffect.Manyearlyinvestigatorsdidnotallowfortheresemblancebetweensibstobedueinparttothesimilarityoftheirsharedenvironmentandlargedominanceeffectswerepostulated(e.g.,in1918Fisherestimatedthedominancevarianceforheightas25%).Instudyingenvironmentallyinfluencedtraitsinhumanswhereenvironmentscannotbeexperimentallyrandomized,theeffectofcommonorsharedenvironmentalinfluencescannotbeignored.Raoetal.[28]foundcorrelationsforheightandweighttodeclinefrom.572and.547fordizygoustwinsto.127and.120forotherfullsibsborn20yearsapartandnotedasteadydeclinerelatedtoagedifferences.Geneticfactorscannotexplainsucheffects,whereascommonenvironmentdoesseemtoofferaplausibleexplanationforthis.Someformulationsincorporateavariable"commonenvironment"withapathtoeachsibling[8,9,22].Weintroduceasimilarbutslightlymoreflexiblemodelwhichminimizesthenumberofassumptionsinvolved.Inthepathdiagramdepictedinfigure6,weintroduceacorrelationcbetweentherelevantenvironmentalinfluencestowhichsiblingsareexposed.Severalpossiblepathswhichcouldinducesuchacorrelationareindicatedbydottedpathsbutonlythecorrelationcisretained.Thenatureofthecorrelationisambiguousinthisformulation,butitisseldom

practicaltoobtainthedatanecessarytodistinguishbetweenthealternativeinterpretationsunlesscandearelarge.Itshouldbenotedthatthisformulationalsoimpliesthatthereisnochangeinvarianceduetoenvironmentalfactorswithassortativemating,sothatcremainsunchanged,althoughthestandardizedregressioncoefficientedecreasesasVpincreaseswithassortativemating.Intherandommatingpopulationsthecorrelationbetweenfullsiblingsisr00*=2r2t02+ceo2,(44)sothatC=[r00-27(I+(I+m))l+m]e2Itshouldbenotedthatcmaydifferindifferentgroupssuchasmonozygoustwins,dizygoustwins,andfullsibs.Furthermore,beingrearedapartdoesnotassurethatc=0.EstimatesofcmaybemadedirectlybyuseofanindexoftherelevantenvironmentassuggestedbyMortonandRaoetal.[8-10],orestimatedusingobservedcorrelationsbetweenrelatives.631 PfiPt~~~~tTT-.01P-FIG.6.-Pathdiagramdepictingsibling-siblingcorrelationwhenacorrelationbetweennontransmissibleenvironmentalvaluesispresent.Dashedlines=possiblesourcesofthecorrelationbetweentheE's.ThresholdCharactersThemultifactorialmodelforthresholdcharactershasbeendescribedbyCrittenden[29],Falconer[30],Reichetal.[25-27],andCurnowandSmith[4].HereitisassumedthatthemanifestationofadichotomoustraitisdeterminedbyanormallydistributedvariableX,termedtheliabilitytodevelopthetrait,withX=T+E;thatis,transmissionofliabilityfromparenttooffspringisdescribedbytheunitarymodel.Itisfurtherassumedthataffectationalstatusisdeterminedbyathresholdsothatanindividualis"affected"(i.e.,manifestsatrait,ifhisscoreonXisabovethethresholdand"unaffected"otherwise).Thecorrelationsbetweenindividualsarethenthecorrelationsbetweentheunderly-ingliabilityscaleXandmaybeestimatedbythetetrachoriccorrelationcoefficient[25,26].ThenormaldeviateXPofthethresholdmaybeestimatedfromthepopulationprevalenceKp.Therandommatingcorrelationsmaybeobtainedfromequations(38)and(39),andtherando

mmatingprevalencemaybeobtainedfromthechangeinvariancegivenbyequation(40).LettingXp*denotethenormaldeviatefortherandommatingprevalenceK,)*wehave[(I1-2r2)(1+M)2-(46)ThevalueofK,*canthenbeobtainedfromthecomplementarydistributionfunctionoftheunitnormalrandomvariableevaluatedatX,.Theincreaseinpopulationprevalenceduetoassortativematingisillustratedin632RICEETAL. MULTIFACTORIALINHERITANCEfigure7withK,*equalto1%andmequalto.4.Themaximumvalueofrisgivenbyequation(16)forr,0*lessthan.2988andbyequation(18)forr,,*greaterthan.2988.Inthelattercase,thecurvesapproachtheasymptotedeterminedbyequation(18)correspondingtoaninfiniteincreaseinthevariance.Figure7showsthattheincreaseinKpissmallforsmallT,increasesmonotonicallywithT,andcanbequitesubstantialformoderatelylargeT.InthecaseofthepolygenicorpseudopolygenicmodelsTis.5.RESEMBLANCEBETWEENOTHERRELATIVESInformationaboutfirst-degreerelativesandmatesinonepopulationisinsufficienttoestimater.Accordingly,informationaboutmultipleclassesofrelativesorheterogene-ouspopulationsisneededinordertoestimatetheparametersoftheunitarymodel.EstimationofTandthecorrelationbetweenrelativeswillbedescribedfortheancestorsanddescendantsofparents-offspring(verticalrelatives)andoffullsibs(fullcollateralrelatives).VerticalRelativesThepathdiagramforthegrandparent-grandchildcorrelationr90isshowninfigure8.Thecorrelationisr90=72(I+m)(1+rTT)t2.(47)Ingeneral,thecorrelationbetweenancestorsanddescendantsofparent-offspringpairsafternadditionalgenerationsisgivenbytheformulartAd=(rp),r"-l(1+rTT)?#-l=Tr'(l+m)(1+rTT)h-lt20.06(48)0.040.02p00.20.40.61,FIG.7.-IncreaseinthepopulationprevalenceKpintermsofTforathresholdcharacterwitharandommatingprevalenceof1%,m=.4andtheindicatedvaluesofrpO*.633 mPbPtt~~~ITTTT41PFIG.8.-Pathdiagramdepictinggrandparent-grandchildcorrelation.wherervi=ro,rV2=rgo,rv3

isthegreat-grandparentcorrelation,etc.NotethatifTisnegative,valuesofthetraitwilloscillate,andforadichotomoustrait,mayappearto"skipgenerations."Recallingequations(35)and(36),rearrangementofequation(48)givesausefulgeneralexpressionforTwhenn�1,rv___1/n1-i1mr,,0Tr11m(49)FullCollateralRelativesSimilarformulaemaybedevelopedfordescendantsoffullsibs.Thepathdiagramforfirst-cousinsandforuncles(aunts)andnephews(nieces)allowingforthecommonenvironmentofthesiblingsisshowninfigure9.Thecorrelationbetweenuncle(aunt)andnephew(niece)israt=rUIN=A2r3(l+r7TT)2t2+ce2mrt2,(50)andthecorrelationbetweenfirst-cousinsisr=2r4(l+r7T)3t2+ce2m2'r2t4.(51)Thecontributionofcommonenvironmenttotheresemblanceofdescendantsoffullsibsmayrequiresomecomment.TheeffectofenvironmentalfactorsEarenottransmissiblebydefinition;nongeneticfactorswhicharetransmittedbetweengenera-tionsareincludedinT.However,thecorrelationbetweenthehomeenvironmentsofsibsinducesagreatercorrelationbetweenthetransmissiblevaluesofthematesofaRICEETAL.634 MULTIFACTORIALINHERITANCEmttT~~~~~TCrrmtItPPFIG.9.-Pathdiagramdepictingthefirst-cousinanduncle-nephewcorrelation.pairofsibsthanwouldbepresentotherwise.Thisincreasedresemblanceisofcoursetransmitted,asshowninfigure9,andissospecifiedintheformulaehere.Regardless,theinfluenceissmallexceptwhenc,e,m,X,andtareallhigh.Whencommonenvironmentisnegligible,thecorrelationbetweencollateralrelativesinintactfamiliesisgiveningeneralasr=2rTn+(1+rT)Vnt2(52)wherenisthedegreeofgeneticrelationshipsothatn=Iforfullsibs,2foruncle(aunt)andnephew(niece),3forfirstcousinsorgreatuncle/nephew,etc.Fromequation(52),anestimateofXsimilartothatobtainedwithverticalrelativesisgivenby=An(1+m)]11n-mr,,(53)However,ifcommonenvironmentisnotnegligible,distinctionsarenecessarybetweencollateralpairsinwhichtheyaredescendantsofonlyoneoreacho

ftwosibs(e.g.,greatunclevs.firstcousin).Ingeneral,thecorrelationbetweenanindividualandthedescendantsofhisfullsibisr={2,rn±+(l+rft)nt2+ce2[r"-'mt2(l+rTT)n2]}(52.1)wheren,thedegreeofgeneticrelationship,is2orgreatersothatCldenotesfullsibs,C2denotesuncleandnephew,C3denotesgreatuncle,etc.Incontrast,thecorrelationbetweendescendantsofeachofapairoffullsibsisgivenasrcn={2rn+1(l+rTT)nt2+ce2[r'n-m2t4(l+r)n-3]}(52.2)wheren,thedegreeofgeneticrelationship,is3orgreatersothatClandC2arethesameasin(52.1),butC3denotesfirstcousins,C4firstcousinsonceremoved,CSsecondcousins,etc.635 PROPERTIESOFTHEUNITARYMODELTheunitarymodelmaybeviewedasanapproximationtothegeneralmodelofcombinedpolygenicandculturalinheritancewhichisdefinedinthebeginningofthispaperanddevelopedfullyelsewherebyCloningeretal.[19].Weknowthatthetwomodelsareequivalentinthethreespecialcasestreatedabove:theculturalmodel,thepolygenicmodel,andthepseudopolygenicmodel.Thegeneralmodelisparameterizedby[3,m,W=VB,/IV,h'=VA/V,andc,andVTisgivenbyVT=b2+h'+2wbh,wherewisthecorrelationbetweenAandBwithinthesameindividual.Anaturalquestiontoaddressishow[3andVT,ofthegeneralmodelrelatetoTandVToftheunitarymodel.Weareinvestigatingthisquestionbygeneratingobservationsunderthegeneralmodelandthenfittingtheunitarymodeltothesimulateddata.Theresultsofonesuchsimulationareshowninfigures10and11.TheparametersofthefullmodelwerechosentobeVI=.8,m=.3,c=.0,togetherwiththeindicatedvaluesof[3.Setsofcorrelations(specifically,parent-offspring,grandparent-grandchild,sibling,uncle,nephew,firstandsecondcousin,andfirstandsecondcousinsonceremoved)weregeneratedforb2from0to.8inincrementsof.1andthenfittedtotheunitarymodel.Asnotedinfigure10,VTwasunderestimatedforalldatasets(exceptforthosewhereweknowtheanswerisexact).Todate,theclaimthatVI,obtainedfromtheunitarymodelisconservativehasbeenb

orneoutbyoursimulations.ItappearsthatVTisunderestimatedbyapproximately2wvbh,andothersimulationshaveshownthatVTisveryclosetothetruevalueswheninissmall.0.900.80VT',0:0Q450.70IX=0.45/30.6000.20.40.60.8b2FIG.10.-ValuesofthetotaltransmissiblevarianceVToftheunitarymodelatthepointofbestfittodatasimulatedunderthefullmodel.ThetrueVTwas.8anddatasetsweregeneratedwithb2=Itob2=.8,withh-=.8-bl-2bhwandwiththeindicatedvaluesoff3.wisdefinedinreference[19].636RICEETAL. MULTIFACTORIALINHERITANCE).bO0.550.6010.50=0.4500.20.40.60.8b2FIG.I1.-ValuesoftheparameterTintheunitarymodelatthepointofbestfitforthesamedatasetsasinfigure10.TherelationshipbetweenTand/3ismorecomplicated,asindicatedinfigure11.WhenVB=.8,weknowr=/3,andwhenVB=0,weknowT=.5.DependingupontherelativeproportionsofV1andVB,Tmaybeanoverestimateorunderestimateof/3.However,whenmis0,agoodapproximationofTisgivenbyT=.5(V,4)/(V7)+/3(VB)/(V7).(54)Finally,theobservationthatT=.5whenVB=0allowsthetestingofthehypothesisthatthereisnoculturaltransmission.Ifr7.5,thenVBcannotbezero.However,asindicatedbythepseudopolygenicmodel,ifTis1/2,wecannotconcludetheVAisnonzero.ThemostimportantpointtonoteisthatthetestofVB=0suggestedhereismorepowerfulindetectingculturalfactorsthantheusualchi-squaregoodnessoffittestagainstanunspecifiedalternativewhichisoftenusedtotestthefitofthepolygenicmodeltoadataset.APPLICATIONSTheunitarymodelcontainsfourparameters(viz.T,m,t,andc),sothatestimatesofatleastfourcorrelationsarenecessaryforparameterestimation.AFORTRANprogramTAU,availableuponrequest,isusedtoprovidemaximumlikelihoodestimatesoftheparametersfromreportedcorrelations.Thisprogramappliestheinversehyperbolictangenttransformation,f(x)=1/2In(1+x)/(1-x),toeachproductmomentestimatorr,andignoringthebiasr/2(N-1),itassumesf(r)-N(f(r),i3)(55)637 whereNisthenumberofpairsusedtoestimateth

ecorrelationr.Givenobservationsr1,72,..randparametervalues0,theloglikelihoodofthedatasetisgivenbykL=1/2{f(F,)-f[r1(O)]}2(N,-3),(56)whereri(O)isthecorrelationpredictedfromthemodel.ThegeneralpurposelikelihoodsearchingprogramMAXLIK[31]isthenusedtoobtainmaximumlikelihoodestimatesoftheparameters0bymaximizingL.Atthepointofbestfit,notethat2L-X2,,whereviskminusthenumberofestimatedparameters.Testingofhypothesesinvolvinglinearconstraintsontheparametersmaythenbeperformedusingalikelihoodratiotest[32]byfirstobtainingthelikelihoodL1,withoutanyconstraints,andthenobtainingthelikelihoodLobyimposingtheconstraints.Theasymptoticdistributionof-21nLo-2(lnL,,-lnL1)(57)isknowntobeachi-squaredistributionwithvdegreesoffreedomwhenvlinearconstraintsareused.Unfortunately,apaucityofdataisreportedonremoterelatives.Thisnodoubtreflectsthefactthatunderthepolygenicmodelonlyestimatesofm,rp0,androoarenecessarytoestimateheritabilityanddominancedeviation.Informationonotherclassesofrelativesisnotnecessaryandstatisticallylessefficient.However,oncenonrandomenvironmentalfactorsareconsidered,suchdataarenotadequate.HumanStatureAlthoughthenecessarycorrelationsarenotavailabletoapplythismethoddirectly,thetwodatasetsdisplayedintableIreflectdifferentdegreesofassortativematingandallowtheestimationofTandt2fromthetwoparent-offspringcorrelations.Thefir;tLdatasetisthatcollectedbyPearsonandLee[33]fromthepopulationofEnglishuniversitystudentsandwasanalyzedbyFisher[2],whoestimatedtherandommatingTABLEIREPORTEDCORRELATIONSFORHUMANSTATUREFROMTwoDATASETSANALYZEDPREDICTEDPREDICTEDPARAMETEROBSERVEDNo.7=.604,t02=.484r=.5,t02=.7281[m.28041,000.2835.2953PearsonandLee[33]!**lrpo.50664,886.5064.5029m.0931200.0848.0070Galton[34].rpO.3355937.3379.3671NOTE.-PredictedcorrelationsatthepointsofbestfitfortheunitarymodelandtheunitarymodelwithT=0.5ar

edisplayedinthelasttwocolumns.638RICEETAL. MULTIFACTORIALINHERITANCEheritabilitytobe75%andthedominancedeviationtobe25%ofthevariance.Theseconddatasetconsistsof200familiescollectedfromtheuppermiddleclassbyGalton[34],thecorrelationsbeingthosereportedbyPearson[35]withthesexescombined.Thecorrelationmwasfoundtobe.2804inthefirstdatasetand.0931inthesecond.Assumingthatt02andTwouldbethesameforeachpopulation,thetwoparent-offspringcorrelationsallowestimationoftherandommatingparametersusingequation(41).Thiswasdoneusingamaximumlikelihoodsearch.Thevaluesofthecorrelationsatthepointofbestfitforbothdatasetsaredisplayedintable1whenrandt02areestimatedsimultaneously.WhenTwasfixedat.5,t02wasestimatedtobe.7281,avalueconsistentwithFisher'sanalysisofPearsonandLee'sdata.Thisvaluecontrastswiththevalueoft02=.484fromtheunitarymodel.Thus,theunitarymodelsupportsthehypothesisthatfamilialfactorsarelessimportantthanpreviouslyindicated.Thelikelihoodratiotestofr=.5isnotsignificant(X21=3.04),despitethedifferenceinthetwoparametersetsandthelargesamplesizes.Thisresultsfromthetwopopulationmethodused,sinceestimationisonlypossiblebecauseofthedifferentialeffectonthechangeofvarianceinthetwopopulations.(Inarandommatingpopulationrpo*=rto2,sothatiftworandommatingpopulationsweresampled,thetwoparameterscouldnotbeestimated.)Forthisreason,wedonotrecommendthismethod,butpresentitonlytoindicatethattheexistingdatamaybedescribedadequatelywithourmodelandthattheconclusionthattransmissionofhumanstatureisintotogeneticshouldbeinvestigatedfurther.Theobservedsiblingcorrelationsare.5433and.4004forthetwodatasets,andusingr=.604,t02=.484,andc=0,thepredictedcorrelationsare.i651and.3929,respectively,sothatcommonenvironmentaleffects(ordominance)wouiUbenegligi-ble.ForT=.5,thepredictedcorrelationswithc=0are.4773and.3664,respectively,sothatcommo

nenvironment(ordominance)wouldberequiredtoexplainthesiblingcorrelations.DISCUSSIONUntilrecently,humangeneticshaspaidlittleattentiontotheimportanceofnonrandomenvironmentintheetiologyorthetransmissionofquantitativephenotypes.Currentinvestigationssupporttheobservationthatmanybehavioraltraitsarehighlyfamilialandaresuggestiveofparenttooffspringtransmission.However,asthepseudopolygenicmodelindicates,suchobservationscannotbeconstruedasproofforgenetictransmissionwhenobservationsaremadeonlywithinintactfamilies.Itisnecessarytoallowforbothgeneticandculturaltransmission,aswellasasystematiceffectduetoasharedenvironmentofrearing,beforeanobservedpatternoffamilialcorrelationsmaybeusedtodeterminethemodeoftransmissionofacomplextrait.Theculturalmodelweproposeispatternedafterthepolygenicmodel.Transmissionfromparenttooffspringisdeterminedbytheadditiveeffectofmanyculturalevents,withthedegreeofparentaldeterminationmeasuredby13.However,for/3greaterthan.5,thismodeldiffersintwoqualitativeaspectsfromthepolygenicmodel.First,thecoefficient2/32inthesiblingcorrelationisgreaterthanthecorresponding13intheparent-offspringcorrelation,sothatr,,,,*isgreaterthanr,,,*,andevenwhenassortative639 matingoccurs,r00canbegreaterthanorequaltorag.Formanytraits,theobservedsiblingcorrelationsare"toolarge,"sothat,incontrasttotheculturalmodel,thepolygenicmodelrequireseitheralargedominancedeviationoralargenontransmissi-blecommonenvironmentaleffecttoexplainthedata.Second,theculturalmodelpredictsthatsecondandthirddegreerelativesaremoresimilarthanwouldbeexpectedunderthepolygenicmodel.Theobservationthatdistantrelativesare"tooalike"issuggestiveofculturaltransmission,andneitherdominancenornontransmissiblecommonenvironmentcanofferasatisfactoryexplanation.Ontheotherhand,where/8islessthan.5,transmissiblefactorsmayplayamajorroleinthee

tiologyofthetraiteventhoughreportedcorrelationsarelow.Ourtreatmentofassortativematingisgeneralinthatthederivationsarevalidevenforculturaltransmission.Inthesecondpaperofthisseries,theseresultswillbeusedtoincludeassortativematingforthegeneralmultifactorialmodelinthecaseofdirectphenotypicassortment.Theimplementationofothertypesofassortativemating,suchasindirectphenotypicassortmentorsocialhomogamy,caneasilybeaccomplishedbyparallelingargumentsusedforthepolygenicmodeltogetherwithourresultswhichallowforculturaltransmission.Whenthresholdcharactersareconsidered,anotherpossibilityisthatassortmentisforthedichotomousphenotypesratherthanfortheunderlyingliabilityvalues.Wearecurrentlyinvestigatingthedifferencesinthesetwodifferentapproaches.Onefurtherquestionrequiressomecomment.IfthehypothesisthatVB=0isrejected,whatistheminimumVBthatcanexplainthedata?Oneanswertothiscanbeobtainedbyfixing/8inthegeneralmodel[19]atitsmaximumvaluetoprovidealowerboundforVB.Theprecisionofsuchanestimatecoulditselfthenbetestedwithfurthersimulation.Theunitarymodelweproposeprovidesacommonmodeltoparameterizethecultural,thepolygenic,andthepseudopolygenicmodelsandprovidesanapproxima-tiontothegeneralmodel.Furthersimulationsareneededtoevaluatetheoperatingcharacteristicsofthismodelmorefully,butourworktodatestronglysupportstheusefulnessofthismodelwhenobservationsareavailableonlyforrelativesrearedwithinintactnuclearfamilies.Theunitarymodelprovidesaframeworktoinvestigatemanyimportantquestionsconcerningthetransmissionofatraitwithoutrequiringanestimateofheritability.Inparticular,issuessuchasindexconstruction,thedetectionofrelevantenvironments,thedetectionofheterogeneoussubformsofatrait,andthedetectionofasinglemajorlocusmaybeapproachedwithinthecontextofthismodel.Observationswithinintactfamiliesareeasilycollectedandstudiesmayber

eplicatedtotesthypothesesgeneratedinanaposteriorimanner.Thisisincontrasttoseparationdatawhereobservationsmadeoncertaintypesofrelatives,forexampleMZtwinsrearedapart,arerareandessentiallynonreplicable.Furthermore,thefittingofamodeltoseparationdatarequirestheassumptionofcommonparametersinquitedifferentrearingenvironments.Ifsuchassumptionsarenotmade,newparametersareintroducedwitheachnewtypeofobservationandthemodelisunderdetermined.Incontrast,theunitarymodelusesonlyonetypeofrearingstructure(viz.,theintactfamily).Forbehavioraltraitstheeventofbeingrearedina640RICEETAL. MULTIFACTORIALINHERITANCEbrokenhomeorofbeingatwinmayitselfbeanimportantfactorinthedevelopmentofthetrait.Accordingly,investigationofthetransmissionofthetraitusingtheunitarymodelprovidesafeasibleandefficientresolutionoftheproblem.IntheunitarymodelnoassumptionismadeastowhatproportionofTisgenetic,andifh2denotestheheritabilityofthetrait,weknowonlythath2St2.Indeed,ifthemodeoftransmissionofatraitisentirelycultural,wecouldhavealarget2withh2=0.ManyinvestigatorsmaketheaprioriassumptionthatT=1/2andt2=h2,orcommentthattheyaremeasuringt2,butstillfixTat1/2.Figures3and4haveshownhowerroneoussuchassumptionsmaybe.Ourmodelprovidesanexplicittestforthepresenceofculturalinheritance(r/1/2)andisanecessaryfirststepintheanalysisofanycomplexdevelopmentaltraitobservedinintactnuclearfamilies.Failuretoallowforculturalinheritanceprecludesitsdiscoveryandmayleadtospuriousresults.SUMMARYAgenerallinearmodeloffamilialresemblanceisdescribedwhichallowsforculturaltransmissionfromparenttooffspring,polygenicinheritance,phenotypicassortativemating,commonenvironment,maternalandpaternaleffects,andthresholdeffects.Threespecialcasesaredescribedindetailwhichareparticularlyusefulwhendataareonlyavailableaboutafewclassesofrelativesrearedinintactfamilies.Thecultural

model,thepolygenicmodel,andthepseudopolygenicmodelsharethecommonfeaturethatallfactorswhicharetransmittedfromparenttooffspringmayberepresentedbyoneparameterwithoutanylossofinformation.Weintroduceanewmodel,termedtheunitarymodel,whichincludesthesemodelsandisappropriatewhencombinedgeneticandculturaltransmissionispresentandwhendataareavailableonlyforindividualsrearedinintactnuclearfamilies.Thebasicpropertiesofthesemodelsareexploredusingpathanalysisandcomputersimulation,includingdescriptionoftherelationshipbetweenparametersunderrandomandassortativemating,rateofapproachtoequilibrium,andconstraintsonthemagnitudeoftheparameters.Generalformulaeforfamilialresemblanceinextendedpedigreesaregivenforanyancestorordescendantofeitherverticalorcollateralrelatives.EstimationproceduresaredescribedandaFORTRANprogramTAU,availableuponrequest,isusedtoprovidemaximumlikelihoodestimatesoftheparametersfromreportedcorrelations.Apowerfultestfordetectingthepresenceofculturaltransmissionissuggestedandappliedtosimulateddataandtodatasetsreportedbyothersforhumanstature,forwhichculturaltransmissionissuggested.Inaddition,itisshownthatthereisnoneedtopostulatedominancetoaccountforavailabledataaboutheight.ACKNOWLEDGMENTSTheauthorsgratefullyacknowledgehelpfuldiscussionandconsultationwithC.C.Li,NewtonMorton,D.C.Rao,PaulVanEerdewegh,DianeWagener,andSewallWrightduringthepreparationofthispaper.However,anyerrorswhichmayremainaretheauthors'responsibility.REFERENCES1.CARTERCO:Polygenicinheritanceandcommondiseases.Lancet1:1252-1256,19692.FISHERRA:ThecorrelationbetweenrelativesonthesuppositionofMendelianinheritance.641 TransRSocEdinb52:399-433,19183.CAVALLI-SFORZALL,BODMERWF:TheGeneticsofHumanPopulations.SanFrancisco,Freeman,19714.CURNOWRN,SMITHC:Multifactorialmodelsforfamilialdiseaseinman.JRStatSoc[A]138(2):131-169,19755.

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