Mixed effects linear models with t-distributions for quantitative genetic analysis: a Bayesian approach

O iginal a icle Mixed e ec s linea models wi h -dis ibu ions o quan i a i e gene ic analysis: a Bayesian app oach Ismo S andén Daniel Gianola a Depa men o Animal Sciences, Uni e si y o Wisconsin, Madison, WI 53706, USA b Animal P oduc ion Resea ch, Ag icul u al Resea ch Cen e - MTT, 31600 Jokioinen, Finland (Recei ed 21 July 1998; accep ed 27 No embe 1998) Abs ac - A Bayesian app oach o in e ences abou pa ame e s o mixed e ec s linea models wi h -dis ibu ions is p esen ed, wi h emphasis on quan i a i e gene ic applica ions. The implemen a ion is ia he Gibbs sample . Da a om a simula ed mul iple o ula ion and emb yo ans e scheme in dai y ca le b eeding wi h non- andom p e e en ial ea men o some cows is used o illus a e he p ocedu es. Ex ensions o he model a e discussed. © In a/Else ie , Pa is mixed e ec s models / Bayesian in e ence / obus es ima ion / Gibbs sampling / S uden ’s -dis ibu ion Résumé - Modèles linéai es mix es a ec dis ibu ions de S uden en géné ique quan i a i e : app oche bayésienne. On p ésen e une app oche bayésienne en ue de l’in é ence conce nan les pa amè es de modèles linéai es mix es a ec des dis ibu ions de S uden , en me an l’accen su les applica ions en géné ique quan i a i e. L’applica ion s’e ec ue g âce à l’échan illonnage de Gibbs. Des données p o enan d’un schéma de sélec ion simulé u ilisan le ans e emb yonnai e chez les bo ins lai ie s en p ésence d’un ai emen p é é en iel de quelques aches son u ilisées pou illus e les p océdu es. Les ex ensions du modèle son discu ées. © In a/Else ie , Pa is modèle mix e / in é ence bayésienne / es ima ion obus e / échan illonnage de Gibbs / dis ibu ion de S uden * Co espondence and ep in s E-mail: [email p o ec ed] 1. INTRODUCTION Mixed e ec s linea models a e used widely in animal and plan b eeding and in e olu iona y gene ics [27]. Thei applica ion o animal b eeding was pionee ed by Hende son [17, 19-21], p ima ily om he poin o iew o making in e ences abou candida es o gene ic selec ion by bes linea unbiased p edic ion (BLUP). Because BLUP elies on knowledge o he dispe sion s uc u e, es ima ion o a iance and co a iance componen s is cen al in p ac ical implemen a ion [14, 18, 29, 32]. Typically, he dispe sion s uc u e is es ima ed using a likelihood-based me hod and, hen, in e ences p oceed as i hese es ima es we e he ue alues (e.g. [8]). Al hough no mali y is no equi ed by BLUP, i is p ecisely when no mali y holds ha i can be iewed as an app oxima ion o he bes p edic o [4, 8, 12, 19]. Mo e ecen ly, Bayesian me hods ha e been ad oca ed o he analysis o quan i a i e gene ic da a wi h mixed linea models [8, 9, 34, 39, 40], and he Bayesian solu ions sugges ed employ Gaussian sampling models as well as no mal p io s o he andom e ec s. I is o p ac ical in e es , he e o e, o s udy s a is ical models ha a e less sensi i e han Gaussian ones o depa u es om assump ions. Fo example, i is known in dai y ca le b eeding ha mo e aluable cows ecei e p e e en ial ea men , and o he ex en ha such ea men canno be accommoda ed in he model, his leads o bias in he p edic ion o b eeding alues [23, 24]. Ano he sou ce o bias in in e ences is an inco ec speci ica ion o he inhe i ance mechanism in he model. I is o en pos ula ed ha he geno ypic alue o a quan i a i e ai is he esul o he addi i e ac ion o alleles a a p ac ically in ini e numbe o unlinked loci and, hus, no mali y esul s [4]. This assump ion is e u ed in an ob ious manne when inb eeding dep ession is obse ed, o when unknown genes o majo e ec a e seg ega ing. Howe e , in he absence o clea ly con adic o y e idence, no mali y is a p ac ical assump ion o make, as hen he machine y o mixed e ec s linea models can be exploi ed. An appealing al e na i e is o i linea models wi h obus dis ibu ions o he e o s and o he andom e ec s. One o such dis ibu ions is S uden ’s , bo h in i s uni a ia e and mul i a ia e o ms. Se e al au ho s [2, 7, 26, 37, 38, 41, 42] ha e s udied linea and non-linea eg ession p oblems wi h S uden ’s -dis ibu ions, bu he e is a sca ci y o li e a u e on andom e ec s models. Wes [41] desc ibed a one-way andom e ec s layou wi h -dis ibu ed e o s and a hea y ailed p io o he andom e ec s. Assuming ha he a io be ween esidual a iance and he a iance o he andom e ec s was known, he showed ha his model could discoun e ec s o ou lie s on in e ences. Pinhei o e al. [30] desc ibed a obus e sion o he Gaussian mixed e ec s model o Lai d and Wa e [25] and used maximum likelihood. They hypo hesized ha he dis ibu ion o he esiduals had he same deg ees o eedom as ha o he andom e ec s, and, also, ha andom e ec s we e independen ly dis ibu ed. The i s assump ion is un ealis ic as i is ha d o accep why wo di e en andom p ocesses ( he dis ibu ions o andom e ec s and o he esiduals) should be go e ned by he same deg ees o eedom pa ame e . The second assump ion is no enable in gene ics because andom gene ic e ec s o ela i es may be co ela ed. In quan i a i e gene ics he andom e ec s o unc ions he eo a e o cen al in e es . Fo example, in animal b eeding p og ams he objec i e is o inc ease a linea o non-linea me i unc ion o gene ic alues which, ideally, akes in o accoun he economics o p oduc ion [16, 28, 33]. He e, i would seem na u al o conside he condi ional dis ibu ion o he andom e ec s gi en he da a, o d aw in e ences. The e a e wo di icul ies wi h his sugges ion. Fi s , i is no always possible o cons uc his condi ional dis ibu ion. Fo example, i he andom e ec s and he e o s ha e independen -dis ibu ions, he condi ional dis ibu ion o in e es is unknown. Second, his condi ional dis ibu ion would no inco po a e he unce ain y abou he pa ame e s, a well-known p oblem in animal b eeding, which does no ha e a simple equen is o likelihood-based solu ion (e.g. [10, 15]). I , on he o he hand, he pa ame e s ( he ixed e ec s and he a iance componen s) a e o p ima y in e es , he me hod o maximum likelihood has some impo an d awbacks. In e ences a e alid asymp o ically only, unde egula i y condi ions, and ini e sample esul s o mixed e ec s models a e no a ailable, which is pa icula ly ue o a model wi h -dis ibu ions. In addi ion, some gene ic models impose cons ain s such ha he pa ame e space depends on he pa ame e s hemsel es, so i would be nai e o apply a egula asymp o ic heo y. Fo example, wi h a pa e nal hal -sib amily s uc u e [6], he a iance be ween amilies is bounded be ween 0 and one- hi d o he a iance wi hin amilies. Mo eo e , maximum likelihood es ima ion in he mul i-pa ame e case has he no o ious de iciency o no accoun ing well o nuisance pa ame e s [3, 8, 13]. A Bayesian app oach o d awing in e ences abou ixed and andom e ec s, and abou a iance componen s o mixed linea models wi h -dis ibu ed an- dom and esidual e ms is desc ibed he e. Sec ion 2 p esen s he p obabili y model, emphasizing a s uc u e sui able o analysis o quan i a i e gene ic da a. Sec ion 3 gi es a Ma ko chain Mon e Ca lo implemen a ion. A Bayesian analysis o a simula ed animal b eeding da a se is p esen ed in sec ion 4. Po en- ial applica ions and sugges ions o addi ional esea ch a e in he concluding sec ion o he pape . 2. THE UNIVARIATE MIXED EFFECTS LINEAR MODEL 2.1. Sampling model and likelihood unc ion Conside he uni a ia e linea model whe e y is an n x 1 ec o o obse a ions; X is a known, ull ank, incidence ma ix o o de n x p o ’ ixed’ e ec s; b is a p x 1 ec o o unknown ’ ixed’ e ec s; Z is a known incidence ma ix o o de n x q o addi i e gene ic e - ec s; u is a q x 1 ec o o unknown addi i e gene ic e ec s ( andom) and e is an n x 1 ec o o andom esidual e ec s. Al hough only a single se o an- dom e ec s is conside ed, he model and subsequen esul s can be ex ended in a s aigh o wa d manne . I is assumed ha u and e a e dis ibu ed inde- penden ly. Suppose he da a ec o can be pa i ioned acco ding o ’clus e s’ induced by a common ac o , such as he d o he d-yea season o cal ing in a ca le b eeding con ex . The model can hen be p esen ed as: whe e m is he numbe o ’clus e s’ (e.g. he ds). He e yi is he da a ec o o clus e i (i = 1, 2, ... , m), Xi and Zi and a e he co esponding incidence ma ices and ei is he esidual ec o pe aining o yi. Obse a ions in each clus e will be modeled using a mul i a ia e - dis ibu ion such ha , gi en b and u, da a in he same he d a e unco ela ed bu no independen , whe eas eco ds in di e en clus e s a e (condi ionally) independen . Le yi !b, u, 62 N ni (X ib + Zi u, 1,,, o, e 2, e ), whe e ni is he num- be o obse a ions in clus e i (i = 1, 2, ... , m), o ’ is a scale pa ame e and . is he deg ees o eedom. I ni = 1 o all i, he sampling model becomes uni a ia e . The condi ional densi y o all obse a ions, gi en he pa ame e s, is Al hough he m dis ibu ions ha e he same , and Qe pa ame e s, hese a e no iden ical. In pa icula , no e ha E(y2 !b, u, Qe, e) Xib + Zi u, and Va (y2!b, u, Qe, e) = hz !./( ! - 2), i = 1,2,...,m, so he mean ec o is peculia o each clus e . Homoscedas ici y is assumed, bu his es ic ion can be li ed wi hou di icul y. When each clus e con ains a single obse a ion, he e o dis ibu ion is he independen -model o Lange e al. [26]; hen, he obse a ions a e condi ionally independen . When all obse a ions a e pu in a single clus e , he mul i a ia e -model o Zellne [42] esul s; in his case, he deg ees o eedom canno be es ima ed. Each o he m e ms in equa ion (3) can be ob ained om he mix u e o he no mal dis ibu ion: wi h he mixing p ocess being: whe e x e is a chi-squa ed andom a iable on Ve deg ees o eedom [26, 38, 41,42]. 2.2. Bayesian s uc u e Fo mally, bo h b and u a e loca ion pa ame e s o he condi ional dis ibu- ion in equa ion (3). The dis inc ion be ween ’ ixed’ and ’ andom’ is equen is , bu om a Bayesian pe spec i e i co esponds o a si ua ion whe e he e is a di e en ial amoun o p io in o ma ion on b and u [8, 13]. In pa icula , he Bayesian coun e pa o a ’ ixed’ e ec is ob ained by assigning a la p io o b, so ha he p io densi y o his ec o would be: in Rp. This dis ibu ion is imp ope , bu lowe and uppe limi s can be assigned o each o he elemen s o b, as in So ensen e al. [34], o make i p ope . The p io dis ibu ion o addi i e gene ic alues u will be aken o be a mul i a ia e -dis ibu ion, and independen o ha o b. F om a quan i a i e gene ics poin o iew his can be in e p e ed as an addi i e, mul i a ia e no mal model (as in [4]), bu wi h a andomly a ying addi i e gene ic a iance. Because he mul i a ia e -dis ibu ion has hicke ails han he no mal, he p oposed model is expec ed o be somewha bu e ed agains depa u es om he assump ions made in an addi i e gene ic e ec s model, so ’gene ic ou lie s’ s emming om nonaddi i i y o om majo genes become, pe haps, less in luen ial in he o e all analysis. All p ope ies o he mul i a ia e no mal dis ibu ion a e p ese ed, e.g. any ec o o scala alued linea combina ion o addi i e gene ic alues has a mul i a ia e -dis ibu ion, he ma ginal dis ibu ions o all e ms in u a e , and all condi ional dis ibu ions a e as well. In pa icula , i he addi i e gene ic alues o pa en s and he seg ega ion esidual o an o sp ing a e join ly dis ibu ed as mul i a ia e , he addi i e gene ic alue o he o sp ing has a uni a ia e -dis ibu ion wi h he same deg ees o eedom. This implies ha he coances y p ope ies o he usual Gaussian model a e p ese ed. We hen ake as p io dis ibu ion: wi h densi y Abo e, q is he numbe o indi iduals included in u (some o which may no ha e da a), A is a known ma ix o addi i e ela ionships, au is a scale pa ame e and u is he deg ees o eedom pa ame e . Hence, Va (ulo, 2, !) = A< ! /( !2), which educes o he a iance-co a iance ma ix o addi i e gene ic alues o a Gaussian model when u -! oo. The scale pa ame e s o 2 and o 2 a e aken o ha e independen scaled in e ed chi-squa e dis ibu ions, wi h densi ies: espec i ely, o a > 0 and o > 0. He e, T’e ( Tu ) is a s ic ly posi i e ’deg ee o belie ’ pa ame e , and Te (T u) can be hough o as a p io alue o he scale pa ame e . These dis ibu ions ha e ini e means and a iances whene e he T pa ame e s a e la ge han 2 and 4, espec i ely. In animal b eeding esea ch, i is common p ac ice o assign imp ope la p io s o he a iance componen s o a Gaussian linea model [8, 9, 39]. I uni o m p io s a e o be used, i is ad isable o es ic he ange o alues hey can ake, o a oid imp op ie y (o en di icul o ecognize, see [22]). He e, one can ake Typically, he lowe bounds a e se o ze o, whe eas he uppe bounds can be elici ed om mechanis ic conside a ions, o se up a bi a ily. P io dis ibu ions o he deg ees o eedom can be disc e e as in Albe and Chib [1] and Besag e al. [2], o con inuous as in Geweke [7], wi h he join p io densi y aken as p( e, !) = p( e )p( u ). In he disc e e se ing, le j, j = 1, 2, ... , d e, and wk, k = 1, 2, ... , d!, be se s o s a es o he esidual and gene ic alues deg ees o eedom, espec i ely. The independen p io dis ibu ions a e: Because a mul i a ia e -dis ibu ion is assigned o he whole ec o u, he e is no in o ma ion con ained in he da a abou ,,. The e o e, equa ion (11) is eco e ed in he pos e io analysis. The e a e a leas wo possibili ies he e: 1) o assign a bi a y alues o Vu and examine how a ia ion in hese alues a ec s in e ences, o 2) o c ea e clus e s o gene ic alues by, e.g. hal -sib o ull-sib amilies, and hen assume ha clus e s a e mu ually independen bu wi h common deg ees o eedom. He e he . pa ame e would be es imable, bu a he expense o igno ing gene ic ela ionships o he han hose om hal -sib o ull-sib s uc u es. Al e na i e (2) may be sui able o dai y ca le b eeding (whe e mos o he ela ionships a e due o si es) o humans (whe e mos amilies a e nuclea ). A hi d al e na i e would be o use (2), hen ind he mode o he pos e io dis ibu ion o u, and hen use (1) as i his mode we e he ue alue. In he ollowing de i a ion, we adop op ion (1). The join p io densi y o all unknowns is hen: wi h ob ious modi ica ions i equa ions (8) and (9) a e used ins ead o equa ions (6) and (7). The join pos e io densi y is ound by combining likelihood equa ion (3) and app op ia e p io s in equa ions (4)-(11), o ob ain: whe e b E !p, u E K,j,o! > 0, Q! > 0 and e E j , j = 1, 2, ... , ,de} i a disc e e p io is employed. The hype -pa ame e s a e 7e , TM! Te , T u and u because we assume his las one o be known. He ea e , we supp ess he dependency on he hype -pa ame e s in he no a ion. 3. THE GIBBS SAMPLING SCHEME A Ma ko chain Mon e Ca lo me hod such as Gibbs sampling is acili a ed using an augmen ed pos e io dis ibu ion ha esul s om mix u e models. The -dis ibu ion wi hin each clus e in equa ion (3) is iewed as s emming om he mix u e p ocesses no ed ea lie . Likewise, he -dis ibu ion in equa- ion (5) can be a i ed a by mixing he ulA, o 2,s2 - N(O, AU2 /82 ) p ocess wi h s2 wu N X2. I ,,. The augmen ed join pos e io densi y is m whe e s, = (se l , ... , sP m ) and N = ! n2. In eg a ion o equa ion (14) wi h ! i=i espec o Se and s2 yields equa ion (13), so hese pos e io s a e ’equi alen ’. The e is a connec ion he e wi h he he e ogeneous a iance models o animal b eeding gi en, e.g. in Gianola e al. [11] and in San C is obal e al. [31]. These au ho s pa i ioned b eeding alues and esiduals in o clus e s as well, each clus e ha ing a speci ic a iance ha a ied a andom acco ding o a scale in e ed chi-squa e dis ibu ion wi h known pa ame e s. The ull condi ional dis ibu ions equi ed o ins umen a Gibbs sample a e de i ed om equa ion (14). Resul s gi en in Wang e al. [40] a e used. Deno e C = !c2!!, i, j = 1, 2,...,p + q, , and = { j, i,j = 1, 2,...,p + q o be he coe icien ma ix and igh -hand side o Hende son’s mixed model equa ions, espec i ely, whe e p + is he numbe o unknowns ( ixed and andom e ec s), gi en he dispe sion componen s Se , s l and he scale pa ame e s Qe and o 2 u The mixed model equa ions a e: bes linea unbiased es ima o (BLUE) o b, and u is he bes linea unbiased p edic o (BLUP) o u. Collec he ixed and andom e ec s in o a’ = (b’, u’) _ (a,, a2 , ... , ap + q) . Le a’ i = (a l , a 2 , ... , ai-1, a i+l , ... , ap + q). The condi ional pos e io dis ibu- ion o each o he elemen s o a is / P+9 B whe e 4i = c- 1 ! - y! c,j’aj L i, j = 1, 2, ... , p + q. This ex ends o blocks B j-1 / B j$i / o elemen s o a in a na u al way. I ai is a sub- ec o o a, he condi ional dis ibu ion o ai gi en e e y hing else, is mul i a ia e no mal wi h mean ai = Ci il i - E Cija i o app op ia e de ini ions o C ij , i and a! as B j-1 / B j54, / ma ices and ec o s. The condi ional pos e io densi y o each o he se is in he o m o a gamma densi y whe e s, _, is s, wi hou S; i’ Equi alen ly, , &dquo;e e / Simila ly, he condi ional pos e io densi y o s! also has he gamma densi y o m The condi ional pos e io dis ibu ion o Qe is a scaled in e ed chi-squa e dis ibu ion wi h o m I a bounded uni o m dis ibu ion is used as p io o Q e, i s condi ional pos e io is he unca ed dis ibu ion: The condi ional pos e io densi y o ou is: dis ibu ions, han hose ob ained wi h a mixed e ec s Gaussian linea model, he cu en pa adigm in quan i a i e gene ics [19, 21, 27]. Ou app oach was illus a ed wi h simula ed da a om a dai y ca le b eeding scheme, whe e cows we e subjec o ai ly p e alen and s ong p e e en ial ea men . A uni a ia e -model o he e o s led o mo e accu a e in e ences abou addi i e gene ic a iance han ei he a he d-clus e ed -model o a Gaussian sampling p ocess. The pos e io dis ibu ions o b eeding alues o some example animals we e sha pe in he uni a ia e -model. Ou model and implemen a ion can be ex ended in se e al espec s. Fo example, i he deg ees o eedom o he dis ibu ion o gene ic alues needs o be assessed, i is possible o clus e he gene ic alues in o ’independen ’ amilies and p oceed as o he esidual a iance. Howe e , such clus e ing would lead o a loss o accu acy in he speci ica ion o he gene ic a iance-co a iance s uc u e, because ela ionships be ween indi iduals in di e en clus e s would no be aken in o accoun . Ano he ex ension would be o ake he deg ees o eedom as con inuous and use a ejec ion algo i hm o a Me opolis- Has ings walk o d aw samples, combined wi h he Gibbs sample o he es o he pa ame e s o he model. Addi ional andom e ec s, such as pe manen en i onmen al e ec s a ec ing all eco ds o a cow, can be inco po a ed, e.g. by aking a uni a ia e -dis ibu ion as p io . Residuals can be clus e ed in di e en manne s. Fo example, clus e ing e o s by si e o ull-sib amilies may cope wi h inadequa e gene ic assump ions, e.g. unknown majo genes may be seg ega ing. In addi ion, i is possible o allow o he e ogeneous a iance in he model wi hou majo di icul y. O he esidual dis ibu ions such as he logis ic o he slash may be conside ed as well. A p esen , i is no ye possible o apply hese me hods o he la ge da a se s used o ou ine gene ic e alua ion in he dai y ca le b eeding indus y, whe e he models can ha e millions o indi idual b eeding alues. Hence, i i is es ablished ha models based on he -dis ibu ion imp o e gene ic e alua ions, compu a ionally simple o as e me hods should be de eloped. One possibili y would be o employ Laplacian app oxima ions o assess he mode o he join dis ibu ion o he dispe sion pa ame e s and hen use some o m o condi ional analysis o ob ain poin p edic o s o b eeding alues. In a model wi h - dis ibu ed andom e ec s, modal es ima es o ixed and andom e ec s, gi en he scale pa ame e s and he deg ees o eedom can be ound using an i e a i e p ocedu e [35]. This equi es sol ing eweigh ed mixed model equa ions se e al imes, as in h eshold models. App oxima e solu ions a e a couple o i e a ions may be adequa e o p ac ical pu poses. This Laplacian-i e a ion app oach would be coun e pa o he s anda d REML-BLUP analysis in linea models unde Gaussian assump ions. Finally, we would like o obse e some in e p e a i e di e ences be ween he Gaussian and he -model om a quan i a i e gene ic poin o iew. He i abili y is he eg ession o geno ype on pheno ype, which is o’!/(<!+o’!) in a Gaussian model. In he -models discussed, his eg ession is wi h he nume a o (and he app op ia e pa o he denomina o ) being equal o o,2i gene ic alues a e assumed o be Gaussian, ins ead o -dis ibu ed, as was he case in ou analysis. The pos e io dis ibu ion o he i abili y can be es ima ed by o ming a sample alue o h2 om he co esponding d aws o ol 2, !, o 2and e. Fo ou simula ion, pos e io mean es ima es o he i abili y we e 0.47 and 0.45 o he Gaussian and he d-clus e ed -models, espec i ely, and 0.19 o he uni a ia e -model, his being closes o he inpu alue. REFERENCES [1] Albe J.H., Chib S., Bayesian analysis o bina y and polycho omous esponse da a, J. Am. S a . Assoc. 88 (1993) 669-679. [2] Besag J., G een P., Higdon D., Menge sen K., Bayesian compu a ion and s ochas ic sys ems, S a . Sci. 10 (1995) 3-66. [3] Box G.E.P., Tiao G.C., Bayesian In e ence in S a is ical Analysis, Wiley, New Yo k, 1973. [4] Bulme M.G., The Ma hema ical Theo y o Quan i a i e Gene ics, Cla endon P ess, Ox o d, 1980. [5] Casella G., Geo ge E.I., Explaining he Gibbs sample , Am. S a . 46 (1992) 167-174. [6] Falcone D.S., McKay T.F.C., An In oduc ion o Quan i a i e Gene ics, 4 h ed., Longman, New Yo k, 1996. [7] Geweke J., Bayesian ea men o he independen S uden - linea model, J. Appl. Econome ics 8 (1993) S19-S40. [8] Gianola D., Fe nando R.L., Bayesian me hods in animal b eeding heo y, J. Anim. Sci. 63 (1986) 217-244. [9] Gianola D., Foulley J.L., Va iance es ima ion om in eg a ed likelihoods, Gene . Sel. E ol. 22 (1990) 403-417. [10] Gianola D., Foulley J.L., Fe nando R.L., P edic ion o b eeding alues when a iances a e no known, Gene . Sel. E ol. 18 (1986) 485-498. [11] Gianola D., Foulley J.L., Fe nando R.L., Hende son C.R., Weigel K.A., Es- ima ion o he e ogeneous a iances using empi ical Bayes me hods: heo e ical con- side a ions, J. Dai y Sci. 75 (1992) 2805-2823. [12] Go ine B., Selec ion on selec ed eco ds, Gene . Sel. E ol. 15 (1993) 91-97. [13] Ha ille D.A., Bayesian in e ence o a iance componen s using only e o con as s, Biome ika 61 (1974) 383-385. [14] Ha ille D.A., Maximum likelihood app oaches o a iance componen es i- ma ion and o ela ed p oblems, J. Am. S a . Assoc. 72 (1977) 320-340. [15] Ha ille D.A., BLUP (Bes Linea Unbiased P edic ion) and beyond, in: Gianola D., Hammond K. (Eds.), Ad ances in S a is ical Me hods o Gene ic Imp o emen o Li es ock, Sp inge -Ve lag, Be lin, 1990, pp. 239-276. [16] Hazel L.N., The gene ic basis o cons uc ing selec ion indexes, Gene ics 28 (1943) 476-490. [17] Hende son C.R., Speci ic and gene al combining abili y, in: Gowan J.W. (Ed.), He e osis, Iowa S a e College P ess, Ames, IA, 1950, pp. 352-370. [18] Hende son C.R., Es ima ion o a iance componen s and co a iance compo- nen s, Biome ics 9 (1953) 226-252. [19] Hende son C.R., Si e e alua ion and gene ic ends, in: P oceedings o he Animal B eeding and Gene ics Symposium in hono o D J.L. Lush, Blacksbu g, VA, Augus 1973, Ame ican Socie y o Animal Science, Champaign, IL, 1973, pp. 10-41. [20] Hende son C.R., Bes linea unbiased es ima ion and p edic ion unde a selec ion model, Biome ics 31 (1975) 423-447. [21] Hende son C.R., Applica ions o Linea Models in Animal B eeding, Uni e - si y o Guelph P ess, Guelph, 1984. [22] Hobe J.P., Casella G., The e ec o imp ope p io s on Gibbs sampling in hie a chical linea models, J. Am. S a . Assoc. 91 (1996) 1461-1473. [23] Kuhn M.T., F eeman A.E., Biases in p edic ed ansmi ing abili ies o si es when daugh e s ecei e p e e en ial ea men , J. Dai y Sci. 78 (1995) 2067-2072. [24] Kuhn M.T., Boe che P.J., F eeman A.E., Po en ial biases in p edic ed ansmi ing abili ies o emales om p e e en ial ea men , J. Dai y Sci. 77 (1994) 2428-2437. [25] Lai d N.M., Wa e J.H., Random e ec s models o longi udinal da a, Bio- me ics 38 (1982) 963-974. [26] Lange K.L., Li le R.J.A, Taylo J.M.G., Robus s a is ical modeling using he dis ibu ion, J. Am. S a . Assoc. 84 (1989) 881-896. [27] Lynch M., Walsh B., Gene ics and Analysis o Quan i a i e T ai s, Sinaue Associa es, Sunde land, MA, 1997. [28] Meuwissen T.H.E., Godda d M., Selec ion o a m animals o non-linea ai s and p o i s, Anim. Sci. 65 (1997) 1-8. [29] Pa e son H.D., Thompson R., Reco e y o in e block in o ma ion when block sizes a e unequal, Biome ika 58 (1971) 545-554. [30] Pinhei o J.C., Liu C., Wu Y., Robus es ima ion in linea mixed e ec s models using he mul i a ia e- dis ibu ion, Bell Labs Technical Repo , 1997, 04/97. [31] San C is obal M., Foulley J.L., Man edi E., In e ence abou mul iplica i e he e oscedas ic componen s o a iance in a mixed linea Gaussian model wi h an applica ion o bee ca le b eeding, Gene . Sel. E ol. 25 (1993) 3-30. [32] Sea le S.R., Casella G., McCulloch C., Va iance Componen s, John Wiley, New Yo k, 1992. [33] Smi h H.F., A disc iminan unc ion o plan selec ion, Ann. Eugenics 7 (1936) 240-250. [34] So ensen D.A., Wang C.S., Jensen J., Gianola D., Bayesian analysis o ge- ne ic change due o selec ion using Gibbs sampling, Gene . Sel. E ol. 26 (1994) 333- 360. [35] S anden I., Robus mixed e ec s linea models wi h -dis ibu ions and applica ion o dai y ca le b eeding, Ph.D. hesis, Uni e si y o Wisconsin, Madison, WI, 1996. [36] S anden L, Gianola D., A enua ing e ec s o p e e en ial ea men wi h S uden - mixed linea models: a simula ion s udy, Gene . Sel. E ol. 30 (1998) 565- 583. [37] Su adha B.C., Ali M.M., Es ima ion o he pa ame e s o a eg ession model wi h a mul i a ia e e o a iable, Comm. S a . Theo y Me h. 15 (1986) 429-450. [38] Ve dinelli I., Wasse man L., Bayesian analysis o ou lie p oblems using he Gibbs sample , S a . Compu . 1 (1991) 105-117. [39] Wang C.S., Ru ledge J.J., Gianola D., Ma ginal in e ences abou a iance componen s in a mixed linea model using Gibbs sampling, Gene . Sel. E ol. 25 (1993) 41-62. [40] Wang C.S., Ru ledge J.J., Gianola D., Bayesian analysis o mixed linea models ia Gibbs sampling wi h an applica ion o li e size in Ibe ian pigs, Gene . Sel. E ol. 26 (1994) 91-115. [41] Wes M., Ou lie models and p io dis ibu ions in Bayesian linea eg ession, J. R. S a . Soc. B Me . 46 (1984) 431-439. [42] Zellne A., Bayesian and non-Bayesian analysis o he eg ession model wi h mul i a ia e S uden - e o e ms, J. Am. S a . Assoc. 71 (1976) 400-405.