Lecture 4 BLUP Breeding Values

Transcription

1 Lctr 4 BLUP Brdng Vals Glhrm J. M. Rosa Unvrsty of Wsconsn-Madson Mxd Modls n Qanttatv Gntcs SISG, Sattl 8 Sptmbr 8 Lnar Mxd Effcts Modl y = Xβ + + rsponss ncdnc matrcs fxd ffcts random ffcts rsdals G ~ MVN,

2 Estmaton of Fxd Effcts y = Xβ + ε wth ε = +, sch that Var[ε] = G + è MLE for β : ˆ β = X V X) X V y ~ MVN β, X V X) ) whr V = G + Prdcton of Random Effcts y Xβ V ~ MVN, G G G E[ y] = E[ ] + Cov[, y = G V Rplacng β by ts stmat: ]Var y Xβ) = G [ y] y E[ y]) G + ) y Xβ) ˆ = G G + ) y Xβˆ)

3 3 = + y y X β G X X X X ˆ ˆ Mxd Modl Eqatons ˆ) ) ˆ Xβ y G + = ˆβ = {X [ + G ) ]X} X [ + G ) ]y BLUP and BLUE: Anmal/plant brdng programs ar basd on th prncpl that phnotypc obsrvatons on rlatd ndvdals can provd nformaton abot thr ndrlyng gnotypc vals h addtv componnt of gntc varaton s th prmary dtrmnant of th dgr to whch offsprng rsmbl thr parnts, and thrfor ths s sally th componnt of ntrst n artfcal slcton programs Mxd Modls n Anmal and Plant Brdng

4 Mxd Modls n Anmal and Plant Brdng Many statstcal mthods for analyss of gntc data ar spcfc or mor approprat) for phnotypc masrmnts obtand from plannd xprmntal dsgns and wth balancd data sts Whl sch statons may b possbl wthn laboratory or grnhos xprmntal sttngs, data from natral poplatons and agrcltral spcs ar gnrally hghly nbalancd and fragmntd by nmros knds of rlatonshps Anmal Modl Cllng of data to accommodat convntonal statstcal tchnqs.g. ANOVA) may ntrodc bas and/or lad to a sbstantal loss of nformaton h mxd modl mthodology allows ffcnt stmaton of gntc paramtrs sch as varanc componnts and hrtablty) and brdng vals whl accommodatng xtndd pdgrs, nqal famly szs, ovrlappng gnratons, sx-lmtd trats, assortatv matng, and natral or artfcal slcton o llstrat sch applcaton of mxd modls n brdng programs, w consdr hr th so-calld Anmal Modl n statons wth a sngl trat and a sngl obsrvaton ncldng mssng vals) pr ndvdal 4

5 Anmal Modl h anmal modl can b dscrbd as: y = Xβ + + y s an n ) vctor of obsrvatons phnotypc scors) β s a p ) vctor of fxd ffcts.g. hrd-yarsason ffcts) ~ N, G) s a q ) vctor of brdng vals rlatv to all ndvdals wth rcord or n th pdgr fl, sch that q s n gnral bggr than n) ~ N, I n ) rprsnts rsdal ffcts, whr s th rsdal varanc h Matrx A h matrx G dscrbng th covarancs among th random ffcts hr th brdng vals) follows from standard rslts for th covarancs btwn rlatvs It can b shown that th addtv gntc covaranc btwn two rlatvs and s gvn by θ, a whr θ s th coffcnt of coancstry btwn ndvdals and, and a s th addtv gntc varanc n th bas poplaton G = A a Hnc, ndr th anmal modl,, whr A s th addtv gntc or nmrator) rlatonshp matrx, havng lmnts gvn by a = θ 5

6 h Matrx A For ach anmal n th pdgr =,,,n), gong from oldr to yongr anmals, compt a and a j j =,,,-) as follows: If both parnts s and d) of anmal ar known: a j = a j = a js + a jd )/ and a = + a sd / If only on parnt.g. d) of anmal s known: a j = a j = a jd / and a = If parnts nknown: a j = a j = and a = Exampl 4 3 Anmal Sr Dam pdgr matrx A 6

7 Anmal Modl In gnral, n anmal/plant brdng ntrst s on prdcton of brdng vals for slcton of spror ndvdals), and on stmaton of varanc componnts and fnctons throf, sch as hrtablty h fxd ffcts ar, n som sns, nsanc factors wth no cntral ntrst n trms of nfrncs, bt whch nd to b takn nto accont.., thy nd to b corrctd for whn nfrrng brdng vals) Anmal Modl Snc ndr th anmal modl G = A and a R = I n, th mxd modl qatons can b xprssd as: X X X ˆβ X + λa û = X y y whr λ = a h = h, sch that: " " ˆβ û = X X X X + λa " X y y 7

8 Condtonal on th varanc componnts rato λ, th BLUP of th brdng vals ar gvn thn by: ˆ = + λa y Xβˆ) hs ar gnrally rfrrd to as Estmatd Brdng Vals EBV) Altrnatvly, som brdrs assocatons xprss thr rslts as Prdctd ransmttng Ablts PA) or Estmatd ransmttng Ablts EA) or Expctd Progny Dffrnc EPD)), whch ar qal to half th EBV, rprsntng th porton of an anmal s brdng vals that s passd to ts offsprng ) h amont of nformaton contand n an anmal s gntc valaton dpnds on th avalablty of ts own rcord, as wll as how many and how clos) rlatvs t has wth phnotypc nformaton As a masr of amont of nformaton n lvstock gntc valatons, EBVs ar typcally rportd wth ts assocatd accracs Accracy of prdctons s dfnd as th corrlaton btwn tr and stmatd brdng vals,.., r = ρû, Instad of accracy, som lvstock spcs gntc valatons s rlablty, whch s th sqard corrlaton of accracy ) ) r 8

9 Prdcton Accracy h calclaton of rqrs th dagonal lmnts of th nvrs of th MME coffcnt matrx, rprsntd as: C = X X X X + λa It can b shown that th prdcton rror varanc of EBV s gvn by: û ρ û,) = Cββ C β C β C PEV = Varû ) = c c whr s th -th dagonal lmnt of, rlatv to anmal. C Prdcton Accracy h PEV can b ntrprtd as th fracton of addtv gntc varanc not accontd for by th prdcton hrfor, PEV can b xprssd also as: c PEV = = r ) sch that rlablty s obtand as: r ) a a, from whch th r = c / a = λc 9

10 hrd hrd Anmal Modl + + = h h y = X β + + λ = = h h ˆβ û " = X X X X + λa " X y y " ), N ~ A = A Brdng vals:, wth Anmal Modl

11 R Cod anmal modl y<-matrxc3,7,35),nrow=3) X<-matrxc,,,,,),nrow=3) toy xampl <-matrxc,,,,,,,,,,,,,,),nrow=3, byrow = RUE) A<-matrxc,,.5,.5,.5,,,,.5,,.5,,,.5,.5,.5,.5,.5,,.5,.5,,.5,.5,),nrow=5) h<-/3 hrtablty a=-h)/h crossprodcts XX<-crossprodX,X) X<-tX) * X<-t) * X <-crossprod,)+a*solva) mxd modl qatons coffcnt matrx and rght hand sd C<-rbndcbndXX,X),cbndX,)) rhs<-rbndtx) * y,t) * y) solton thta.hat <- solvc) * rhs h = α = 3 ĥ = 9 ĥ = 348 û = 4. û =. û3 = 4. û 4 =. û5 =. Anmal Modl h anmal modl can b xtndd to modl mltpl corrlatd) trats, mltpl random ffcts sch as matrnal ffcts and common nvronmntal ffcts), rpatd rcords.g. tst day modls), and so on Exampl Mrod 996, pp74-76): Wanng wght kg) of pglts, progny of thr sows matd to two boars:

12 A lnar modl wth th fxd) ffct of sx, and th random) ffcts of common nvronmnt rlatd to ach lttr) and brdng vals can b xprssd as X: Wght Assmng that =, = 5 and = 65 c, th MME ar as follows: X X X X W ˆβ X y X + A λ W û = y W X W W W + Iλ ĉ W y whr λ = = 3.5 and λ = = 4. 3! y = Xβ + + Wc + Sx Brdng vals Common nvronmnt c Rsdal h BLUEs and BLUPs nvrtng th nmrator rlatonshp matrx) ar: Mrod xampl