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Genomicanalyseswithemphasisonsingle-step

Condensednotesforshort-coursetaughtin2015inPoznan,PolandIgnacyMisztaletal.,11/20/14-11/29/17IntroductionInthelastcoupleofyears,theanimalbreedingresearchfocusedongenomicselection.EarlystudiesfocusedonthedesignofSNPchipsformanyspeciesanddevelopedbasicmethodologiesbasedeitheronSNPestimationorGBLUP.SuccessivestudieslookedatthenumberofgenotypedanimalsandsizeofSNPchipforareasonableaccuracy,optimalSNPselection,andSNPweighting.Laterstudiesponderedimputationwithlowdensitychipsandgainswithhighdensitychips,bestchoiceofanimalsforgenotyping,validationmethods,etc.Nowthemosttrendingtopicistheuseofsequencedata.Theanimal-breedinggroupattheUniversityofGeorgia(UGA)haslongbeenactiveintoolsforcommercialgeneticevaluationacrossspecies.Whengenomicselectionappeared,thegroupfocusedondevelopmentoftechnologythatwouldbesuitableforacommercialapplication.Themethodwassingle-stepGBLUP,aBLUPwithadifferentrelationshipmatrix.WhilethebasictheoryforssGBLUPissimple,therearemanydetails.Inparticular,detailsthatseemtobeunimportantforonespeciesbecomeparamountinanotherspecies.ThelastmajorstepsinthedevelopmentofssGBLUPwerealgorithmsforinversionofgenomicrelationshipmatrixandthepedigreerelationshipmatrixforasubsetofgenotypedanimals.Thesedevelopmentsremovedlimitstothenumberofgenotypedanimals.Thesuccessofthefirstalgorithmraisesdoubtsaboutmeaningfulaccuracygainswithhigh-densitychips(dimensionalityofgenomicinformationlessthan15,000)andpavesthewayforgenomicselectionbecomingaroutineprocedurelikeBLUP.Single-stepGBLUPhasbeenappliedbysomeofthelargestcompanies/institutionsintheUS.Asaresult,UGAhasaccesstoperhapsthemostcomprehensivedatasetsanywhere.Withaccesstodatasetsandpracticalproblems,thegroupsolvedmanypuzzles,developedmanytools,anddiscoveredmanysolutions.Thepurposeofthismaterialistopresentanoutlineofdevelopmentsinthegenomicselectionfromapracticalpointofview,foruseinanewshortcoursetaughtbyMisztal,AguilarandLourencoin2015inPoznan,Poland.Duetoscarcityoftime,nearlyallreferencesinthisdraftaretopapersassociatedwithUGAandcollaborators.Seethosepapersforreferencesandnamesofthe“UGAteam.”Formoreadditionalinformationpleasesee2016coursenotesaswellasBLUPF90manualavailableatnce.ads.uga.edu.

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Models,Methods,andProgramsGoodgenomicpredictionsrequireafewcomponents:

1. Appropriatemodelsfortraitsofstudy2. Appropriatemethodsforparameterestimation3. Appropriatemethodsforgeneticevaluation4. Softwarethatcanimplementmixedmodelsforbothparameterestimationandgenetic

predictionHereweassumethatallmodelsareanalyzedbymixedmodelmethodology.Goodmodelsareaprerequisiteforanygeneticevaluationincludinggenomic.Badmodels=badGEBVs.Allmodelsareapproximations.Manyeffectsinfluencethetraitsofinterest.Givinglargedatasets,anyeffectislikelytobestatisticallysignificantalthoughmanymaybeunimportantinpractice.Wecandefinepracticallyequivalentmodelsassuchwherecorrelationsof(G)EBVforselectioncandidatesareatleast0.98.Then,ourdesiredmodelisassimpleaspossible(parsimonious)thatgivesessentiallythesamepredictionsasmorecomplicatedmodels.Pleasenotethatverycomplexmodelsmaygivebadpredictionsduetotraitsnotfollowingthemodelassumptionsorduetocomputinginstability.Also,commonlyusedstatisticalcriteriaformodelselectionsuchasBICoftenleadtocomplexmodelsandnottothebestpredictionsasdeterminedbysomesortofcross-validation.CommentsonModelsSingle-trait(repeatabilitywithrepeatedrecords)animalmodel.Basicmodelwhenothertraitsarenotavailableorareweaklycorrelated.MultipletraitmodelAmodelthataccountsforcorrelationsamongtraits(geneticandenvironmental).Importantwhensometraitsaremissing,whentherecordingissequentialandwhengeneticcorrelationsarenotcloseto0.Inclusionoftraitsunderselectioninthemodelhelpstoremovebiasesfromanalysesofalltraits.Randomregressionmodel(RRM)Usefulforanalysesoftraitsthatarerecordedcontinuouslyovertime(e.g.,milkproductionorbodyweight)orothervariablesliketemperature-humidityindexorherd-level.ThekeypartofRRMistheuseofseveralparameterstodescribevariabilityofcontinuoustraits.Functionsthatusesuchparametersmayberegularpolynomials(simplebutwithpoornumericalstability),Legendreorthogonalpolynomials(betternumericalproperties),orsplines(lessvariabilityatextremes).Typeandorderoffunctionsdeterminetheshapeofthecovariancetrajectories.Inparticular,toohighpolynomialorderinRRMshowsmodelingartifactswhere(co)variancecurveshavelargefluctuationsandarenotsupportedbybiologicalormanagementreasons.One-wayofsimplifyingRRMistheuseoflinearsplinesastheyareequivalenttomultiple-traitmodelswithtraitscorrespondingtopointsatknots(Misztal,2006)andcross-validationpropertiesaregood(Bohmanovaetal.,2008).

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Unknownparentgroups(UPG)Ifthebasepopulationisheterogeneous,underselection,themeritofmissingparentsmaybeuneven.Then,pedigreescanbeaugmentedby“phantomparents”viaunknownparentgroups.Groupscanmodelthemeritofunknownparentsbyyearofbirth,sex,pathofselection,geneticoriginorbreed.UseofUPGisdangerousduetothepossibilityofconfoundingorlargeSE(UPGarefixedeffects).Therefore,whenUPGareused,thesolutionsofUPGshouldbeexaminedforexcessivefluctuations,trends,orjustcommonsense.Onewayofreducingtheconfounding/fluctuationsistreatingUPGasrandom.Inthesimplestcase,thisisbytreatingUPGasanimals(var(UPG)=additivevariance).UPGandhistoricaldataInmanyanalyses,theuseofmorethan2-3historicalgenerationsdoesnotinfluenceGEBVoftheyoungestgeneration(selectioncandidates)ormayevenimproveit(Lourencoetal.,2014).Thisisduetoanumberofreasonsincluding1)decaysofgenomicpredictions,2)changesintraitsovertime,3)imperfectmodelingoffixedeffectsovertime.So,oneofthesimplestwaystolimitcomputationsisbytruncatingphenotypesandpedigrees.TruncatingdataalsolimitsoreveneliminatestheneedforUPG.CommonfeaturesofsoftwareSoftwareformixedmodelsshouldgenerallysupport:- missingtraits- differentmodelspertrait- heterogeneousresidualvarianceApproachesforvariancecomponentestimation(VCE)VCEisusuallyaccomplishedbyREMLorbyBayesiananalysesviaGibbssampling.Forgeneralproperties,seepaper“Reliablecomputinginestimationofvariancecomponents.”Belowaresomeofthemorepopularfeaturesofcommonalgorithms.GeneralREMLprogramsrelyonsparsematrixsoftware.Theyhaveapproximatelyaquadraticcostwiththenumberofanimalsandthecubiccostwiththenumberoftraits.Theyarelikelytofailwhenthenumberoftraitsistoolargeorresultingcovariancematricesareclosetosingularity.AI-REMLisfastwithsimplermodelsbutoftencrasheswithmodelsclosetotheparameterspace.EM-REMLismorestablebutveryslowandtherearenogoodestimatesofSE.EM-REMLisfasterwithless“missinginformation.”EM_REMLgets“stuck”withRRMifthestartingparametersaresmall.

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SpecialversionofREML(canonicaltransformation)canbeverystableandinexpensivewithalargenumberoftraitsbutaddmodellimitations.SeeKarinMeyer’spageformanyoptionsinREMLforcomplicatedmodels.BayesianmethodsviaGibbssampling(BAGS)maybesloworfastdependingonoptimizations.Usuallythenumberofrequiredsamplesishigherwithmodelcomplexityandwiththenumberofparameters.Forinstance,areducedanimalmodelmayrequire10timesfewersamplesthanaregularanimalmodel.AnalyzingtheoutputofBAGSisanecessity.AnoptimizedBAGS(e.g.,gibbs1f90andlaterversions)isresistanttonon-positivedefinitematricesandcanbeveryfastwithlargenumberoftraitsifmodelspertraitarenottoodifferent.Complexmodels(e.g.,threshold)aremucheasiertoimplementviaBAGSthanviaREML.ApproachesfornationalgeneticevaluationFornationalevaluations,thenumberofeffectsinthemodelcanbesolargethatthemixedmodelequationswouldnotfitinmemoryevenifweusethelargestcomputeravailable.Subsequently,solutionsarecomputedby“matrixfree”or“iterationondata”approaches,wherethedataisreadeveryroundofiterationandcoefficientsofmixedmodelequationsarerecreated.Insteadofbeingstored,thesecoefficientsareusedimmediatelytocreatequantitiesusedbyiterationmethods.Aparticularlyefficientyetsimpletoimplementiterationmethodispreconditionedconjugategradient(Tsurutaetal.,2001).Thismethodcanusecoefficientsofmixedmodelequationsinanyorder.Whysingle-stepforgenomicevaluation?Currentmethodsrequirede-regressionsandcreationofanindexcombiningdifferentsourcesofinformation(e.g.,parentaverage,directgenomicvalue,pedigreeprediction).Asbothrelyonaccuracies,whenaccuraciesareapproximatedboththederegressionsandtheindexareapproximate.Additionally,deregressiononamixofhighandlowaccuracyanimalscancreatedouble-counting(Legarraetal.,2014;Lourencoetal.,2015).DerivationofHThederivationsfollowLegarraetal.(2009)andAguilaretal.(2010).LetAbeanumeratorrelationshipmatrixandletthegeneticvariancebesetto1.0.Letindices1refertoungenotypedand2togenotypedanimals.𝑣𝑎𝑟 𝑢 = 𝐴𝜎()

𝐴 = 𝐴** 𝐴*)𝐴)* 𝐴))

Basedonconditionaldistributions:𝑢*|𝑢)~𝑁(𝐴*)𝐴))/*𝑢), 𝐴** − 𝐴*)𝐴))/*𝐴)*)

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𝑢)~𝑁 0, 𝐺 𝑢* = 𝐸 𝑢*|𝑢) + 𝜀= 𝐴*)𝐴))/*𝑢) + 𝜀Calculatevariancesandcovariances:𝑣𝑎𝑟 𝑢* = 𝑣𝑎𝑟 𝐴*)𝐴))/*𝑢) + 𝜀 = 𝑣𝑎𝑟(𝐴*)𝐴))/*𝑢)) + 𝑣𝑎𝑟(𝜀)= 𝐴*)𝐴))/*𝐺𝐴))/*𝐴)* + 𝐴** − 𝐴*)𝐴))/*𝐴)*Rearranging:= 𝐴** + 𝐴*)𝐴))/*𝐺𝐴))/*𝐴)* − 𝐴*)𝐴))/*𝐴)*= 𝐴** + 𝐴*)𝐴))/*𝐺𝐴))/*𝐴)* − 𝐴*)𝐴))/*𝐼𝐴)*= 𝐴** + 𝐴*)𝐴))/*𝐺𝐴))/*𝐴)* − 𝐴*)𝐴))/*𝐴))𝐴))/*𝐴)*Therefore,𝑣𝑎𝑟 𝑢* = 𝐴** + 𝐴*)𝐴))/* 𝐺 − 𝐴)) 𝐴))/*𝐴)*𝑣𝑎𝑟 𝑢) = 𝑣𝑎𝑟 𝑍𝑎 = 𝐺𝑐𝑜𝑣 𝑢*, 𝑢) = 𝑐𝑜𝑣 𝐴*)𝐴))/*𝑢), 𝑢) = 𝐴*)𝐴))/*𝑣𝑎𝑟 𝑢) = 𝐴*)𝐴))/*𝐺Finally:

𝐻 = 𝑣𝑎𝑟 𝑢* 𝑐𝑜𝑣 𝑢*, 𝑢)𝑐𝑜𝑣 𝑢), 𝑢* 𝑣𝑎𝑟 𝑢)

= 𝐴** + 𝐴*)𝐴))/* 𝐺 − 𝐴)) 𝐴))/*𝐴)* 𝐴*)𝐴))/*𝐺𝐺𝐴))/*𝐴)* 𝐺

=𝐴 + 𝐴*)𝐴))/* 𝐺 − 𝐴)) 𝐴))/*𝐴)* 𝐴*)𝐴))/* 𝐺 − 𝐴))𝐺 − 𝐴)) 𝐴))/*𝐴)* 𝐺 − 𝐴))

Whichcanbesimplifiedto:

=𝐴 + 𝐴*)𝐴))/* 00 𝐼

𝐼𝐼 𝐺 − 𝐴)) 𝐼 𝐼 𝐴))/*𝐴)* 0

0 𝐼

Theinversecanbederivedfromdistributionsknowingthat:

𝑢*|𝑢)~𝑁 𝐴*)𝐴))/*𝑢), 𝐴** − 𝐴*)𝐴))/*𝐴)* = 𝑁 𝐴*)𝐴))/*𝑢), 𝐴** /*

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TypesofgenomicrelationshipmatrixThemostpopularmatrixwasdefinedbyVanRaden(2008).ItisbasedonaSNPmodel:

𝑦? = 𝜇 + 𝑍𝑎 + 𝑒,𝑣𝑎𝑟 𝑎 = 𝐼𝜎B),𝑣𝑎𝑟 𝑒 = 𝐼𝜎C)whereZisamatrixofgenecontentadjustedforallelefrequenciesand𝑎isavectorSNPeffects.Whenuisavectorofbreedingvalues:

𝑢 = 𝑍𝑎, 𝑣𝑎𝑟 𝑢 = 𝐺𝜎(), 𝐺 = 𝑍𝑍D ∗ 𝜎B)/𝜎()Alternately,

𝐺 =𝑍𝑍D

𝑝?(1 − 𝑝?)

ManyotherGhavebeenproposedbutnoneseemstoprovidemoreaccuratesolutions.PleasenotethatZadjustedforallelefrequenciesis

𝑍 = 𝑀 − 2𝑃

whereelementsofMare0,1or2andPisamatrixofallelefrequencies.TheSNPmodelandGBLUPhavesamesolutionsregardlessofgenefrequencies(StrandenandChristensen,2011)butssGBLUPisaffectedbyP.ThepropertiesofGdependongenefrequencies.Withcurrentallelefrequencies,themeanofthediagonalelementsofGis1.0andthatofoff-diagonalsis0.Withequalallelefrequencies(0.5),themeansofdiagonalandoff-diagonalelementscanbeashighas1.0and1.5.DecompositionofGEBVByanalyzingonelineofMMEwecanfindthedecompositionofGEBV:

𝑢? = 𝑤*𝑃𝐴 + 𝑤)𝑌𝐷 + 𝑤O𝑃𝐶 + 𝑤Q𝐷𝐺𝑉 + 𝑤S𝑃𝑃, 𝑤? = 1

p (u1, u2 ) = p (u1 |u2 )p (u2 )

∝ exp[−0.5(u1 −A12A22−1u2 ′) A11(u1 −A12A22

−1u2 )]exp[−0.5 ′u2G−1u2 ]

= exp −0.5 ′u1 ′u2⎡⎣

⎤⎦

A11 −A11A12A22−1

−A22−1A21A

11 G−1 +A22−1A21A

11A12A22−1

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥u1

u2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎛

⎝⎜⎜

⎞

⎠⎟⎟

= exp −0.5 ′u1 ′u2⎡⎣

⎤⎦A11 A12

A21 G−1 +A22 −A22−1

⎡

⎣⎢⎢

⎤

⎦⎥⎥

u1

u2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎛

⎝⎜⎜

⎞

⎠⎟⎟

.

!!H−1 = A11 A12

A21 G−1 +A22 −A22−1

⎡

⎣⎢⎢

⎤

⎦⎥⎥= A−1 +

0 00 G−1 −A22

−1

⎡

⎣⎢⎢

⎤

⎦⎥⎥

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wherePAisParentAverage,YDisYieldDeviation(phenotypesadjustedformodeleffectsotherthanadditivegeneticanderror),PCisProgenyContribution,DGVisthegenomicprediction(comingfromG-1orindirectlyfromSNPeffects),PPisthepedigreepredictionbasedonthesubsetofgenotypedanimalsfromA(comingfromA22

-1 ).Whenbothparentsareknownandeachprogenyhasaknownmate,theweightsare:

𝑤* = )TCU

; 𝑤) = UV WTCU

; 𝑤O = UX )TCU

; 𝑤Q = YZZ

TCU ; 𝑤S =

/B[[ZZ

TCU

𝑑𝑒𝑛 = 2 + 𝑛^/𝛼 + 𝑛`/2 + 𝑔?? − 𝑎))??

wherenristhenumberofrecords,αisthevarianceratio(residualovergeneticvariance),andnpisprogenysize.withvaluesforDGVandPP:

DGVi=− 𝑔?fj,j≠i uh

𝑔??;PPi=

− 𝑎))?f

j,j≠i uh𝑎))??

Thedecompositionforyounganimals(w2=w3=0)isthesameasinVanRadenetal.(2009b)exceptthattheeffectsareestimatedjointly.PPaccountsforpartofPAexplainedbyDGV.Inparticular,whenA=A22,PAandPPcancelout.Whenoneortwoparentsaregenotyped,PPwillincludeafractionofPA.Whenagenotypedanimalisunrelatedtoagenotypedpopulation,PP=0.Alternately,DGVandPPcanbecombinedintoagenomicindex(GI)asinVanRadenandWright(2013):

𝑤Q𝐷𝐺𝑉 + 𝑤S𝑃𝑃 = 𝑤j𝐺𝐼

GI=(𝑔?fj,j≠i −𝑎))

?f )uh𝑔?? − 𝑎))??

; 𝑤j =𝑔?? − 𝑎))??

𝑑𝑒𝑛

GIexcludestheeffectofPAfromDGV.SeeLourencoetal.(2015b)formoredetails.Assumingthatdeviations(GEBVs–EBVs)andEBVsareuncorrelatedleadstothefollowingapproximation[19]:

𝑔?? − 𝑎))?? ≈ (𝑔?f − 𝑎)),?f)) ≈ 𝑛Y − 1 Δ?)f,fo?

where𝑛YisthenumberofgenotypedanimalsandΔ?)istheaverageofthesquareddifferencebetweengenomicandpedigreerelationshipsacrossallindividualswithindividuali(Misztaletal.,2012).Inpractice,Δ?)isabout0.04

2(Wangetal.,2014).

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QualitycontrolforGQualitycontrolofGisacriticalpartofgenomicselectionaspoorrelationshipscouldleadtomanytypesofbiasesandlossesofaccuracy.CallingrateforSNPandanimalsRawgenotypesincludeeitherthecorrectSNPoranerrorcodestatingthatanSNPcouldnotbereliablyread(ortyped).CallingrateforSNPremovesspecificSNPmarkersfromallgenotypedanimalswherethatSNPcouldnotbereliablytypedinalargenumberofcases.CallingrateforanimalsremovesgenotypesofsuchanimalswherelessthanathresholdSNPsarecorrectlytyped.Usuallythethresholdis90-95%correcttyping.ParentalexclusionsandparentidentificationParent-progenygenotypesshouldbecompatible.Inpractice,thismeansthatsuchpairsshouldnothaveexclusivegenotypes(aainparentandbbinprogeny).Forunrelatedanimals,thenumberofSNPwithoppositegenotypesisupto12.5%(Tsurutaetal.,2009).Forparent-progenypairs,thenumberisusuallylessthan3%withrawgenotypesbutcandropbelow0.1%(afewSNPwith50kchip)afterimposingthecallingrateedits.Ifthereisexclusionforanyparent-progenypair(>3%exclusions),possibilitiesincludepedigreeorgenotypingerror.Incaseofwrongpaternity,onesolutionistofindacompatibleparent(canbedonebysoftwarelikeSEEKPARENTF90).DistributionsofdiagonalsofGUndercorrectscalingandwithcurrentallelefrequencies,alldiagonalsofGshouldhaveaquasi-normaldistributionwithameanof1.0andSD<0.1.Toosmallortoolargediagonals(<0.6or>1.5)indicateeitherpoorgenotypingqualityorananimalfromanotherlineorbreed(Simeoneetal.,2011).DifferencesbetweenmatchedGandA22ForproperlymatchedGandA22,differencesareusually<0.1.Largedifferencesindicatepedigreeorgenotypingproblems,orinsufficientpedigree.Inpopulationswithmissingpedigrees,thereisanextravariationofthedifferenceduetounequalpedigreelength(Wangetal.,2014).ChromosomeeditingSNPsinthesexchromosomesandSNPsthatareinunmappedpositions(usuallycalledchromosome0)contributelittleifanytoGEBVandareroutinelyeliminated.HeritabilityofgenecontentWithcorrectgenotypes,heritabilityofeachSNPasadependentvariable(y)inananimalmodelis100%(Fornerisetal.,2015).Underspecialgenotypingerrors,includingthosefrombadimputation,theheritabilityislower.Inpractice,lessthan0.99averageheritabilityforallSNPindicatesproblems.Populationstratificationusingprincipalcomponents

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Genotypesofapopulationcanbeshowngraphicallyasaplotofthetwolargestprincipalcomponents.Whileasinglepopulationclusterhasanovalshape,separateclustersindicateseparatelinesorbreeds.

Principalcomponentplotforasimulatedpopulationcomposedoftwopurebreds(AAandBB)andonecrossbredline(F1).MatchingGandA22NormalprocedureformatchingGandA22includeonlyadjustmentstoGforthesameaveragesofdiagonalandoff-diagonalelements.

G=(1-λ/2)G*+11’λ

whereGistheadjustedgenomicrelationshipmatrix,1isavectorofones,andλisWright’sFSTthatiscalculatedas(Vitezicaetal.,2011):

λ= 1n2( A22i,j–ji Gi,jji )

wherenisthenumberofelementsinA22andG.

Whenthebasepopulationisheterogeneous(pedigreesmissingacrossgenerations),adjustmentsaredonetoanaveragepedigreelength,andGistoobigforanimalswithshortpedigreeandtoosmallwithlongpedigree(Chengetal.,2011;Vitezicaetal.,2011).ThisresultsinpoorconvergenceratewhensolvingssGBLUPbyiterationandinupward/downwardbiasesorinflation(Misztaletal.,2012).SeveraloptionsexistformatchingGandA22.OneoptionistomodifytheH-1matrixbyusingaweightforA22-1(omega)smallerthan1.0.

H-1=A-1+0 00 G-1–𝛚A22

-1

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AnotheroptionistointroducenonzerorelationshipsbetweenunknownparentslikeinVanRadenorviametafoundersconcept(Legarraetal.,2015).Thelastchoiceistocutphenotypesandpedigreesto2-3generations.Cuttingdatatoafewgenerationsdidnotdecreaseaccuracy(Lourencoetal.,2014)andreducestheimpactofoldpedigrees.Correlationsbetweendiagonalandoff-diagonalelementsofGandA22Intypicalanalyzes,thecorrelationsbetweenGandA22isabout0.3±0.1fordiagonalsand0.6±0.15foroff-diagonals.Thecorrelationsdependonthedepthofpedigree.Thecorrelationsbetweenthediagonalsaretreatedasthosebetweenpedigreeandgenomicinbreeding.Whilepedigree-basedinbreedingdependsonthedepthofpedigree,thegenomicinbreedingindirectlyincludesallpastinbreeding.Ingeneral,newinbreedingismoredetrimentalthanoldinbreeding,whereeffectsofdeleteriousgenesaremostlyremovedbyselection.Recentinbreedingwiththegenomicinformationcanbeassessedbyrunsofhomozygosity.MatchingofGandA22incrossbred/multibreedpopulationsUndercrossbreeding,ideallyGwouldbematchedtoA22forallbreedcombinations.Inparticular,blocksofGduetopurebredbreedcombinationswouldbe0.However,manualmanipulationofblocksofmatricesoftenmakesthematrixnon-positivedefinite.Inaterminalcrossmodelinpigs,thebestapproachforGwastouseaveragegenefrequenciesacrossallanimals,withoutadditionalsteps(Lourencoetal.,2016).Inamultibreedevaluationofsheep,wherephenotypeswereavailableoncrossbredsbutnotoneverypurebred,Gwasgeneratedusinganaverageallelefrequencybutmodelingincludedexplicitunknownparentgroups(UPG).ImportanceofmatchingofGandA22fordifferentspeciesandgenotypingscenariosTheimpactofincorrectmatchingislargeinsimulatedpopulations,smallerinpigsandchicken,andwasalmostnoneindairywhenthereferencepopulationincludedhighaccuracybullsonly.Theimpactofthegenomicinformationisnegligibleonhigh-accuracyanimalsandsomescalingcancelsoutforyounganimals.Also,theimpactofscalingisstrongerunderstrongselection.Simulatedpopulationsusuallyincludemanygenotypedgenerations,eachwithrelativelysmallnumberoflow-accuracyanimals,andwithselectiononasingletrait.Therefore,anyproblemswithscalingwillinfluencetheoldergenerationsandsubsequentlytheyounggeneration.Inrealpopulations,selectionisusuallymultitraitandnotverystrongonanysingletrait.ValidationReliability(R2)ofGEBVwitholderdataInpopulationswithhighaccuracymalesandlargenumberofprogeny,validationincludespartialandcompletedatacomparisons.Apartialdatasetexcludesprogenyoftheyoungestmales.ComparisonsaremadewithGEBVorEBVobtainedwithpartialdataandEBVorDYDofamaleobtainedwithcompletedata.Whenthenumberofprogenyislarge,thecomparisoninvolvesPA+DGV-PPandPC.ThedecompositionofGEBVindicateswhensuchapproachisappropriate.

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PredictabilityWhenyoungestanimalshavemainlyownrecords,therealizedaccuraciescanbecomputedasCorr(gebv,y-Xb)/h,wheregebvispredictionforagivenanimalobtainedwithoutitsphenotype(partialdata),y-Xbisphenotypeofthatanimaladjustedforfixedeffects(basedoncompletedata),andhisthesquarerootofheritability(Legarra,2008).Predictabilityseemsveryobjectivebutishardtoapplyforcomplexmodels(e.g.,maternaleffects).Whenheritabilityislow,accuraciescanovertheupperlimitfortraditionalblup(0.70)andssGBLUP(1.0);inthiscase,itisrecommendedtousepredictivity(Corr(gebv,y-Xb))insteadofaccuracy.Negativeorverylowvaluescanbeobservedforsmallgenotypedpopulations.X-foldcross-validationsWhenthenumberofgenotypedanimalsissmall,cross-validationisdonebycreatingmultiplesubsetsofgenotypedanimals(folds),andcomparingpredictionsobtainedwithphenotypesforallbutonesubset,againsty-Xbforthatsubset.E.g.,seepapersbySaatchietal.(2011)orKachmanetal.(2013).Thecross-validationcangenerateveryhighcorrelationsduetostrongrelationshipsacrosssamples.Thecorrelationsareamorerealisticestimateofaccuracywhensubsetsarechosentobelessrelated,e.g.,byK-meansclustering.SNPweightingandGWASinsingle-stepTheGmatrixusedinsingle-stepcanbeweighted,withweightsobtainedelsewhere,e.g.,byBayesianorLassomodels.WeightsandsubsequentlyManhattanplotscanbeobtainedfromsingle-stepdirectlybybacksolvingDGVtoSNPeffects(Wangetal.,2012;Wangetal.,2014).FollowingWangetal.(2012),theideathatGBLUPisequivalenttoSNP-BLUPwasextendedtothessGBLUPapproach.VanRaden(2008)showedthatfromtheselectionindexequationforGBLUP:

𝑢 = 𝐆 𝐆 + 𝐑𝜎B)

𝜎C)

/*

(y − 𝐗𝛃)

whereRisadiagonalmatrixaccountingforheterogeneousresidualvariance.If𝑢|𝑎 = 𝐙𝑎,replacingthefirstGby𝐙′,weightedbytheratioofSNPtoadditivedirectvariances,wouldallowthecalculationofSNPeffects(a):

𝑎 = 𝐙′k 𝐆 + 𝐑𝜎B)

𝜎C)

/*

(y − 𝐗𝛃)

wherekis𝜎B)/𝜎(),and𝜎B)istheSNPvariance.Itcanbeassumedthat𝜎B) = 𝜎() 2 𝑝?(1 − 𝑝?).Therefore,kcanbereducedto1 2 𝑝?(1 − 𝑝?).

Assumingthat𝑤 = 𝐆 + 𝐑 yz[

y{[

/*(y − 𝐗𝛃),𝑎 = k𝐙′w,andtherefore,𝑢 = 𝐆w.Inthisway,

w =𝐆/*𝑢,andfinally,theSNPeffectscanbecalculatedas:𝑎 = k𝐙′𝐆/*𝑢

as𝑉𝑎𝑟(𝑎) = 𝐃,theconditionalmeanofSNPeffectsgiventheGEBVis:𝑎|𝑢 = k𝐃𝐙′𝐆/*𝑢

Thus,givenGEBVfromssGBLUPareavailable,SNPeffectsarecalculatedas(Wangetal.,2012):

12

𝑎 = k𝐃𝐙′𝐆/*𝑢AssumingthatDisadiagonalmatrixwithweightsforSNP,theiterativeprocessis:

1. SettheweightmatrixD=I2. G=ZDZ’/k(kscalingfactor;detailsomitted)3. Runsingle-step

4. FromGEBV,obtainDGVasDGVi=/ YZ~j,j≠i ��

YZZ

5. ConvertDGVtoSNPeffectsa=DZ’inv(ZDZ’)DGV6. CalculateSNPvarianceanduseasweightdi=2piqiai27. NormalizeDforthesamesum(D)8. Goto2

Usuallybestweightsareobtainedafter1-2rounds.Step3canbedoneonce(ifDGVchangelittleacrossrounds)oreveryround.Formulain6canbedifferent.SNPvariancescanbeplottedasManhattanplots.IntestsusingsimulateddatasetstheestimateofSNPeffectsweresimilartothosebyBayesB.However,thebestestimateswerenotforSNPeffectofQTLbutforaclusterofnearbySNPs.ThisillustrateslimitedresolutionofGWASinpopulationswithsmalleffectivepopulationsize.GWASbysingle-stepallowsforanalysisinmodelswhereobtainingderegressedproofsisdifficult(e.g.,maternal,randomregressionorreactionnorm).TherearenoknownmethodsatthistimetoassessSNPsignificanceundersingle-step.Weightingseemstoimprovetheaccuracyfordatasetswithsmallnumberofgenotypedanimals,butmarginalornoimprovementisobservedforlargegenotypedpopulations(>10k;Lourencoetal.,2017).Indirectpredictioninsingle-stepviaSNPeffectsWhenmanyyounganimalsaregenotypedandfastpredictionisessential,theirpredictioncanbeobtainedindirectly:

1. Runsingle-stepusingonlyanimalswithinformation2. PredictSNPeffectsfromDGVasabove3. UseSNPeffectsforpredictions,possiblyaddingparentaverage.

Whenthenumberofgenotypedanimalsinthereferencepopulationislarge,parentaverageaddsverylittletopredictionsandcanbeomitted(Lourencoetal.,2015).

13

ApproximationsofaccuraciesAccuraciesingenomicmodelscanbeapproximatedbycalculatingtheamountofinformationduetogenomics(Misztaletal.,2013).Ingeneraltheinformationdiforanimaliiscomposedofinformationduetophenotypes,pedigrees,andgenomicinformation

𝑑? = 𝑑?`� + 𝑑?

`CT + 𝑑?YCU

Withreliabilitiescalculatedas

𝑟𝑒𝑙? =𝑑?

𝛼 + 𝑑?

wherealphaisvarianceratiofortheanimalmodeliftheinformationisintermsofeffectivenumberofrecords,orforthesiremodeliftheinformationisintermsofeffectivedaughters.Approximately,thegenomicaccuracyis:

𝑑?YCU~ 𝑔?f − 𝑎)),?f

)

f,fo?

𝑎𝑐𝑐f)

whereaccjis“independent”accuracyofanimalj.Independentmeansnotsharedwithotheranimals.OneproposedwayofobtainingaccuracyisbyapproximatingthelefthandsideofMMEbythefollowingandsubsequentinversion:

𝐿𝐻𝑆YCU = 𝐷`� + 𝐷`CT + 𝐼 + 𝐺/* − 𝐴))/*LinearadjustmentstoD’smayimprovetheapproximation.Computingsingle-stepwithalargenumberofgenotypedanimalsREMLandYAMSEfficiencywithREMLcomputationsinanimalmodelsisbasedonMMEbeingsparseandsubsequentuseofsparsematrixsoftware.Withgenomics,MMEcontainslargedenseblocksduetoG,andsparsematrixcomputationsarelessefficientwithdensematricesthandensematrixtechniques.Denseblocksalsooccurwithlargemultiple-traitandrandom-regressionmodels.Onesolutionistheuseofacomputationalpackagethatrecognizesdenseblocksandrearrangecomputationsaccordingly,allowingparallelcomputationsforlargedenseblocks.ApackagethatimplementsthistechniqueiscalledYAMSandwasdevelopedbyYutakaMasuda.Whenappliedto(AI)REMLF90forcomplexmodels,theprogramsrunupto20timesfasterwithindividualcomputationsandupto100timesfasterwithsparseinversion.Withefficientmatrixfactorizationandinversion,otheroperationsbecomebottlenecks.Thereislittleimprovementforverysimplesingle-traitmodelswithnodenseblocks.StructureofgenotypedpopulationsInitialgenotypingfocusesonhighaccuracyanimals.Whenallhighaccuracyanimalsaregenotyped,genotypingmovestolower-accuracyanimalsthatprovidemuchlessinformation.Withregardtoyounganimals,manyaregenotypedbutfewareselected.Forinstance,over1.8

14

millionHolsteinshavebeengenotypedby2017,butonly25karehighaccuracybulls,<50karecowswithphenotypeeligibleforanationalevaluation,andtherestareanimalsunimportantforgenomicselection,exceptforpossiblereductioninbiasduetopre-selection.Subsequently,computingchoicesmayeitheruseallgenotypesinanalysesorselectonlytheimportantonesforthemainanalysisandindirectpredictionsfortherest.EfficientinversionofGbyAPYalgorithmTherankofGforlargepopulationsasshownbytheproportionofeigenvaluesexplaining99%ofthevariationislimitedduetosmalleffectivepopulationsize(Pocrnicetal.,2016).Sucharankwasfoundtobearound10kinHolsteinsandAngus,6kinpigs,and4kinbroilerchicken.Thisisduetoalimitednumberofindependentchromosomesegments(ICS),orlimitednumberofindependentSNPcluster.SuchafactcanbeusedtoderiveaninverseofGefficiently(Misztaletal.,2014;Fragomenietal.,2015;Misztaletal.,2015;Lourencoetal.,2015;Misztal2016;Masudaetal.,2016;Bradfordetal.,2017).LetabeavectorofeffectsofICS(oruncorrelatedSNPsegments).Divideindividualsintocoreindividualsdenotedascandother(non-core)individualsdenotedasn.Then = +c c cu Z a ε and = +n n nu Z a ε .Whenthenumberofcore

animalsisthesameasthenumberofICS: a ≈Zc−1uc and = +n n nu Pu ε ,wherePisamatrixthat

relatesBVsofnon-coretocoreindividuals.Denote:

2var b bb bcu

c cb ccs

é ù é ù= =ê ú ê ú

ë û ë û

u G GG

u G G,

Usingtherecursionequations, - -

- - --é ù é ù= + -é ùê ú ê ú ë ûë û ë û

1 11 1 1 .cc cc cn

nc ccG 0 G G

G M G G I0 0 I

Theaboveinversehascubiccost(andquadraticmemory)forcoreanimalsandlinearcost(andlinearmemory)fornoncoreanimals.Thisinverseisalsosparse.Intests,APYinversionofa570kx570kmatrixforHolsteinstook<2hofcomputingtimeandusedlessthan100Gbytesofmemory(Masudaetal.,2015).TheAPYinverseistherealinverse,andsimulationsshowedslightlyhigheraccuracyofGEBVcomparedtousingtheregularinversionofG(withblending),whichcanbepartlyduetoeliminationofnoisecontainedintheregularG-1.

WhileG-1foralargenumberofanimalsissparse,A22-1canbedense(FauxandGengler,2013).Thismatrixcanbecomputedindirectly(StrandenandMantysaari,2014):

𝐴))/* = 𝐴)) − (𝐴*))′ 𝐴** /*(𝐴*))

whereallmatricesontherighthandsidearesubmatricesofsparseA-1andcanbestoredexplicitly.InthePCGalgorithm,onlyaproductofamatrixbyavectorisneeded.ThenA22-1qcanbecomputedas

𝐴))/*𝑞 = 𝐴))𝑞 − 𝐴*) D[ 𝐴** /*[𝐴*)𝑞]]

15

Withsomeexceptions(solvingstepwithA11),computationsincludeonlyproductsofamatrixbyavector.See(Masudaetal.,2016)fordetails.

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Simeone,R.,I.Misztal,I.Aguilar,andZ.Vitezica.2012.Evaluationofamulti-linebroilerchickenpopulationusingasingle-stepgenomicevaluationprocedure.J,Anim.Breed.Genet.

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