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Derivation, Calculation, and Use of National Animal Model Information

ABSTRACTNew terms and definitions were deve-loped to explain national USDA geneticevaluations computed by an animal mod-el. An animals ITA combines informa-

tion from its own records and records ofall its relatives through a weighted aver-age of 1)average of parents evaluations,2) half of its yield deviation, and 3)average across progeny of twice progenyevaluation minus mates evaluation.Yield deviation is a weighted average ofa cows lactation yields minus solutionsfor management group, herd-sire, andpermanent environmental effects. Bullsdo not have yield deviations; however, aweighted average of daughter yield devi-ations adjusted for mates merit can pro-vide a useful, unregressed measure ofdaughter performance. Reliability is thesquared correlation of predicted and truetransmitting ability. An animals parents,own records, and progeny each contrib-ute amounts of information measured indaughter equivalents. Reliability ofUSDA evaluations then is computed as(total daughter equivalents)/(total daugh-ter equivalents 14).(Key words: animal model, genetic eval-uation, reliability)

Abbreviation key: D E = daughter equivalent,DYD = daughter yield deviation, MCC =Modified Contemporary Comparison, MCD =modified contemporary deviation, MD = man-agement group deviation, PA = parent average,PC = progeny contribution, PPA = predictedproducing ability, RE = record equivalent,REL = reliability, TA = transmitting ability,YD = yield deviation.

Received September 4 1990.Accepted January 8, 1991.

P. M. VanRADEN and G. R. WIGGANSAnimal Improvement Programs LaboratoryAgricultural Research Service USDABeltsville, MD 2o7052350INTRODUCTION

Animal model evaluations use informationfrom all known relationships among animals top r d c t each animals genetic merit. New ter-minology and explanations were needed toprovide animal model information to users.Simple explanations were provided by Wig-gans and VanRaden (9), but derivations ofsome terms were not given.Methods to s u m m a r k accuracy providedby the additional sources of information in-cluded in evaluations also were needed. Powell5) presented formulas to measure accuracyobtained by incorporating daughter and soninformation into Modified ContemporaryComparison MCC) cow evaluations, but theseformulas assumed that all daughters had equalnumbers of records, that all sons had equalnumbers o f daughters, and that there was noadjustment for merit of mates. Meyer (3) ap-proximated accuracy reasonably well in an ani-mal model by adjusting diagonals for off-diag-onal elements. Misztal and Wiggans 4)

obtained more precise measures of the accu-racy of individual evaluations using an itera-tive procedure that was computationally af-fordable but lacked easy interpretation.This article explains how PTA, yield devia-tion 0 .aughter yield deviation DYD),reliability (REL), and daughter equivalents(DE) are calculated and how they interrelate.

MATERIALS AND METHODSModel

mal model can be represented asIn matrix notation, the current USDA ani-y = Mm + Za + ZAsg + Pp + C c + e

where y represents s t a n d a r d i i d mdk, fat, orprotein yield; m, , g p, and c are vectors ofeffects for management group, random portionof additive genetic merit, unknown-parentgroup, permanent environment, and herd-sire1991 J Dairy Sci 142737-2146 2737


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2738 VanRADEN and WGGANSinteraction, respectively; M, Z, ZA , P, and C tionship matrix among all animals in a, andare incidence matrices for these effects; and e R-1 is a diagonal matrix with diagonals equalis error. The mauix AB relates animals to to wlen, lactation length weights (8). Mixedunknown-ancestor p u p s and is equivalent to model equations were given in scalarform forAloQ as reported by Wiggans et al. (7). this model (8) and in matrix notation for aVectors a, p, c, and e are mutually uncor- similar model with var(e) = 147). withrelated with variances A , I


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DERIVATION OF NIM L MODEL INFORMATION 2739parents and progeny. Records wthno management group mates are deleted, and consequently such YD are not reported. However, ifgroup sizes are small, information from parentsand progeny will have some influence on thecows YD andMD ecausemanagement groupsolutions are adjusted for average genetic mer-it, which includes the cows own genetic merit.Averages of management group solutionsand MD are not reported for individual cowsbut can be constructed from variables provid-ed. Subtraction of twice an animals PTA( P T A d from the animals PPA gives thetotal of solutions for permanent environmentand herd-sire interaction. This total can beadded to YD to obtain MD:

M D = YD + PPA - 2PTA-Weighted average of management group solu-tions for a cow can then be computed as theweighted average of her standardized yieldminus MD.Predicted Transmitting Abilities

The matrix A- has nonzero off-diagonalsonly for an animals parents, pro eny, andzero if an animals parents or mates are un-known (6). Because PTA are elements of Qdivided by 2, division of both sides of Equa-tion [l] by 2 and transfer of off-diagonal termsof A- to the right side of Equation [l] gives

mates (Z),and coefficients of AgA- are non-

kaqpar(1PTAsire+ E A d a m )+ [diag(ZR-lz)~YD/2+ -5kaQprogC2PTAprog - mAmte P I

where Sparequals 1 if both parents are known,2/3 if one is known, and ly2 if neither isknown, and ~~~quals 1 if progenys otherparent is known and 2/3 if unknown. Appropri-ate genetic group solutions divided by 2 re-place TAsbe, FT&. and ITAmte if any areunknown. An analogous formula using breed-ing value instead of FTA was given by Wig-gans et al. (8).Equation 121 can be simplified by definingparent average PA) as average of parentsPTA and progeny contribution PC) asweighted average of twice FTAprog minusmAmte or

Then, substituting ZWlen 2k,4pa + .5& mgfor diag(ZR-Z + A-llcJ Wlm lactationallength weight) and dividing both sides ofEquation [2] by this quantity gives

where wl, w2, and w3 are weights that sum to1.The numerator of w1 is 2k,qp,; the numera-tor of w2 is Cwl,, for the cow; and the numera-tor of w3 is .5k,+g The denominator of al lthree w is the diagonal of ZR-lZ + A-llg,which equals the sum of the three numerators.These weights were derived directly from themixed model equations but may be interpretedmore easily if numerators and denominatorsare each divided by k .Evaluations computed by an animal modelare interpreted more easily if the three compo-nents of Equation [4] are reported along withPTA-. The term 2EAF0 - ITAm, isreported by USDA individualfy for daughtersof bulls and is labeled contribution to bull onthe Bull Evaluation and Daughter List. Contri-butions to bull are used directly to calculatethe bulls FTA. However, when interpretingthese contributions, it must be rememberedthat PTApmg is also a function of RAW;i.e., daughter contribution to bull includessome information contributed by the bull to hisdaughter.To illustrate, suppose the mixed modelequations include some progeny with norecords or descendants. For such progeny,PTA would equal the progenys PA orFTLz PTAm&, and contribution to

bull then would equal 2[(FTA- RAmte)/21 - which reduces to PTA-. Suchprogeny obviously do not contribute new in-formation but simply reflect back informationreceived. Similarly, progeny providing littlenew information to their sires evaluationlikely will have contributions close to his PTA.Because Equation [4] is solved iteratively,an animals FTA may ffect and be affected byall relatives. The YD of daughters are muchless dependent on the bulls PTA, and aweighted average of YD of daughters adjustedfor PTA can be used to construct FTA ofbulls that do not have grandprogeny.

Journal of Dairy Science Vol. 74, No. 8, 1991


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VanRADEN and WIGGANS740Daughter Yield Deviat ions ny's PTA equation. Thus, PC is a regressed

&d not indekdent measure of progeny per-formance because PTA- is directly includedin ITAproc A more independent and un-DYD.

PrOg)nformation from YD of progeny (YDis included in PTA- only indirectly afterprogeny's progeny and parents by the proge-YDprOg is combined with information from the regressed of progeny is

The FTA equation of any daughter without progeny can be written as

where wallows PC to be expressed as

and w2 are w1 and w2 of progeny. Substitution of Equation [5] into Equation [3]lprog pmg

Because these progeny have no progeny of their own, w3Therefore,

equals 0 and w equals 1-w .pmg l 2ptos

PC = Qprog[(l - w2P g mAanim wbg(YDprog - PTAmate)l/rxlprog= RA- + Mprogwb(-pTA..m + m p r o g - flAmte)I/rQ,g. 161

Substituting Equation [6] into Equation [4] and accumulating all terms involving PTA- to theleft side gives

(1 - w3 + ~3 progw+flprog)~Aanim = WIPA + WZ(YD/~) ~3 progw ,(YDprog- FTAwteWiprog

Next, by replacing 1 w3 with w1 w2, removing the common denominator of the w fromboth sides, and defining DYD as

PTA- can be rewritten asPTA- = XlPA xz YDl2) + x@YD

where XI , x2, and x3 are weights that sum to 1.Numerators of x1 and qual numerators ofw1 and w2; numerator of xg is .5ka r0gw2pl.which was derived as the numerator of w3times ~progw2FogEq,,rog.he denominator forall three x is the sum of the numerators. Be-cause w is always less than l , x3 is alwaysless than w3, which reflects that DYD is anunregressed measure of progeny performance,whereas PC is a regressed measure. The DYDmay be helpful in explaining evaluations and

2ptos

also as a dependent variable in statistical testsand calculation of conversions across coun-tries.Currently, DYD is provided to the dairyindustry for bulls with 10 or more daughters,but not for cows. For bulls with granddaugh-ters, DYD does not include all informationfrom descendants because information fromgranddaughters and sons is excluded. For eachdaughter with daughters of its own, w iscalculated as if granddaughter information didnot exist and is not the actual w2 of the bull'sdaughter. For al l daughters of the bull, wused in calculating DYD is

2 P g

2og



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DERIVATION OF ANIMAL MODEL INFORMATION 2741Otherwise, presence of granddaughters wouldreduce weight given to that daughters YD.Values of w and x for a specific animalprovide only an imprecise measure of howmuch information came from its PA, YD, andPC or DYD. The three terms used to constructFTA may have large part-whole correlations,and important factors such as parent REL,number of management group mates, and mateREL do not enter into calculation of either wor x. More precise measurement of the influ-ence of PA, YD, nd PC can be obtained byexamining their contributions to REL.Reliability

The measure of accuracy of an evaluation,REL, s the squared correlation of an animalspredicted and true transmitting abilities TA).An equivalent definition of REL is variance ofthe animals PTA divided by variance of itsTA. Selection is ignored for computing REL.i.e., correlations and variances are derivedfrom unselected rather than selected populationparameters. Exact REL usually must bereplaced by approximations if matrices are toolarge to invert.Like R A ,RELcan account for informationfrom all relatives by only directly includingterms for parents, self, and progeny adjustedfor mates. Misztal and Wiggans 4) accuratelyestimated REL by adding information fromthese three sources. They expressed informa-tion in record equivalents (RE) because this isthe natural unit of information for a cow inmixed model equations. Interpretation of R E isdifficult because only one sex has records,because RE increase nonlinearly with addi-tional records as the result of permanent envi-ronmental effects, and because large numbersof records are not biologically possible.Daughter equivalents were selected for theUSDA animal model implementation becauseof simpler interpretation and the tradition ofrelating accuracy of sire evaluations to numberof daughters. One DE is the amount of infor-mation contributed to a parent by a standarddaughter. For USDA evaluations, a standarddaughter was defined as having one record, aninfinite number of management group mates,and the other parent wth perfect REL. TotalDE for an animalDws the sum of DEfrom PA (DEpA), OW yield (DEyleld), andprogeny adjusted for mates (ZDqOgmte):

D L DEPA + Dqield + rn og-mate-Each animals RE @EL& is calculated

from DE-:

where lQ is a variance ratio calculated as 4 -2h2)/h2, which can be interpreted using siremodel terms as the ratio of error to sire vari-ance with dam variance removed from errorvariance or (< $ + + .5


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2742 VanRADEN and WIGGANSwhere n is number of records, and ryie1d isyield repeatability. Actual formulas to calcu-latey ccount for lactation lengthweights o the cow and her management groupmates and also number and average REL ofsires of management group mates. After ac-counting for these effects with formulas givenby Wiggans et al. (8),

Dqeld = kd( eld/(l - q e l d ) .Daughter Equivalents from Progeny Ananimals DEp..g-m;lte from each progeny iscalculated from the REL provided by thatprogeny, assuming that it is the animals only

source of information wL):

where mirogs the progenys REL includinginformation from its yield and its progeny butnot from its parents, and RELLte s matesREL with DE from this progeny excludedThen DEprog-mate for each of the animalsprogeny is calculated as

The equations for R E L L and DEpmg-mtecan be derived (see Appendix) from formulasof Misztal and Wiggans (4) and also fromselection index procedures as follows. Yieldinformation from a progeny that has noprogeny of its own is summarized asYDpmg-ITA,~. TO simplify calculations, let miogbe a YDpog in which true values for manage-ment group, permanent environment, and herd-sire interaction are subtracted instead ofpredictions of these effects. Then

where TA- and TAmte are true TA of theanimal and its mate, s is Mendelian samplingof progeny, and q s an element of e. If miroe

were the progenys only source of information,then mirngould be calculated as varianceof the progenys breeding value divided byvariance of miog,r

LetPTALe a prediction of TA- fromjust this progeny and IT&k be a predictionof TAmte from a l l information excluding thisprogeny. Because Cov(TA-, mifogIT Jquals V W A ~ . EL that theanimal receives from just this progeny adjustedfor mate wL) an be calculated as

A more convenient expression may be

These formulas to calculate REL from a sumof DE provide good approximations (4) but arenot exact because some covariances assumedto be 0 may not be, e.g., if relatives arecompared in the same management group orherd or if an animal has several progeny by thesame mate.Daughters in the Same Herd Inclusion ofherd-sire interaction in the animal model limitsinformation contributed to a sire by daughterslocated in any one herd. Formulas to accountfor herd-sire interaction in USDA-DHIA ani-mal model evaluations were given by Wigganset al. (8), but these included an assumption ofno adjustment for merit of mates. Adjustmentfor mates can be included in formulas of Wig-gans et al. (8) by subtracting REL,Lk4/4from variance assigned to progeny information[specifically from o, in the daughter weight



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DEFSVATION OF AMMAL MODEL IMPORMATION 2743formula of Wiggans et al. (S)]. With this ad-justment, formulas of Wiggans et d. (8) can bereexpressed as

where d accumulates information from daugh-ters in one herd. If daughters of an infinitenumber of other sires are present in each man-agement group, d can be calculated as

Actual formulas adjust for number of management group mates and number and averageREL of their sires by replacing Z,wlen by a sumof lactation weights 8) times4 he termC+ .54+4 ould be factored out but achievesscaling so that d is simply the number ofdaughters if each daughter is a standarddaughter.

Replacement of 4 ,


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2744 VanRADEN and WGGANSTABLE . Example daughter equivalents @E) contributed to cow reliability (REL) by various sources of information.Relative Information available DEParents1 Sire with 70 REL, dam with 3046 REL 4.7

selfz 1 lactation record 4.7

Daughte9 1 lactation record 1.0

son3 daughter with 1 lactation record 2

sire with 99 REL, dam with 50 REL 8.3sire with 99 REL. dam with 99 REL 14.03 lactation records 7.85 lactation records 9.03 lactation records 1.55 lactation records 1.7

1.84.45.4Evaluation with 99% REL. 7.010 daughters in 10 herds, each with 1 lactation record50 daughters in 50 herd, each with 1 lactation record100 daughters in 100 herds, each with 1 lactation record

lparcnt RFL excluding informaton contributed by this offspring.2~acta tion ecords with infinite numtm of management group mates.3 ~ t h aparent assumed to have 99% REL.

DEyleld, and DEprog-mate in Table 1. Thesevalues were computed with assumptions thatparent REL are adjusted for the animals con-tribution and of perfect REL for mate, but theyusually are good approximations of DE even ifthese assumptions are not met. Examples ofDEpmmte contributed to sire if daughters arein the same herd are in Table 2. Sires RELthen is obtained from DE from all herds plus

Calculation of FTA, DYD, andRELwi ll bedemonstrated for an example cow, althoughDYD is reported only for bulls. Suppose theDEPA.

TABLE . Example daughter equivalents DE)ontributedto sire reliability (REL) by daughters m the same herd.Number of DEdaughters contributed DE/in herd to sire REL daughter

1 1.0 1.002 1.7 .865 3.0 .610 4.1 .425 5.2 250 5.7 .11100 5.9 06

Calculated by 1fi.16+ (.84/d)], where d is number ofstandard daughters in the herd and a standard daughter hasone record, infinite management group mates, and theother parent with perfect R F L

example cow has a YD based on threerecordsof +loo0 kg, a PA of -500 kg, and twodaughters with one record each. The firstdaughter has a YD of -300 kg, and her siresPTA is -200 kg. The second daughter has aY D of +400 kg and an unknown sirq anunknown-group solution of +lo0 kg is substi-tuted for her sires PTA. The cows DYD canthen be computed as

DYD = [.2174(-3o0 + 200).2941(2/3)(4OO - lOO ]/1.2174 + .2941(2/3)1= 90 kgwhere .2174 and .2941 are w for the firstand second daughters, respectively. The cowsPTA can then be computed from PA, YD, andDYD as

2 P B

FTA = .516(-500) .431(1000/2)+ .053(90)= -38 kgwhere X I = S16, x2 = .431, and x3 = .053.Because the two daughters have no progeny oftheir own, their PTA can be obtained fairlyeasily and qu a l -126 kg and 81 kg, respec-tively. Then, the cowsPC s



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DERIVATION OF ANI= MODEL INFORMATION 2745PC = { [2(-126) 2001 + (2/3)[2(81)- 1oO)ll/[~ - (2/3)1= -6 kg.

The cows PTA also can be computed as aweighted average of PA, YD, and PC with theweights in Table 3. For comparison, values ofx and DE provided by parents, self, andprogeny expressed as fractions of total DE alsoare given. Values of DEPA, DQeld, andDEprog - were 8.3, 7.8, and 1.93, respec-tively. Assumptions were that the cows damhad 50 REL after removing cows contribu-tion to dam and that the cows sire and the sireof her first daughter both had 99 REL. Thesecond daughter with unknown sire contri-buted .93 DE, a situation not included in Table2. The cows total DE is 18.03, which resultsin REL- = 18.03/(18.03 14) = .56.Implementation

The FTA, F SL, and deviation variables forabout 14 million cows and 100,OOO bulls havebeen computed semiannually since the firstrelease of animal model information in July1989. The new evaluations and terminologygenerally have been accepted well by the dauyindustry. A few complaints were received.Some dairy producers disliked that REL forbulls has a more limited range than did R epeatability and that PTA cannot be approxi-mated easily from data available on the farm.Others would have preferred that MD ratherthan YD be reported.

CONCLUSIONSAnimal model evaluations combine infor-mation from an animal and all relatives usingoptimum statistical techniques but can be ex-plained easily without matrix algebra. Thecows own information is summarized by herYD, a weighted average of yields adjusted foreffects other than genetic merit and error. Eachcows PTA combines information from her

YD with information from her parents (PA)and her progeny (adjusted for genetic merit ofmates) through a simple weighted average. Ifprogeny do not have progeny of their own,ITA also can be computed as a weightedaverage of PA, half of YD, and DYD.The total amount of information providedby records of the animal and all its relatives is

TABLE 3. Weights assigned to parent, self, and progenyinformation and proportion of daughter equivalents @E)contributed by each for an example cow.weightingfactor Parents Self r o a YW .445 .370 .I85X .516 .431 .os3DP,/CotalDE .460 .433 .1w

summarized by RELthrough a simple functionof total DE from parents, own yield, andprogeny adjusted for mates. As with ITA,information from more distant relatives is in-corporated through parents and progeny.Daughters located in the same herd provideless information and fewer DEp.og - mate totheir sire than if each were located in a m e r -eat herd. Computed REL do not equal trueREI., but do account for estimation of manage-ment group effects, herd-sire interaction, andcorrection for merit of mates. Further refinements such as reduction of DEprog- mate con-tributed to dam if daughters are in the samemanagement group and inclusion of maternalgranddaughter information for bulls also maybe possible.

REFERENCES1Dicldnsoq P. N.,H. D. orman, R. L. Powell, L. G.Waite, and B.T. McDaniel. 1976. procedures used to

calculate the USDA-DHIA Modified ContemporaryComparison Page 18 in USDA Prod. Res. Rep. No.165, Washbgton, DC.2 Henderson, C. R 1976. A simple method for comput-ing the inverse of a numerator relationship matrixused in prediction of breeding values. Biometrics 32:69.3Meyer. K . 1989. Approximate accuracy of geneticevaluation under an nim l model. Livest. Prod. Sci.21:87.4 Misztal, I., and G. R.Wiggans. 1988. ApproximatiOnof prediction m o r variances in large-scale animalmodels. J. Dairy Sci. Il(Supp1. 2):27.5 Powell, R L. 1981. Possible effects of embryo trans-fer on evaluation of cows and bulls. J. Dairy Sci. 64:2476.6 Westell, R A., R L.Quaas, and L.D. Van Vleck1988. Genetic groups in an animal model. J. DairySci. 71:1310.

7Wiggaw. G. R. I. Misaal, and L. D. Van VI&1988. Animal model evaluationofA r n e mi yieldwith all lactations, had-sire interaction, and groupsbased on unknown parents. J. Dairy Sci. 71:1319.8 Wiggans, G. R., I. Missal, and L. D. Van Vleck.1988. Implementationof an ni m l model for geneticevaluation of d iry cattle in the United States. J. DairyJournal of Dairy Science Vol. 74, No. 8, 1991


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2746 VanRADEN and WIGGANSSci. 7l(Suppl. 2):54. Thus,9 Wiggans, G. R.,and P. M. VanRaden. 1989. USDA-DHIA animal model genetic evaluations. Natl. Coop.DHI Progr. Handbook, Fact Sheet H-2. Washington,Dc.

=as4 = a(REL,* + mi)/(4 - RELf - RELi).

APPENDIXSimilarly, RE contributed to sire s by progenya (RE ) can be calculated assa

Formulas for D E ~ And DEprog- mate areequivalent to the appendix formulas of Misztaland Wiggans (4) for RE from parents andprogeny. hoofs follow. Formulas of Meyer (3)for information from parents are identical tothose of Misztal and Wiggans 4), but Meyers(3) formulas for information from progeny didnot adjust for =Late.

Let ba represent total amount of informationfor animal a expressed in RE rather than inDE. Define qa as RE for a with parents contri-butions excluded, q, as RE of as sue with thecontribution of progeny a excluded, q d as thecorresponding statistic for progeny as dam,and a s the ratio

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