Principal component and factor analytic models in international sire evaluation

RESEARCH Open Access P incipal componen and ac o analy ic models in in e na ional si e e alua ion Anna-Ma ia Ty ise ä 1* , Ka in Meye 2 , W F eddy Fikse 3 , Vincen Duc ocq 4 , Je e Jakobsen 5 , Ma in H Lidaue 1 and Esa A Män ysaa i 1 Abs ac Backg ound: In e bull is a non-p o i o ganiza ion ha p o ides in e na ionally compa able b eeding alues o globalized dai y ca le b eeding p og ammes. Due o di e en ai de ini ions and models o gene ic e alua ion be ween coun ies, each biological ai is ea ed as a di e en ai in each o he pa icipa ing coun ies. This yields a gene ic co a iance ma ix o dimension equal o he numbe o coun ies which ypically in ol es high gene ic co ela ions be ween coun ies. This gi es ise o se e al p oblems such as o e -pa ame e ized models and inc eased sampling a iances, i gene ic (co) a iance ma ices a e conside ed o be uns uc u ed. Me hods: P incipal componen (PC) and ac o analy ic (FA) models allow highly pa simonious ep esen a ions o he (co) a iance ma ix compa ed o he s anda d mul i- ai model and ha e, he e o e, a ac ed conside able in e es o hei po en ial o ease he bu den o he es ima ion p ocess o mul iple- ai ac oss coun y e alua ion (MACE). This s udy e alua ed he u ili y o PC and FA models o es ima e a iance componen s and o p edic b eeding alues o MACE o p o ein yield. This was es ed using a da ase comp ising Hols ein bull e alua ions ob ained in 2007 om 25 coun ies. Resul s: In o al, 19 p incipal componen s o nine ac o s we e needed o explain he gene ic a ia ion in he es da ase . Es ima es o he gene ic pa ame e s unde he op imal i we e almos iden ical o he wo app oaches. Fu he mo e, he esul s we e in a good ag eemen wi h hose ob ained om he ull ank model and wi h hose p o ided by In e bull. The es ima ion ime was sho es o models i ing he op imal numbe o pa ame e s and p olonged when unde - o o e -pa ame e ized models we e applied. Co ela ions be ween es ima ed b eeding alues (EBV) om he PC19 and PC25 we e uni y. Wi h ew excep ions, co ela ions be ween EBV ob ained using FA and PC app oaches unde he op imal i we e ≥0.99. Fo bo h app oaches, EBV co ela ions dec eased when he op imal model and models i ing oo ew pa ame e s we e compa ed. Conclusions: Gene ic pa ame e s om he PC and FA app oaches we e e y simila when he op imal numbe o p incipal componen s o ac o s was i ed. O e - i ing inc eased es ima ion ime and s anda d e o s o he es ima es bu did no a ec he es ima es o gene ic co ela ions o he p edic ions o b eeding alues, whe eas i ing oo ew pa ame e s a ec ed bull ankings in di e en coun ies. Backg ound Ac i e in e na ional ade o semen and emb yos o dai y ca le has c ea ed a need o global compa isons o gene ic me i o si es. The In e na ional Bull E alua ion Se ice, In e bull, was es ablished in 1983 o espond o his need. In e na ional b eeding alues o dai y bulls a e cu en ly es ima ed h ee imes a yea and hey a e exp essed in he uni s o each membe coun ies and a e ela i e o each coun y’s own base g oup o animals [1]. In o de o accu a ely pe o m he e alua ions, eli- able gene ic pa ame e s, i.e., a iance componen s and gene ic co ela ions, a e equi ed. Daugh e g oups in di e en coun ies a e assumed o be gene ically co ela ed bu en i onmen ally unco e- la ed. The e o e, each biological ai unde e alua ion is ea ed as a di e en ai o each coun y pa icipa ing in he in e na ional si e e alua ion. Typically, some coun ies a e e y highly co ela ed. The mul i- * Co espondence: [email p o ec ed]i 1 Bio echnology and Food Resea ch, Biome ical Gene ics, MTT Ag i ood Resea ch Finland,31600 Jokioinen, Finland Full lis o au ho in o ma ion is a ailable a he end o he a icle Ty ise ä e al.Gene ics Selec ion E olu ion 2011, 43:33 h p://www.gsejou nal.o g/con en /43/1/33 Gene ics Selec ion E olu ion © 2011 Ty ise ä e al; licensee BioMed Cen al L d. This is an Open Access a icle dis ibu ed unde he e ms o he C ea i e Commons A ibu ion License (h p://c ea i ecommons.o g/licenses/by/2.0), which pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly ci ed. dimensionali y and high gene ic co ela ions c ea e se - e al p oblems such as o e -pa ame e ized models, inc eased sampling a iances and an inc eased p obabil- i yo pa ame e s obeou side hebounda ieso he pa ame e space, e.g. [2]. Fo es ic ed maximum likeli- hood (REML) es ima ion, hese, in u n, complica e maximiza ion o he likelihood and hus, exace ba e he ime needed o es ima e a iance componen s. The numbe o coun ies pa icipa ing in he in e na ional Hols ein si e e alua ion o p o ein yield in 2011 is 28. This equi es es ima ion o a 28 × 28 (co) a iance (VCV) ma ix desc ibed by 406 pa ame e s, i he gene ic (co) a iance ma ix is conside ed o be uns uc- u ed. The cu en p ac ice is o es ima e his ma ix by pe o ming a numbe o sepa a e analyses conside ing selec ed sub-se s o coun ies [3,4]. The esul ing es i- ma es a e hen combined o build up he comple e VCV ma ix. Typically, his esul s in a non-posi i e de ini e ma ix and a “bending”p ocedu e is applied o ensu e ha he o e all ma ix is alid [5]. P incipal componen (PC) and ac o analy ic (FA) models p o ide a highly pa simonious s uc u e o he VCV ma ix compa ed o he s anda d mul i- ai model, e.g. [6,7] and hey ha e, he e o e, a ac ed con- side able in e es o hei po en ial o ease he bu den o he es ima ion p ocess o mul iple- ai ac oss coun- y e alua ions (MACE) [8]. Bo h app oaches decompose he gene ic co a iance ma ix in o pe aining ma ices o eigen alues and eigen ec o s. Each eigen ec o , i.e., PC, o ms a linea combina ion o he ai s, while he co e- sponding eigen alue gi es he a iance explained. PC a e independen o each o he . The aim o he PC me hod is o de ec all necessa y componen s explaining a ia ion in mul i-dimensional da a wi hou loosing any impo an in o ma ion. The i s PC explains he maximum amoun o gene ic a ia- bili y in he da a and each successi e PC explains he maximum amoun o he emaining a iabili y. Fo highly co ela ed ai s, only he leading PC ha e p ac i- cal in luence on gene ic a ia ion and PC wi h a negligi- ble e ec can be omi ed wi hou impai ing accu acy o es ima ion. Fu he mo e, he pa ame e educ ion esul s in a ank educ ion and in a educ ion o he dimension o he mixed model equa ions. The FA me hod is ela ed o he PC me hod bu i s app oach is di e en . The ai s s udied a e assumed o be linea combina ions o a ew la en a iables, e e ed o as common ac o s. Any a iance no explained by hese is modelled sepa a ely, i.e. as ai -speci ic, by i ing co esponding speci ic ac o s. Due o he pa i ioning o a iance in o common and ai -speci ic a iance, he numbe o ac o s needed o explain he a iabili y in he da a is no mally no ably smalle han he numbe o PC needed in he PC app oach. Fu he , since he ac o s a e assumed o be unco ela ed, subs an ial spa si y o he mixed model equa ion (MME) is gained compa ed o he s anda d uns uc u ed mul i a ia e analysis. Howe e , he esul ing (co) a iance ma ix is o ull ank i all ai - speci ic a iances a e non-ze o. Fu he mo e, ac o axes can be o a ed. No mally, his is done o ease hei in e - p e a ion, bu i also makes i possible o use he Cho- lesky pa ame e iza ion ha enhances he con e gence a e o maximum likelihood es ima ion, e.g. [7,9]. Madsen e al. [10] we e he i s o sugges he use o educed ank co a iance ma ices o MACE. Ins ead o using s anda d expec a ion-maximiza ion algo i hm o REML es ima ion o a iance componen s o MACE, hey s udied he easibili y o exploi ing an a e age- in o ma ion (AI) algo i hm ha is known o be as and e ec i e. They de eloped an AI-REML algo i hm, which e alua es o each ound, whe he o no he VCV ma ix is posi i e de ini e. I a non-posi i e de ini e ma ix is encoun e ed, he o iginal VCV ma ix is decomposed and all eigen alues less han he ope a- ional ze o a e eplaced wi h a small posi i e numbe . Thus, hei me hod is no a eal educed ank me hod in he sense ha small o nega i e eigen alues would ha e been emo ed. In u n, Lecle c e al. [11] s udied bo h PC and FA app oaches o a sub-se o well-linked base coun ies, pe o ming dimension educ ion o his sub-se and hen es ima ing gene ic co ela ions be ween he emaining and he base coun ies, keeping he gene ic co ela ions among he base coun ies ixed. When applying he app oach p oposed by Lecle c e al. [11], special emphasis should be placed on selec ion o sui able base coun ies. Män ysaa i [12] in oduced a bo om-up PC app oach ha begins wi h a sub-se o coun ies and adds he emaining coun ies sequen ially. By examining in each s ep whe he o no he new coun y inc eases he ank o he gene ic VCV ma ix, he bo om-up app oach only i s PC wi h non-negligible eigen alues and hus a oids o e -pa ame e ized models. While his o iginal s udy was pe o med wi h a simula ed da ase , ecen wo k has demons a ed he use ulness o his app oach o es ima e he a iance componen s o MACE [13,14]. Typically, he con en ional PC analysis is done a e he comple e VCV ma ix has been es ima ed. Then, he ma ix is decomposed and i possible, i s dimension is educed. Ki kpa ick and Meye [15] sugges ed he di ec es ima ion o he leading p incipal componen s (di ec PC). Howe e , his equi es he app op ia e ank o be known o o be es ima ed p io o he a iance componen analysis. Simila ly, a VCV ma ix imposing a FA s uc u e can be es ima ed di ec ly [6]. Howe e , a oo s ingen pa ame e educ ion should be a oided since selec ing oo low a ank can lead o biased es i- ma es o gene ic pa ame e s [14,15]. This is, because he Ty ise ä e al.Gene ics Selec ion E olu ion 2011, 43:33 h p://www.gsejou nal.o g/con en /43/1/33 Page 2 o 10 numbe o a ailable pa ame e s is no longe su icien o desc ibe he (co) a iance s uc u e o he model ade- qua ely, and pa o he gene ic a iance will be e-pa - ioned in o he esidual a iance. Fu he mo e, wi h mo e han one ma ix o be es ima ed, he educed ank es ima o can be inconsis en , i.e. pick up he w ong subse o PC [2]. The isk o his happening when ela- i ely ew PC a e conside ed is high. Bo h di ec PC and FA app oaches ha e been applied o bee ca le da ase s and ha e demons a ed hei po en ial obeused o la ge,mul i- ai da ase s,e.g.[16,17].In addi ion, he di ec PC app oach p o ed o be an appeal- ing me hod o es ima e a iance componen s o MACE in a ecen s udy [14]. The objec i es o his s udy a e o e alua e he u ili y o he ac o analy ic app oach o a - iance componen es ima ion o MACE and o assess he impac o al e na i e pa ame e iza ions, bo h PC and FA, o p ac ical p edic ion o b eeding alues wi h MACE. Me hods Da ase P o ein yield da a om he Augus 2007 In e bull Hol- s ein e alua ion we e used. A si e model wi h si e- ma e nal g andsi e pedig ee o 106 003 indi iduals was employed. The da ase comp ised 116 941 de- eg essed b eeding alues om 25 coun ies [18]. The numbe o bulls pe coun y a ied om 145 o 23 380, wi h a mean o 4 678 (Table 1). Bulls we e mainly used in one coun y; only 8% o he bulls (7 621) we e used in mo e han one coun y and 0.3% o he bulls (286) in mo e han 10 coun ies. Common bulls we e de ined as bulls wi h daugh e s in bo h coun ies, wi hou es ic ions on he coun y o o igin. The numbe o common bulls a ied d ama ically be ween coun ies, anging om ze o o 1 194. The numbe o common bulls was smal- les be ween he F ench Red Hols ein and o he coun- ies (min 0, max 73, mean 9) and la ges be ween he USA and o he coun ies (min 6, max 1 044, mean 410). Fo a mo e de ailed desc ip ion o he da a, see [14]. Random eg ession MACE si e model The classical MACE model o he i h si e, deno ed as: yi=Xib+Ziui+εi , (1) and he andom eg ession (RR) MACE model, deno ed as: Table 1 Va iances ± s anda d e o s o p o ein yield om he ac o analysis i ing 9 ac o s and om he PC analysis i ing 19 PC Coun y Numbe o bulls FA9 PC19 Common Coun y speci ic Combined Canada 7 028 113.2 ± 2.2 8.5 ± 1.2 121.7 ± 1.9 121.3 ± 1.9 Ge many 16 734 66.2 ± 1.3 6.0 ± 1.0 72.2 ± 0.8 72.2 ± 0.8 Denma k-Finland-Sweden 8 900 61.9 ± 1.1 4.1 ± 0.7 66.0 ± 1.0 66.0 ± 1.0 F ance 11 127 76.9 ± 1.5 7.4 ± 1.0 84.3 ± 1.2 84.4 ± 1.2 I aly 6 322 81.6 ± 1.8 4.5 ± 1.1 86.1 ± 1.4 86.1 ± 1.4 The Ne he lands 9 696 73.4 ± 1.4 5.6 ± 0.9 79.0 ± 1.1 78.9 ± 1.1 USA 23 380 315.3 ± 4.6 15.9 ± 3.1 331.2 ± 3.4 331.1 ± 3.4 Swi ze land 715 51.3 ± 2.0 0.0 ± 0.0 51.3 ± 2.0 51.9 ± 2.0 G ea B i ain 4 361 54.9 ± 1.1 0.0 ± 0.0 54.9 ± 1.1 54.9 ± 1.1 New-Zealand 4 253 21.6 ± 0.5 0.0 ± 0.0 21.6 ± 0.5 21.6 ± 0.5 Aus alia 4 950 20.6 ± 0.8 4.9 ± 0.6 25.5 ± 0.6 25.6 ± 0.6 Belgium 634 38.2 ± 2.1 4.7 ± 0.9 42.9 ± 2.0 43.0 ± 2.0 I eland 1 260 19.5 ± 0.9 1.4 ± 0.5 20.9 ± 0.7 20.9 ± 0.7 Spain 1 499 50.0 ± 1.5 3.0 ± 0.5 53.0 ± 1.4 52.8 ± 1.4 Czech Republic 2 036 80.3 ± 2.8 0.0 ± 0.0 80.3 ± 2.8 80.1 ± 2.8 Slo enia 196 7.9 ± 0.8 0.0 ± 0.0 7.9 ± 0.8 8.1 ± 0.9 Es onia 472 55.6 ± 4.3 5.0 ± 2.8 60.6 ± 3.5 61.1 ± 3.5 Is ael 773 76.7 ± 4.1 0.0 ± 0.0 76.7 ± 4.1 76.1 ± 4.1 Swiss Red Hols ein 1 162 46.8 ± 2.1 2.2 ± 1.1 49.0 ± 1.9 48.0 ± 1.8 F ench Red Hols ein 145 76.9 ± 8.6 0.0 ± 0.0 76.9 ± 8.6 80.4 ± 9.1 Hunga y 1 898 64.4 ± 2.4 8.6 ± 1.3 73.0 ± 2.2 72.9 ± 2.2 Poland 5 071 31.4 ± 2.0 0.6 ± 1.8 32.0 ± 0.8 32.0 ± 0.8 Sou h A ica 920 38.3 ± 2.3 0.0 ± 0.0 38.3 ± 2.3 37.8 ± 2.2 Japan 3 177 63.8 ± 1.6 0.0 ± 0.0 63.8 ± 1.6 64.3 ± 1.6 La ia 232 15.7 ± 3.2 7.0 ± 2.8 22.7 ± 2.3 23.1 ± 2.3 Ty ise ä e al.Gene ics Selec ion E olu ion 2011, 43:33 h p://www.gsejou nal.o g/con en /43/1/33 Page 3 o 10 y i =Xib+ZiVνi+εi , (2) a e equi alen bu di e en ly pa ame e ized models. In bo h (1) and (2), y i is a n i ec o o na ional de- eg essed b eeding alues o bull iand bis a ec o o coun y/ ai e ec s. In (1), u i is a ec o o di e en in e na ional b eeding alues o bull iand in (2), ν i is a ec o o eg ession coe icien s o bull i.X i and Z i deno e incidence ma ices assigning obse a ions o espec i e e ec s. Decomposing he × gene ic co( a - iance) ma ix o si e e ec s, Va (u i )=Gin o G=VDV T wi h Da ma ix o eigen alues and V he co esponding ma ix o eigen ec o s, gi es Va (ν i )=D. In (1) and (2), ε i is a n i ec o o esiduals wi h Va (ε i )=diag(g jj l j / EDC ij ), whe e g jj is he si e a iance, λj=(4−h2 j )/h 2 j wi h h 2 j he he i abili y o coun y jand EDC ij he e ec- i e daugh e con ibu ion o bull iin coun y j.In(2), he b eeding alues o bull iha e o be back- ans- o med o ge u i =Vν i . Fo he es ima ion o a iance componen s, we did no g oup animals wi h unknown pa en age in o gene ic g oups, bu o p edic ion o he b eeding alues, gene ic g oups we e used. PC app oach The RR MACE model acili a es pa ame e educ ion, when Ghas eigen alues close o ze o. Then, he p inci- pal componen s wi h he smalles eigen alues can be omi ed wi hou impai ing he accu acy o es ima ion. In ha case, Gcan be desc ibed as G1=V1D1V T 1 ,whe e D 1 is × and con ains he leading eigen alues and V 1 is he × ma ix o he co esponding eigen ec- o s, wi h < .Now, he andom eg ession coe i- cien s, ν ∗ i , a e p edic ed o each bull and he b eeding alues can be back- ans o med: u i∼ =V1D1ν ∗ i . FA app oach Fo he FA app oach, u i is di ided in o ec o s o com- mon ac o s, δ i ,wi hVa (δ i )=I,andcoun yspeci ic e ec s, τ i ,wi hVa (τi)=F=diag{σ2 τi j} .Thisgi esu i = Lδ i +τ i ,wi hLdeno ing he ma ix o ac o loadings [19]. The FA ep esen a ion o he MACE model is exp essed as: yi=Xib+Zi(Lδi+τi)+ε i (3) The FA app oach models Gas he sum o wo e ms: he common (co) a iances and he ai -speci ic a - iances, i.e., G=LL T +F. The numbe o pa ame e s can no exceed ( +1)/2, hus < ac o s explain he com- mon co a iances, e.g. [7]. I all coun y-speci ic a - iances a e non-ze o, he esul ing model will no be o educed ank, bu is desc ibed e y pa simoniously wi h p= + - ( - 1)/2 [7]. Models Con a y o he cu en p ac ice o da a sub-se ing o MACE a iance componen analysis o p o ein yield in Hols ein [3], all he da a in his s udy was included in a single VCV analysis o each model in es iga ed. Fo he PC app oach, es ima es o G om se e al i s we e ob ained om a p e ious s udy [14]. The app op ia e i was chosen by pe o ming se e al analyses encompass- ing a i s , in o med guess o he co ec ank, which was ob ained by decomposing he (co) a iance ma ix p o ided by In e bull and by s udying he magni ude o he eigen alues. Nex , we examined Akaike’sin o ma- ion c i e ion (AIC), log L and beha iou o he PC om analyses using successi e numbe s o PC o de e - mine he app op ia e ank. Fo he model wi h an op i- mal i , AIC should each i s minimum alue and he inc ease o he Log Likelihood beyond he op imal i is expec ed o be ma ginal. Fu he mo e, he magni ude o he leading PC and he sum o he eigen alues should be s abilized, i.e. no change alue as he numbe o PC i ed is inc eased. I his we e no he case, i would be an indica ion ha he e was s ill no able e-pa ioning o he gene ic a iance in o he esidual a iance, i.e. ha oo ew PC had been i ed [2,16,17]. Fo he di ec PC app oach, ank 19 (PC19) was selec ed as bes [13,14]. Fo compa ison, analyses we e also ca ied ou using oo low a ank (PC15) and ull ank (PC25). Fo he FA app oach, successi e analyses i ing om se en o 12 ac o s we e ca ied ou and he bes model was chosen ollowing he same p inciples as o he PC app oach. A model i ing nine ac o s (FA9) was chosen as bes and esul s om he model i ing oo ew ac- o s (FA7) a e p esen ed o compa ison. In addi ion, √  alues, de ined as he squa e oo o he a e age squa ed de ia ion o he es ima ed gene ic co ela ions [17,14], we e calcula ed o indica e he di e ences in he es ima es o he gene ic co ela ions be ween each es ed i and he e e ence model (PC19) o compa ison. √ =    2  i=1  j=i+1 ( ij,m− ij,19)2 ×( −1) , (4) whe e is he numbe o ai s, ij,m is he es ima ed gene ic co ela ion be ween ai s iand j om an analy- sis i ing m ac o s. Es ima ed ac o s o he FA9 model we e o a ed o ease hei in e p e a ion. The pu pose o o a ing ac o s is o load a iables as unambiguously as possible o he ac o s. In ha case, each ac o has only a small g oup o a iables wi h s ong loadings. Ro a ions can be clas- si ied in o wo g oups: o hogonal and oblique o a ions. In o hogonal o a ion, ac o s do no co ela e wi h each o he : Ty ise ä e al.Gene ics Selec ion E olu ion 2011, 43:33 h p://www.gsejou nal.o g/con en /43/1/33 Page 4 o 10 G=LTTTLT+F=LLT+F , (5) whe e Tis an o hogonal ans o ma ion ma ix. In oblique o a ion, axes do no emain pe pendicula . In his case, he o a ion uses a gene al non-singula ans- o ma ion ma ix ins ead o he o hogonal ans o ma- ion ma ix [19]. By allowing ac o s o co ela e wi h each o he , i is easie o clus e a iables and simpli y hei in e p e a ion. In his s udy, we pe o med he oblique p omax o a ion. Calcula ions we e ca ied ou using he R S a s package [20]. A e o a ion, he ma ix was so ed and he smalles loadings (wi h a cu -o o 0.2) we e hidden o u he ease in e p e a ion. The numbe o pa ame e s was 271, 305, 326, 180 and 215 o PC15, PC19, PC25, FA7 and FA9, espec i ely. Va iance componen s we e es ima ed by es ic ed max- imum likelihood, using an a e age in o ma ion algo- i hm as implemen ed in WOMBAT [21]. The gene ic co ela ions ob ained om he In e bull es un p eced- ing Augus 2007 e alua ion we e used o compa ison. Analysis o es ima ed b eeding alues Consequences o applying he ob ained a iance compo- nen s o he mo e pa simonious PC and FA models o he p ac ical p edic ion o b eeding alues wi h MACE we e s udied by moni o ing he co ela ions be ween es ima ed b eeding alues (EBV) om he di e en PC and FA models. Fo his, EBV we e p edic ed unde he ollowing models: PC25, which is equal o he classical MACE model, PC and FA models wi h he op imal i (PC19 and FA9) and PC and FA models wi h oo low a i (PC15 and FA7). Fu he mo e, co ela ions be ween EBV om PC15 and PC19, om FA7 and FA9, and om PC19 and FA9 o each coun y we e conside ed o ou subg oups: A) bulls used only in hei own coun y, B) bulls used in hei own coun y and ab oad, C) bulls used only ab oad, and D) impo ed bulls. B eeding alues we e ob ained using a p econdi ioned conjuga ed g adien i e a ion on da a algo i hm as implemen ed in MiX99 [22]. Resul s and Discussion Selec ion o he FA model The in o ma ion used o model selec ion o he FA app oach is collec ed in Table 2. Based on AIC, i ing 9 ac o s was bes , al hough he di e ence be ween FA9 and FA10 was e y small. Mean alues o he gene ic co ela ions om he di e en i s we e p ac ically iden- ical, al hough he e we e some di e ences in he dis i- bu ions o he es ima es om he di e en i s. In e es ingly, based on he √  alues, gene ic co ela- ions om FA12 we e closes o he es ima es om he di ec PC analysis unde he op imal ank 19, bu no o he gene ic co ela ions om he op imal i (FA9). Inspec ion o he sum o he eigen alues de i ed om he a iance due o common ac o s (Table 2) and he coun y-speci ic a iances om he di e en i s e ealed ha some e-pa ioning o he gene ic a iance occu ed wi h dec easing i . Pa o he a iance due o common ac o s was mo ed in o he coun y-speci ic a iance. As a consequence, he numbe o coun ies wi h ze o coun- y-speci ic a iance dec eased om 13 ( i 12) o i e ( i 7) and he sum o he coun y-speci ic a iances inc easedby57%.Excep o he i s eigen alue, he dis ibu ion o he a iance due o common ac o s o he indi idual eigen alues emained, howe e , qui e con- s an be ween he i s. The i s eigh eigen ec o s om he op imal i (FA9) and he wo b acke ing i s, i.e. FA8 and FA10, a e shown in Figu e 1. As migh be expec ed om he almos iden ical AIC alues o FA9 and FA10, all hei eigen ec o s we e i ually iden ical. Howe e , eigen ec- o s om analyses i ing eigh ac o s s a ed o de ia e om hose i ing nine and 10 ac o s om he second eigen ec o onwa ds, wi h a subs an ial de ia ion o he eigh h eigen ec o . The pa e n o he eigen ec o s om FA7 de ia ed e en mo e om hose o he op imal i Table 2 Cha ac e is ics o he analyses i ing om se en o 12 ac o s Fi 7 Fi 8 Fi 9 Fi 10 Fi 11 Fi 12 -1/2 AIC a -11 -17 0 -1 -8 -12 Log L b -79 -67 -33 -18 -10 0 No o pa ame e s 180 198 215 231 246 260 Sum o eigen alues c 1541 1585 1602 1608 1619 1631 E1 d 85.9 83.1 82.6 82.3 82.0 81.3 E2 4.7 4.9 4.8 4.7 4.3 4.4 E3 3.4 4.5 3.8 3.8 3.9 3.8 E4 2.4 2.6 2.9 2.9 2.7 2.8 E5 1.5 1.7 1.9 1.9 1.9 1.9 E6 1.2 1.4 1.5 1.5 1.7 1.6 E7 0.9 1.1 1.1 1.1 1.1 1.2 E8 0.7 0.8 0.9 0.8 0.9 E9 0.7 0.7 0.7 0.7 E10 0.4 0.6 0.6 E11 0.3 0.4 E12 0.3 g , min e 0.16 0.05 0.13 0.12 0.07 0.06 g , max e 0.93 0.94 0.94 0.94 0.94 0.94 g , mean e 0.69 0.69 0.69 0.69 0.69 0.68 √  0.039 0.038 0.023 0.022 0.019 0.017 a Akaike’s in o ma ion c i e ion, exp essed as de ia ion om highes alue b Maximum Log Likelihood, exp essed as de ia ion om highes alue c De i ed om he a iance due o common ac o s d Eigen alues 1 o 12 o LL T , exp essed as p opo ion (in %) o o al e Gene ic co ela ions: minimum, maximum and mean alues Squa e oo o he a e age squa ed de ia ion o he gene ic co ela ions. The es ima es ob ained unde he di ec PC ank 19 model we e used as he es ima es o compa ison. Ty ise ä e al.Gene ics Selec ion E olu ion 2011, 43:33 h p://www.gsejou nal.o g/con en /43/1/33 Page 5 o 10 ( esul s no shown), indica ing ha i ing oo ew ac- o s was associa ed wi h inaccu a e es ima ion o he di ec ions o he PC. In addi ion, o e i ing PC had ha dly any in luence on es ima ion o he di ec ions. This was no only he case o FA10, bu also o FA11 and FA12 ( esul s no shown). Resul s also indica ed ha he las eigen ec o s o all es ed i s we e inaccu- a ely es ima ed since hei pa e n de ia ed no ably om he pa e ns o he eigen ec o s o he models i - ing mo e PC. Based on he simula ion s udies by Ki k- pa ick and Meye [15] and Meye [16], inaccu a e es ima ion o he las eigen ec o s was caused by la ge sampling a iances. Howe e , his is o mino p ac ical impo ance since he magni ude o he las eigen alues is negligible compa ed o ha o he leading eigen alues, i.e. he las p incipal componen s con ibu e li le o he es ima e o he gene ic co a iance ma ix (Table 2). The ma ix o o a ed ac o loadings is gi en in Table 3. E en wi h he o a ion, hei in e p e a ion was no easy. In mos cases, he possible in e p e a ion seemed o be connec ed wi h he ac i e ade o bulls be ween some coun ies and hus, wi h he s ong gene ic links c ea ed be ween hem, see, e.g. [23]. Is ael, Sou h A ica and Japan impo bulls p edominan ly om he USA, whe eas he F ench Red popula ion has only ew links wi h he USA ( ac o 3). Fu he mo e, he highes p o- po ion o impo ed bulls in Es onia, Poland and La ia comes om Ge many ( ac o 5). USA, F ance, I aly, Spain and Hunga y ha e, in u n, s ong links among o he s, mainly due o he ade o bulls om USA ( ac- o 7), whe eas he Ne he lands is a popula ading pa ne wi h coun ies like Ge many, Denma k, Finland, Sweden, Belgium and I eland ( ac o 8). New-Zealand, Aus alia and I eland we e posi i ely weigh ed coun ies in ac o 9. The common ea u e o hese is ha hey all a e g azing coun ies. Va iances and gene ic co ela ions Es ima es o gene ic a iances om FA9, PC19 and PC25 we e almos iden ical (FA9 and PC19 in Table 1), excep o some di e ences be ween app oaches o F ench Red Hols ein (PC19: 80.4 ± 9.06, PC25: 80.6 ± 9.16, FA9: 76.9 ± 8.60). The di e ences in es ima es and hei high s anda d e o s can be a ibu ed o he low numbe o he bulls (145) in his popula ion (Table 1). Fo FA9, he e was subs an ial a ia ion in he amoun o he coun y speci ic a iance. On a e age, he p opo - ion o he o al gene ic a iance a ibu ed o coun y speci ic e ec s was 5%, wi h he highes p opo ions o Aus alia (19%) and La ia (31%). Unde he op imal i ●●●●●●●●●●●●●●●● ● ● ●●●●● ● ● ●●●●●●●●● ●●●●● ●● ●●●●●● ● ● ● CAN DEU DNK FRA ITA NLD USA CHE GBR NZL AUS BEL IRL ESP CZE SVN EST ISR CHR FRR HUN POL ZAF JPN LVA −0.1 −0.05 0 0.05 0.1 ● ● i 8 i 9 i 10 Fi s ●●●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ●●●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● CAN DEU DNK FRA ITA NLD USA CHE GBR NZL AUS BEL IRL ESP CZE SVN EST ISR CHR FRR HUN POL ZAF JPN LVA Second ● ●●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● CAN DEU DNK FRA ITA NLD USA CHE GBR NZL AUS BEL IRL ESP CZE SVN EST ISR CHR FRR HUN POL ZAF JPN LVA Thi d ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● CAN DEU DNK FRA ITA NLD USA CHE GBR NZL AUS BEL IRL ESP CZE SVN EST ISR CHR FRR HUN POL ZAF JPN LVA Fou h ●● ● ● ● ●● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● CAN DEU DNK FRA ITA NLD USA CHE G BR NZL AUS BEL IRL ESP CZE SVN EST ISR C HR FRR HUN POL ZAF JPN LVA −0.1 −0.05 0 0.05 0.1 Fi h ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● CAN DEU DNK FRA ITA NLD USA CHE G BR NZL AUS BEL IRL ESP CZE SVN EST ISR C HR FRR HUN POL ZAF JPN LVA Six h ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● CAN DEU DNK FRA ITA NLD USA CHE G BR NZL AUS BEL IRL ESP CZE SVN EST ISR C HR FRR HUN POL ZAF JPN LVA Se en h ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● CAN DEU DNK FRA ITA NLD USA CHE G BR NZL AUS BEL IRL ESP CZE SVN EST ISR C HR FRR HUN POL ZAF JPN LVA Eigh h Figu e 1 Fi s eigh s anda dized eigen ec o s om ac o analysis unde i s 8, 9 and 10. Coun y codes: Canada (CAN), Ge many (DEU), Denma k-Finland-Sweden (DFS), F ance (FRA), I aly (ITA), The Ne he lands (NLD), Uni ed S a es o Ame ica (USA), Swi ze land (CHE), G ea B i ain (GBR), New Zealand (NZL), Aus alia (AUS), Belgia (BEL), I eland (IRL), Spain (ESP), Czech Republic (CZE), Slo enia (SVN), Es onia (EST), Is ael (ISR), Swiss Red Hols ein (CHR), F ench Red Hols ein (FRR), Hunga y (HUN), Poland (POL), Sou h A ica (ZAF), Japan (JPN), La ia (LVA). Ty ise ä e al.Gene ics Selec ion E olu ion 2011, 43:33 h p://www.gsejou nal.o g/con en /43/1/33 Page 6 o 10 (FA9), in nine o he 25 coun ies/popula ions he gene ic a iance was o ally explained by he common a iance. These coun ies/popula ions we e Swi ze land, G ea B i ain, New Zealand, Czech Republic, Slo enia, Is ael, F ench Red Hols ein, Sou h A ica and Japan. As shown in Figu e 2, es ima es o he gene ic co e- la ions o he FA and he di ec PC app oaches unde he op imal i we e in good acco dance. Fu he , In e bull and he di ec PC ull ank es ima es p e- sen ed o compa ison (Figu e 2), as well as he es i- ma es om he bo om-up PC app oach [13,14], we e consis en wi h hese es ima es. S anda d e o s o he es ima es om he di ec PC ull ank model we e la - ge compa ed o hose ob ained unde he op imal i PC and FA models. Thus, pa ame e educ ion using ac o analy ic, di ec and bo om-up PC models wo ked well o a iance componen es ima ion o MACE. Fu he mo e, compa ed o he analyses using unde - o o e -pa ame e ized models, op imal i esul ed also in he sho es unning imes: FA7 14.5 days, FA9 3.5 days, FA11 31.5 days, and PC15 21.5 days, PC19 9 days, PC25 16.5 days. Consequences o he PC and FA models o es ima ed b eeding alues Co ela ions be ween EBV om he PC and FA app oaches a e in Tables 4 and 5. Resul s om he com- ple e da a a e in Table 4 and esul s om he subse s o da a in Table 5. EBV om PC19 and PC25 we e iden i- cal. This was expec ed, gi en ha he eigen alues om 21 o 25 unde he ull ank model we e ze o [14]. The esul shows ha applying a PC model wi h he op imal i has no p ac ical consequences on anking o he bulls. EBV co ela ions be ween PC15 and PC19 we e lowe han hose be ween PC19 and PC25, demons a - ing ha he use o oo low a ank a ec ed he es ima es. Resul s also indica ed ha he p edic ion o he EBV migh be mo e sensi i e when using oo low a i unde he di ec PC app oach han unde he ac o analy ic app oach (Tables 4 and 5). Table 3 Ro a ed ma ix o ac o loadings om he FA9 analysis Fac o s Coun y F1 F2 F3 F4 F5 F6 F7 F8 F9 G ea B i ain -0.92 Czech Republic 0.93 Is ael 0.24 0.89 0.26 Sou h A ica 0.23 0.95 Es onia 0.61 Poland 0.32 0.55 Swi ze land -0.60 Swiss Red Hols ein -0.55 USA 0.26 -0.85 Ge many -0.20 0.28 0.22 -0.54 The Ne he lands -0.55 New-Zealand 0.20 0.82 Canada -0.29 -0.20 Denma k- Finland- Sweden -0.28 F ance -0.27 -0.20 I aly -0.34 Aus alia 0.43 Belgium -0.40 I eland -0.23 0.21 Spain -0.20 Slo enia 0.22 -0.20 F ench Red Hols ein 0.32 -0.39 Hunga y -0.20 Japan 0.25 La ia 0.23 -0.37 Figu e 2 Es ima es o gene ic co ela ions o p o ein yield om FA9 and PC19 analyses. Summa y s a is ics o In e bull and he PC ull i es ima es a e p esen ed o compa ison. Quan . Re e s o quan ile. Ty ise ä e al.Gene ics Selec ion E olu ion 2011, 43:33 h p://www.gsejou nal.o g/con en /43/1/33 Page 7 o 10 In mos cases, EBV co ela ions om he FA and PC app oaches unde he op imal i we e uni y o close o uni y (Tables 4 and 5). EBV co ela ions we e less han 0.99 only o Slo enia, he F ench Red Hols ein popula- ion and La ia in he comple e da ase (Table 4). These we e he coun ies/popula ions wi h he lowes numbe o eco ds and weak ies wi h he o he coun ies. The mean numbe o common bulls be ween F ench Red Hols ein and he o he coun ies was as low as 9, and hose o La ia and Slo eniawe e29and32, espec- i ely. Thus, he esul s indica e ha he use o he FA app oach unde he op imal i has no p ac ical conse- quenses on he anking o bulls. All EBV co ela ions we e uni y in he subg oup A (bulls used only in hei own coun y). In all s udied subg oups, EBV co ela ions om he FA9 and PC19 we e uni y o close o uni y, excep in subg oup C (bulls used only ab oad) o Slo enia, F ench Red Hols ein and La ia (< 0.99, Table 5). Co ela ions be ween EBV om PC15 and PC19 and om FA7 and FA9 ended o be lowe han hose be ween FA9 and PC19, bu hey we e s ill e yhigh.Theonlyexcep ionswi hco ela ionso 0.99 o g ea e occu ed in subg oup C o Is ael, F ench Red Hols ein and La ia. I was expec ed ha subg oup C would be he mos challenging g oup o analyse since he EBV we e p edic ed based on co e- la ed in o ma ion only. Resul s ag eed well wi h a p e- ious s udy, in which he FA and PC app oaches unde he educed ank RR MACE models we e applied, bu in which he a iance componen s used o he p edic- ions we e p o ided by In e bull [24]. Using he PC19 model educed he numbe o equa- ions in he mixed model by 24% compa ed o PC25. Fo simplici y, he FA model was implemen ed as a s anda d mul i a ia e model using Gwi h a FA s uc- u e, ins ead o an ex ended FA model discussed by Thompson e al. [6]. Due o his, model FA9 p o ided no ad an age o using mo e spa se coe icien ma ix in he MME. Times equi ed o sol ing mixed model equa ions anged om 5 min (PC19) o 7 min (FA9). Thus, di e ences in he compu ing imes we e o no p ac ical signi icance. Conclusions The andom eg ession ep esen a ion o MACE acili- a es exploi a ion o p incipal componen o ac o Table 4 Co ela ions be ween EBV in he comple e da a: analyses wi h op imal and oo low a i wi hin app oaches, and analyses wi h op imal i s be ween app oaches Coun y PC15 PC19 FA7 FA9 FA9 PC19 Canada 0.999 1.000 1.000 Ge many 1.000 1.000 1.000 Denma k-Finland-Sweden 1.000 1.000 1.000 F ance 1.000 1.000 1.000 I aly 1.000 1.000 1.000 The Ne he lands 1.000 1.000 1.000 USA 1.000 1.000 1.000 Swi ze land 0.999 1.000 1.000 G ea B i ain 1.000 1.000 1.000 New-Zealand 0.995 0.997 0.999 Aus alia 1.000 1.000 1.000 Belgium 0.997 1.000 1.000 I eland 0.997 0.999 0.999 Spain 1.000 1.000 1.000 Czech Republic 0.993 0.997 0.999 Slo enia 0.994 0.993 0.979 Es onia 0.995 1.000 0.996 Is ael 0.985 0.985 0.993 Swiss Red Hols ein 0.999 1.000 0.999 F ench Red Hols ein 0.999 0.988 0.988 Hunga y 0.999 1.000 1.000 Poland 0.999 1.000 0.999 Sou h A ica 0.992 0.995 0.998 Japan 0.999 0.999 1.000 La ia 0.982 0.993 0.977 Ty ise ä e al.Gene ics Selec ion E olu ion 2011, 43:33 h p://www.gsejou nal.o g/con en /43/1/33 Page 8 o 10 analy ic app oaches o a iance componen es ima ion and p edic ion o b eeding alues o in e na ional si e e alua ions. Bo h PC and FA allow a educ ion o he numbe o pa ame e s o be es ima ed, and bo h me h- ods bene i om he mo e pa simonious a iance s uc- u e. Gene ic pa ame e s om di e en app oaches we e e y simila when he op imal numbe o PC/ ac o s was i ed. Compu ing ime o es ima ion o a iance componen s was sho es unde he op imal i . O e i - ing inc eased he s anda d e o s o he es ima es, bu had no isible impac on he es ima es o on p edic ion o he b eeding alues. Fi ing oo low a numbe o pa ame e s a ec ed, in u n, bull ankings in di e en coun ies. Acknowledgemen s The s udy was a pa o he coope a ion p ojec o In e bull Cen e and MTT Ag i ood Resea ch Finland. Au ho de ails 1 Bio echnology and Food Resea ch, Biome ical Gene ics, MTT Ag i ood Resea ch Finland,31600 Jokioinen, Finland. 2 Animal Gene ics and B eeding Uni , Uni e si y o New England, A midale NSW 2351, Aus alia. 3 Depa men o Animal B eeding and Gene ics, SLU, Box 7023, S-75007 Uppsala, Sweden. 4 UMR 1313 INRA, Géné ique Animale e Biologie In ég a i e, 78352 Jouy-en- Josas Cedex, F ance. 5 In e bull Cen e, Depa men o Animal B eeding and Gene ics, SLU, Box 7023, S-75007 Uppsala, Sweden. Au ho s’con ibu ions AMT pe o med he s a is ical analyses and w o e he i s d a o he manusc ip . KM modi ied he WOMBAT so wa e o he needs o his s udy. JJ and WFF p o ided he da ase s. EAM, MHL, KM, VD, WFF and JJ supe ised he s udy and con ibu ed o w i ing he manusc ip . All au ho s ha e ead and app o ed he inal manusc ip . Compe ing in e es s The au ho s decla e ha hey ha e no compe ing in e es s. Recei ed: 17 Janua y 2011 Accep ed: 23 Sep embe 2011 Published: 23 Sep embe 2011 Re e ences 1. In e bull. [h p://www.in e bull.o g/]. Table 5 Co ela ions be ween EBV in ou subg oups: analyses wi h op imal and oo low a i wi hin app oaches, and analyses wi h op imal i s be ween app oaches Subg oup B a Subg oup C b Subg oup D c Coun y PC15 PC19 FA7 FA9 FA9 PC19 PC15 PC19 FA7 FA9 FA9 PC19 PC15 PC19 FA7 FA9 FA9 PC19 Canada 1.000 1.000 1.000 0.999 1.000 1.000 1.000 1.000 1.000 Ge many 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 Denma k-Finland-Sweden 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 F ance 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 I aly 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 The Ne he lands 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 USA 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 Swi ze land 0.999 1.000 1.000 0.999 1.000 1.000 0.998 1.000 1.000 G ea B i ain 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 New-Zealand 0.998 0.999 1.000 0.994 0.997 0.999 0.998 0.999 1.000 Aus alia 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 Belgium 0.999 1.000 1.000 0.997 1.000 1.000 0.999 1.000 1.000 I eland 0.999 1.000 1.000 0.997 0.999 0.999 1.000 1.000 1.000 Spain 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 Czech Republic 0.998 0.999 1.000 0.993 0.997 0.999 0.998 0.999 1.000 Slo enia 0.998 0.998 0.994 0.994 0.993 0.979 0.999 0.998 0.996 Es onia 0.999 1.000 1.000 0.995 1.000 0.996 0.999 1.000 1.000 Is ael 0.995 0.990 0.996 0.985 0.985 0.993 0.995 0.990 0.996 Swiss Red Hols ein 1.000 1.000 0.999 0.999 1.000 0.999 1.000 1.000 1.000 F ench Red Hols ein 1.000 1.000 1.000 0.999 0.988 0.988 1.000 1.000 1.000 Hunga y 0.998 1.000 1.000 0.999 1.000 1.000 0.999 1.000 1.000 Poland 0.999 1.000 1.000 0.998 0.999 0.999 0.999 1.000 1.000 Sou h A ica 0.997 0.998 0.999 0.992 0.995 0.998 0.997 0.998 0.999 Japan 0.999 1.000 1.000 0.999 0.999 1.000 1.000 1.000 1.000 La ia 0.996 0.999 0.996 0.982 0.993 0.977 0.998 0.999 0.998 a Subg oup B: bulls ha e been used in hei own coun y and ab oad b Subg oup C: bulls ha e been used only ab oad c Subg oup D: impo ed bulls Ty ise ä e al.Gene ics Selec ion E olu ion 2011, 43:33 h p://www.gsejou nal.o g/con en /43/1/33 Page 9 o 10