Mauricio A. Elzo, University of Florida, 1996, 2005, 2006, [20-1] ANIMAL BREEDING NOTES CHAPTER 20 ADDITIVE GENETIC GROUP MODELS

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1 Maurc A. Elz, Uvrt f Flra, 996, 5, 6,. [] ANIMAL BEEING NOES CHAPE AIIVE GENEIC GOUP MOELS h ml tu utl w hav all aum that th xpct valu f th BLUP f th amal tc valu wa zr,.., E[u] =. h wa a aumpt ma wh th BLUP wa rv. It ma that a prr all amal ar cr t qual. Al, th frmat u t valuat a amal rr twar zr. Hwvr, m ca E[u]. h ca happ wh: () Amal cm frm varu tcall tct ppulat. () Amal wr r vr a umr f ar wth a ppulat that urwt tc cha u t lct vr tm. h ca ca hal a a tc rup factr t th ml. h rup trat, hwvr, wll ffr ca () a (). Atv rup ml (AGM) wll xpla a th rmatral rar ml (SMM). Staar Atv Gtc Grup Ml (SAGM) h SAGM ca u, fr xampl, wh amal l t ffrt r r ffrt cutr, a thr mrat acr th varu r ppulat. A lar ml fr uch ca : E[] = u = u A var = = vctr f tc rup,

2 [] = cc matrx rlat lmt f u t lmt f, a = A ¼ h rma trm ar a f fr th SMM. mark: () h av lar ml aum qual atv tc a rual varac a cvarac all tc ppulat, whch ma t tru. () Sr ar t wth tc rup. clu tc rup th av ml, a a utract,.., = ( B u) Lt = u B hu, th SAGM : E[] = = A var = h uual MME fr th SAGM ar: A h quat fr th fx ffct ar uuall ar. hu, th MME fr th SAGM cm:

3 [3] M M M M M A M M = B B B ( B ) B Bcau r ar t wth tc rup, th prramm trat t frm th MME vlv: () cmput th M matrx, () umm th apprprat rw f M t ul th M matrx, () umm th apprprat clum f M t frm th M matrx, (v) cmput M, (v) a apprprat rw f M t ta M, a (v) cmput th { } u uaa alrthm a A B u Hr rul. h jctv f th SAGM cul : () t cmpar ppulat,.., w wat t cmput: { B } E[ B ] = B () t rak r wth a ppulat,.., w t cmput: { ŝ j ŝ j} () t cmpar r acr ppulat,.., w mut ta: {u j B u j }

4 [4] u j = ŝ j u j = ŝ j Uuall th ar cmput a vat frm a a ppulat f a a l ppulat r a wht avra f all f th ppulat vlv. I uch ca, E[ ] = B a h vctr f prct f atv tc valu f amal u,.., û = ĝ ŝ ca ta rctl mf th ar MME fr th SAGM a fllw: () Lt P = I I P B = I I () Prmultpl th LHS a th HS f th ar MME (P B ), (P B ) = I I t ta: M A M A M () Irt P B P tw th LHS a th vctr f ukw f th MME (),.., ptmultpl th LHS P B, a prmultpl th vctr f ukw P. h rult t f quat call th mf MME fr th SAGM. h quat ar:

5 [5] A A A M A u M h quat cvr much fatr tha th uual MME fr th SAGM (Va Vlck a wr, 985). Multpl trat vr f th mf MME wr frt mplmt at Crll Uvrt th m98 (Nrth Eat ar Sr Evaluat a th Amrca Smmtal Sr Evaluat). mark: () Frm th frt quat f th mf MME: A B B = A B B u = (A B ) B A B u.., rup ffct ar lar cmat f th u th BLUP f (a a vat frm a ) : ĝ = (A B ) B A B û () Frm th c quat f th mf MME: BA B B (M A B B ) u = M û = (M A B B ) B M (M A B B ) B A B B ĝ A th amut f frmat pr amal cra, th c trm wll t zr. Al, t that: (M A B B ) B A B B = (M A B B ) B [(M A B B ) B M] = B (M A B B ) B M û = (M A B B ) B M( B ĝ ) ĝ A th amut f frmat pr amal cra, th mprtac f th rup cmpt

6 [6] cra. () Prmultpl th f th MME f th SAGM (.., th umf MME) wth ar [BI ] l: A B B ŝ = A B ŝ =.., th wht um f th ŝ j um t zr wth tc rup, th wht ar th lmt f th vr f th umratr rlathp matrx. If thr l ppulat, =, thu, A B ŝ = A B û = a wh A = I, ŝ = f thr ar vral tc rup, ŝ = f thr l tc rup. A prf that A B û = a l ppulat (uaa, 986) th fllw. h MME fr a l ppulat ar:, a (M A B B ) û = M M = B () B M = I B () B M = B () B = Ntc that ( = umr f rcr vctr ) th clum pac f. hu, ca

7 [7] wrtt a k, = k M = km = Al, =, = umr f r M = M = hu, (M A B B ) û = M A B û = û = f A = I th û um t zr "wth" th vrall ma f a l ppulat. h SAGM ca al u t accut fr ffrc th ma tc valu f amal r ffrt ar, rat, tc. Hwvr, a mr raltc ml th accumulat rup ml. Accumulat Atv Gtc Grup Ml (AAGM) h AAGM wa truc hmp (979). h AAGM fr a r ml, u calar tat, : j = x j k E[ j ] = x j k a k k j a k k var j a = I

8 [8] = vctr f fx ffct, a k = atv tc rlathp tw amal a k, k = atv tc rup whch amal k l t, a k = ummat vr all actr f amal. hu, ta f hav rup ffct fr th th amal a th SAGM, th AAGM cta a wht um f th tc rup ffct f all th actr f a amal at t that amal rup,.., fr th r ml av: ak = r tc rup k k rar tc rup 3 rat rar tc rup h atv tc valu f r : u = k a k k rvat f th AAGM u matrx tat Lt a lar ml fr vctr : E[] = u = E[u]

9 [9] u A var = But A = var (I ½ P ) (I ½ P ¼ P m ) fr ml clu r a am fr ml clu r a m hu, th fllw quvalt AAGM ml ca w cr: E[] = ( B )u = ( B u) = φ = E[φ] var = h MME fr th quvalt AAGM ml ar: Cmpt f th accumulat rup call that fr th SMM, φ = (I B P B 3P m ) u = B u

10 [] Cqutl, th accumulat rup wll clu all th tc ffct cta φ wh xpctat ffrt frm zr. hu, () f th r a th m f amal ar tf, = ½u ½u ½ u ¼u ½ = u ½ u m m = ¼ um ¼ um ½ ¼um = ¼ um ½ ˆ E = E [¼ um ] E [½ φ ] E [φ ] Atv tc valu f ull m Slct f ull am Slct f th ull thmlv () f th r f amal tf l, = ½ u ½ u ½ u = ½ u E = E[½ u ] E[ ]

11 [] E = E [¼ um ] E [¼ um ] E [½ φ ] E [φ ] Atv tc valu f ull m Atv tc valu f ull m Slct f ull am Slct f th ull thmlv () f th matral rar f amal tf l, = ½ u ½ u ¼ u ½ u ½ u m = ½ u m = ½ u ¼ um ½ E = E [½ u ] E [¼ um ] E [½ φ ] E [φ ] Atv tc valu f ull r Atv tc valu f ull m Slct f ull am Slct f th ull thmlv (v) f thr th r r th m f amal ar tf, = ½ u ½ u

12 [] E = E [½ u ] E [½ u ] E [φ ] Atv tc valu f ull r Atv tc valu f ull Slct f th ull thmlv am Crtra u t ctruct rup (a) Actr tf,.., th rm ca, thr ar fur rup: rup = r a m tf, rup = r tf l, rup 3 = m tf l, a rup 4 = thr r r m tf. () I at t th rup crtr (a) w ca a tm, tu, r, r, tc. Fr tac, f thr wr l mr crtr: tm a l rat, thr wul ht rup all. I trm f th AAGM ml w hav that: E [ ] = matrx that rlat th φ t tc rup accr t crtra (a) a () av. hu, th cmplt pcfcat f th quvalt AAGM ml : E[] = φ = φ var = σ

13 [3] But, u = φ. hu, = u E[] = u var = σ A h quvalt AAGM ml ca rwrtt a fllw: = ( u) = (u ) =, = u E[] = var = σ A wth MME: A h quat fr th AAGM ar tcal t th f th SAGM, wth uttut fr, a u th fact that A = a A B = B B B. h trafrm MME t cmput vctr u a ta f vctr a ca ta u a mlar prcur t th utlz wth th SAGM. hu, () Lt

14 [4] P = I I I, P B = I I I, a (P B ) = I I I () Prmultpl th LHS a th HS f th AAGM MME (P B ), a th ptmultpl th LHS f th rult MME P B a prmultpl th vctr f ukw P. h rult MME ar th mf MME fr th AAGM: u A h MME fr th AAGM ar mpl t prram (uaa, 986). Uuall th fx ffct ar ar, th mf MME fr th AAGM cm: M u M A M = B B B ( B ) B B. h MME ca ult a fllw: () Cmput a halfbtr th matrx M, () Cmput a tr th vctr M, a () Cmput a halftr th matrx. A h ctrut f a ull t th umatrc () ar:

15 [5] m f ull /6 r f ull /8 ull B/4 ull rup /4 /4 B/ / B m f ull r f ull ull ull rup lathp tw th SAGM a th AAGM h SAGM ; = u E[] =, = cc matrx rlat r t rup u A var = σ h AAGM : E[] = u =, = cc matrx rlat rual tc trm (φ j ) t rup u A var = I th SAGM, ull ar plac t rup. hu, E[u j ] = O th thr ha, rual tc trm (φ j ) ar rup th AAGM. hu, But E[φ j ] =

16 [6] φ = B u u = φ E[u j ] = t j t j = lmt f th th rw f t, a = (I B P) B = (I B P B 3P m ) B f r a am ar clu th ml f r a m ar clu th ml Lt * = t j h, fr th SMM w hav that th lmt f ar: t j = ½ t, j m ¼ t m, j t = * = () ½ t ¼ t j j, k k mj mj, k k k k * * * = ½ m ¼ j E[u j ] = ½ E[u ] m ¼ E[um ] j j = E[uj] ½ E[u ] m ¼ E[um ] r j j E [ ] = E[uj] ½ E[u ] m ¼ E[um ] j j j j j j m j j j th SAGM a th AAGM wll quvalt l f th pr f all ull ar th am a all ull l t th am rup. If, th rak f ull wll th am th SAGM j j j

17 [7] a th AAGM. mark: () h cmpt f th rual tc trm φ j wll p th actr tf a vual. () h xpct valu f th φ j wll trm th xpct valu f th atv tc valu f th actr f vual j clu φ j a th xpct valu f th Mla ampl trm (.., th ) l t th actr a t amal j. () h xpct valu f th Mla ampl trm (.., th ) rflct th ccurrc f lct a rup f amal. Lt = tc rup valu f th amal th th tc rup, th th tc rup ma f t, fr tac, all amal l t th th (tm r) ucla. Bcau t tmal, ( B a ) u t ta a uqu ĝ ull valuat. Ntc that f thr l (tm r) ucla ( B a ) = ( a B a ) =. h, (a) f th th tc rup ulct, E[ ] j j = j th amal frm th th tc rup () f thr wa lct f amal th th tc rup, E[ j] (lct )

18 [8] E[ ] j lct = atv tc valu f th lct rup f amal frm th th (tm r) ucla, a Δ = tc lct ffrtal fr th th tc rup. Exampl: If ull f r ar mat t am f r, th xpct valu f th f thr pr qual t: () E[ j ] = = ½ E[ ] ½ E[ ] j j r am = E[ j] E[ j ] = ½ ½ f th part f amal j ar tf a th r a th am tc rup ar ulct, () E[ ] = E[ j] E[ j ] j = ½ ½ f th part f th amal j ar kw a th partal tc rup ar lct.

19 [9] Urta th lut ta u th AAGM Cr th MME wthut ar th,.., A u () Grup lut Frm th 3 r quat f th MME: B B B û = B B ĝ ĝ = ( B ) B B B û B = (I B P) fr ml wth r a am: AM, AM, SM B = (I B P B 3P m ) fr th SMM B = (I B P ) fr th SM φˆ = B û = {û j δ { ûj δ { ûj δ j j j ½ ½ û ½ û δm ¼ ûm ½ j û j j } δ j j û j } j } fr th AM,AM,SM fr thsmm fr thsm B = aal matrx f rup wht,.., th j,j, whch wr f th chaptr al wth A a A B. Ntc that th f all th amal a rup th am cau f th rup trat a actr tf. = cc matrx that um th trm j, j ˆ wth a rup, j

20 [] ( B ) B = aal matrx wth th wht fr ach rup. Ntc that cau B aal, () ( B ) B = j,j = umr f amal rup, a () B B û = B B û ĝ = ( ) B B B û = ( ) B B û ĝ = ûj ½ û ½ û m ¼ ûm j= j j j j j j = ûj PIj j= kj = f actr k f amal j kw a u th ml a thrw, a PI j = pr x f amal j. () Prct f atv tc valu Frm th c quat f th MME: B ( B A B B ) û B B B B ĝ = B ( B A B B ) û = B ( B ) B B B ĝ û = ( B A B B ) B B ( B ) ( B A B B ) B B B B ĝ A a xampl, cr th th part a SMM. I th ca, A B ha at mt 3 zr valu

21 [] th th rw,.., ¼ m ½ Al, B B B ĝ p p p p pp ĝ ĝ I p p p p pp ĝ ĝ p p p p pp ĝ ĝ ĝ hu, ĝ û ½ û ¼ û r * m * ĝ ¼ û ½ û û r m * _ * r r û * * * ¼ û ½ û ĝ r m * Numrcal xampl fr th SAGM a th AAGM Cr a SMM u th xampl u fr th rct ffct ml.

22 [] () h SAGM : = h MME a th vctr f lut fr th SAGM ar: α 4 A Smmtrc

23 [3] ŝ ŝ ŝ ŝ 5 4 3

24 [4] () h AAGM : = h trafrm MME (.., trm f, a u = ) a th vctr f lut fr th AAGM ar: u α 4 α 4 α 4 α 4 A u u u u Smmtrc

25 [5] û û û û mark h AAGM cr hr plac all amal t tc rup. A ffrt trat t a l a amal (.., amal wthut tf part) t tc rup. Fr a cmprhv crpt f th rup trat, pla rfr t uaa (988). frc uaa,. L Pral Cmmucat. Amal Scc 7. Crll Uvrt, Ithaca, N.Y. uaa,. L Atv tc ml wth rup a rlathp. J. ar Sc. 7 (Suppl. ): hmp, Sr valuat. Bmtrc 35: Va Vlck, L.. a. J. wr Cmpar f tratv prcur fr lv quat fr r valuat. J. ar Sc. 68:64.