MIXED EFFECTS MODELS

Transcription

1 MIED MODELS BLUP

2 MIED EFFECS MODELS Up to ow w dcd modl cldg fxd ffct ol. Frqtl, howvr, lar modl cota alo factor who lvl rprt a radom ampl of a poplato of all pobl factor lvl. Modl cotag both fxd ad radom ffct ar calld mxd ffct modl. Lar mxd ffct modl hav b wdl d aal of data whr rpo ar cltrd arod om radom ffct, ch that thr a atral dpdc btw obrvato th am cltr. For xampl, codr rpatd marmt ta o ach bjct logtdal data, or obrvato ta o mmbr of th am faml a gtc td.

3 LINEAR MIED EFFECS MODEL β rpo cdc matrc fxd ffct radom ffct rdal Grall, t amd that ad ar dpdt from ach othr ad ormall dtrbtd wth zro-ma vctor ad varac-covarac matrc G ad, rpctvl,..: G ~ MVN, Ifrc rgardg mxd ffct modl rfr to th tmato of fxd ffct, th prdcto of radom ffct, ad th tmato of varac ad covarac compot, whch ar brfl dcd xt.

4 ESIMAION OF FIED EFFECS Lt β ε, whr ε E[ ε ] E[ ] E[ ] E[ ] Var[ ε ] Var[ ] Var[ ] Var[ ] G Sch that ~ MVN β, V, whr V G Udr th crcmtac, th MLE for β : β V V ~ MVN β, V

5 ESIMAION OF FIED EFFECS A G ad ar grall ow, a tmat of V d tad ch that th tmator bcom: V V β h varac-covarac matrx of ow approxmatd b. β Not: bad dowward a a coqc of gorg th varablt trodcd b worg wth tmat of covarac compot tad of thr tr ow paramtr val. Approxmatd cofdc rgo ad tt tattc for tmabl fcto of th tp ca b obtad b g th rlt: V V K β ], [ D N F ra ϕ ϕ K β K K V K β K whr rfr to a F-dtrbto wth dgr of frdom for th mrator, ad dgr of frdom for th domator, whch grall calclatd from th data g, for xampl, th Sattrthwat approach. ], [ D N F ϕ ϕ ra N K ϕ ϕ D

6 ESIMAION PREDICION OF RANDOM EFFECS I addto to th tmato of fxd ffct, vr oft gtc trt alo o prdcto of radom ffct. I lar Gaa modl ch prdcto ar gv b th codtoal xpctato of gv th data,.. E[ ]. Gv th modl pcfcato, th jot dtrbto of ad : E[ ] β V ~ MVN, G E[ ] Cov[, G V ]Var β G G G From th proprt of mltvarat ormal dtrbto, w hav that: [ ] E[ ] G β h fxd ffct β ar tpcall rplacd b thr tmat, o that prdcto ar mad bad o th followg xpro: G G β

7 MIED MODEL EQUAIONS h olto ad dcd bfor rqr. A V ca b of hg dmo, pcall amal brdg applcato, t vr grall comptatoall dmadg f ot fabl. Howvr, Hdro 95 prtd th mxd modl qato MME to tmat β ad mltaol, wthot th d for comptg. h MME wr drvd b maxmzg for β ad th jot dt of ad, xprd a: β û V G β β G G, β / / xp,, p G β β G G, β ],, log[p β G G β β β V h logarthm of th fcto :

8 MIED MODEL EQUAIONS h drvatv of rgardg β ad ar: Eqatg thm to zro gv th followg tm: G β β β G β β β G whch ca b xprd a: ow a th mxd modl qato MME.

9 BLUE ad BLUP Ug th cod part of th MME, w hav that: G β β G o that: β G G It ca b how that th xpro qvalt to ad, mor mportatl, that th bt lar bad prdctor BLUP of. û β β G β G G β ] [ } ] [ { Ug th rlt to th frt part of th MME, w hav that: V V β Smlarl, t how that th xpro qvalt to, whch th bt lar bad tmator BLUE of β.

10 BLUE ad BLUP It mportat to ot that ad rqr owldg of G ad. h matrc, howvr, ar rarl ow. h a problm wthot a xact olto g clacal mthod. h practcal approach to rplac G ad b thr tmat ad to th MME. Not that f G ad ar ow, th varac covarac matrx of th BLUE ad BLUP : β û Var G β Ĝ β ~ ~ β G ~ ~ If G ad ar ow ad thr val ar rplacd th MME b om ort of pot tmat ad, th w olto ad of th tm: Ĝ > ~ ~ Var G β ar o logr BLUE ad BLUP olto, a th ar ot v lar fcto of th data. It how alo that grall:

11 ESIMAION OF VARIANCE COMPONENS Codr th data t blow, rlatd to obrvato of half-b faml of rlatd r. h followg modl ca b d to rprt th data: µ j ANOVA Etmato whr j rprt th photpc trat obrvato of prog j j,,, faml, µ a ma, a ffct commo to all amal havg r, ad j a rdal trm. j h r ffct qvalt to th tramttg ablt whch qal to ohalf addtv gtc val of r, a o-half of t g ar radoml tramttd to ach of t prog. h rdal trm j rfr to addtoal gtc ffct ch a th ffct of dam ad vromtal compot. d It amd that ~, ad ~,. j d Sr

12 ESIMAION OF VARIANCE COMPONENS h ovrall ampl ma gv b: ANOVA Etmato From th modl ttg dcd bfor w hav that: E[ ] µ ad Var[ ] j j. N whr ad j ar th r-pcfc ma. j j N j h aal of varac ANOVA approach cot of a orthogoal dcompoto of th total m of qar SS to btw cla or, or ca, r ad wth cla or rdal compot. h corrctd trm of th gral ma SS gv b: N SS j j

13 SS ESIMAION OF VARIANCE COMPONENS j j [ j j ANOVA Etmato B addg ad btractg wth th parth, th SS ca b xprd a: ] j j j It that th lat part of th xpro qal to zro, o that SS ca b wrtt a two compot, gv b: SSS j ad RSS j j whch ar th r ad th rdal m of qar, rpctvl. h SSS trm mar th varato of ach prog faml arod th ovrall ma, whl th RSS trm mar th xtra varato rlatd to ach obrvato arod t r avrag.

14 ESIMAION OF VARIANCE COMPONENS ANOVA Etmato It ca b how that th xpctato of th m of qar trm ar: E[SSS] N N o that th ANOVA tmator of th r ad rdal varac compot ar gv b: N [SSS ] ad RSS N N rpctvl. I th pcfc ca of balacd data,.. th am prog z for all r, N / ad th ANOVA tmator bcom: ad E[RSS] N SSS ad RSS

15 ESIMAION OF VARIANCE COMPONENS I gral, th ANOVA approach wor wll for mpl modl ch a a o-wa trctr or balacd data ch a data from dgd xprmt wth o mg data, bt th ar ot dcatd for mor complx modl ad data trctr ch a tho grall fod amal brdg. A mbr of mthod hav b propod for tmatg varac compot mor complx caro, ch a th xpctd ma qar approach of Hdro 953, ad th mmm orm qadratc bad tmato Rao 97a, 97b, bt maxmm llhood bad mthod ar crrtl th mot poplar o, pcall th rtrctd or rdal maxmm llhood REML approach, whch attmpt to corrct for th wll-ow ba th clacal maxmm llhood ML tmato of varac compot. h two mthod ar brfl dcrbd xt.

16 ESIMAION OF VARIANCE COMPONENS Maxmm Llhood ML Etmator Maxmm llhood tmat of th varac compot ca b obtad b maxmzg th log-llhood L β, G, wth rpct to ach lmt of G ad, aftr rplacg β b g β V V. Altratvl, G,, ad β ca b tmatd mltaol b maxmzg thr jot log-llhood wth rpct to th varac compot ad th fxd ffct. Stadard rror ca th b obtad b th vr of th tmatd Fhr formato matrx. h approach provd a tmator for th varac-covarac matrx of β whch ta to accot th xtra varablt rlatd to th tmato of th varac compot.

17 A a mpl xampl of maxmm llhood tmato of varac compot, codr th balacd ca.., cotat prog z half-b faml data t dcd prvol, ad th lar modl: µ j j wth th am dfto a bfor, bt wth th addtoal ampto of ormalt of both th r ad th rdal ffct,..: d d ~ N, ad ~ N, j

18 O a matrx otato, th modl ca b xprd a: whr rprt th vctor of obrvato of prog.., rlatv to r ; ad rprt - dmoal colm vctor of ad, rpctvl; ad d th vctor of rdal aocatd wth prog. µ ] [ ],,, [

19 h vctor of obrvato ha th a mltvarat ormal dtrbto wth ma vctor ad varac-covarac matrx gv b, ad t dt fcto from whch th llhood fcto obtad ca b wrtt a: whr a matrx of, ad th Krocr prodct. ] [ µ N µ N I I / N N /,, p π µ I J I µ µ xp N N I J π µ µ xp N N N N J I J

20 h log-llhood fcto ca b wrtt th a: l µ,, N log log j j µ B tag th drvatv ad ttg thm to, th followg olto ar obtad: µ, RSS ad from whch maxmm llhood tmat of th varac compot ar obtad, xcpt f <, whch ca th tmat t to zro. Not th dffrc btw th maxmm llhood ad th ANOVA tmator of. It wll ow that maxmm llhood tmat of varac compot ar bad dowward a th do ot ta to accot th dgr of frdom d for tmatg th fxd ffct. SSS

21 ESIMAION OF VARIANCE COMPONENS Rdal Maxmm Llhood REML Etmator Aothr altratv llhood-bad mthod for frrg varac compot mxd modl th rtrctd or rdal maxmm llhood approach REML, whch corrct th ba aocatd wth maxmm llhood tmat b tag to accot th dgr of frdom d for tmatg th fxd ffct. h REML approach for tmato of varac compot maxmz th llhood fcto of a t of rror cotrat d L, whr L a [ p] fll-ra matrx wth colm orthogoal to th colm of th cdc matrx. h vctor d th follow a mltvarat ormal dtrbto wth ll ma vctor ad varac-covarac matrx L VL L G L. Not that th dtrbto of d do ot dpd o β.

22 h rdal llhood fcto for th varac compot th: L G, π p / / L VL xp d L VL d Aothr approach for obtag th rdal llhood fcto for th varac compot b tgratg th fxd ffct ot of th fll llhood fcto,..: L G, L β, G, dβ a lltratd th followg xampl.

23 Rcall th balacd half-b faml data t, ad t aocatd llhood fcto: It rdal llhood th: whch qal to: whr. N N,, L π µ µ j j xp µ µ d,, L, L N N π µ µ d xp xp j j N N, L λ π xp xp j j λ π µ λ λ

24 B tag th drvatv wth rpct to λ ad, ad b g th varac proprt of maxmm llhood tmator, th followg olto ar obtad: RSS ad whch ar th REML tmat of th varac compot, xcpt f <,.. f SSS < RSS. SSS A xplct form of ML ad REML tmator ar oft ot avalabl for mor complx mxd ffct modl, ML ad REML tmat ar grall obtad b tratv approach ch a th xpctato-maxmzato EM algorthm ad Nwto- Rapho-bad procdr.

25 MIED MODELS IN ANIMAL BREEDING Amal brdg program ar bad o th prcpl that photpc obrvato o rlatd dvdal ca provd formato abot thr drlg gotpc val. h addtv compot of gtc varato th prmar dtrmat of th dgr to whch offprg rmbl thr part, ad thrfor th all th compot of trt artfcal lcto program. Ma tattcal mthod for aal of gtc data ar pcfc or mor approprat for photpc marmt obtad from plad xprmtal dg ad wth balacd data t. Whl ch tato ma b pobl wth laborator or grho xprmtal ttg, data from atral poplato ad agrcltral pc ar grall hghl balacd ad fragmtd b mro d of rlatohp.

26 ANIMAL MODEL Cllg of data to accommodat covtoal tattcal tchq.g. ANOVA ma trodc ba ad/or lad to a btatal lo of formato. h mxd modl mthodolog allow ffct tmato of gtc paramtr ch a varac compot ad hrtablt ad amal brdg val whl accommodatg xtdd pdgr, qal faml z, ovrlappg grato, x-lmtd trat, aortatv matg, ad atral or artfcal lcto. o lltrat ch applcato of mxd modl amal brdg, w codr hr th o-calld Amal Modl tato wth a gl trat ad a gl obrvato cldg mg val pr amal.

27 ANIMAL MODEL h amal modl ca b dcrbd a: β whr: a vctor of obrvato photpc cor β a p vctor of fxd ffct.g. hrd-ar-ao ffct ~ N, G a q vctor of brdg val rlatv to all amal wth rcord or th pdgr fl, ch that q gral bggr tha rprt rdal ffct, amd rdal varac. ~ N,R I, whr th

28 HE MARI A h matrx G dcrbg th covarac amog th radom ffct hr th brdg val follow from tadard rlt for th covarac btw rlatv. It that th addtv gtc covarac btw two rlatv ad gv b θ, whr a θ th coffct of coactr btw dvdal ad, ad a th addtv gtc varac th ba poplato. Hc, dr th amal modl, G A a, whr A th addtv gtc or mrator rlatohp matrx, havg lmt gv b a θ.

29 ANIMAL MODEL I gral amal brdg trt o prdctg brdg val for lcto of pror dvdal, ad o tmato of varac compot ad fcto throf, ch a hrtablt. h fxd ffct ar, om, ac factor wth o ctral trt trm of frc, bt whch d to b ta to accot.., th d to b corrctd for wh frrg brdg val. Sc dr th amal modl G A a ad modl qato ca b xprd a: R I, th mxd λa β λ whr, ch that: a h h β λa

30 Codtoal o th varac compot rato λ, th BLUP of th brdg val ar gv th b λa β. h ar grall rfrrd to a tmatd brdg val EBV. Altratvl, om brdr aocato xpr thr rlt a prdctd tramttg ablt PA or tmatd tramttg ablt EA, whch ar qal to half th EBV, rprtg th porto of a amal brdg val that pad to t offprg. h amot of formato cotad a amal gtc valato dpd o th avalablt of t ow rcord, a wll a how ma ad how clo rlatv t ha wth photpc formato. A a mar of amot of formato lvtoc gtc valato, EBV ar tpcall rportd wth t aocatd accrac. Accrac of prdcto dfd a th corrlato btw tr ad tmatd brdg val,.., r ρû,. Itad of accrac, om lvtoc pc gtc valato rlablt, whch th qard corrlato of accrac. r

31 PREDICION ACCURACY h calclato of ρ û, rqr th dagoal lmt of th vr of th MME coffct matrx, rprtd a: C ββ β λa C C It how that th prdcto rror varac of EBV gv b: PEV Varû whr c th -th dagoal lmt of C, rlatv to amal. h PEV ca b trprtd a th fracto of addtv gtc varac ot accotd for b th prdcto. hrfor, PEV ca b xprd alo a: ch that c r, from whch th rlablt obtad a: a PEV r c a C C β û r c / a λc

32 hrd hrd ANIMAL MODEL h h β

33 ANIMAL MODEL ~ N, A Brdg val:, wth A β α λa h h h α 3 ĥ ĥ û û û û û

34 h amal modl ca b xtdd to modl mltpl corrlatd trat, mltpl radom ffct ch a matral ffct ad commo vromtal ffct, rpatd rcord.g. tt da modl, ad o o. EAMPLE: Mrod 996, pp74-76 Codr th followg data t o th wag wght g of pglt, whch ar prog of thr ow matd to two boar:

35 A lar modl wth th fxd ffct of x, ad th radom ffct of commo vromt rlatd to ach lttr ad brdg val ca b xprd a : Wght Sx β Wc Rdal Commo vromt Brdg val Amg that, 5 ad 65, th MME ar a follow: c W W β A λ W W W W Iλ c W whr λ 3.5 ad λ c

36 h BLUE ad BLUP vrtg th mrator rlatohp matrx ar:

37 Harvll ad Callaa 989

38 pdgrmm: A R pacag for fttg gralzd lar mxd modl amal brdg Vazqz t al.