Gene ics
Selec ion
E olu ion
Knü e al. Gene ics Selec ion E olu ion 2013, 45:24
h p://www.gsejou nal.o g/con en /45/1/24
RESEARCH Open Access
Impac o p io speci ica ions in a
sh inkage-inducing Bayesian model o
quan i a i e ai mapping and genomic
p edic ion
Timo Knü 1, Esa Lää ä2and Mikko J Sillanpää1,2,3,4*
Abs ac
Backg ound: In quan i a i e ai mapping and genomic p edic ion, Bayesian a iable selec ion me hods ha e
gained popula i y in conjunc ion wi h he inc ease in ma ke da a and compu a ional esou ces. Whe eas
sh inkage-inducing me hods a e common ools in genomic p edic ion, igo ous decision making in mapping s udies
using such models is no well es ablished and he obus ness o pos e io esul s is subjec o misspeci ied
assump ions because o weak biological p io e idence.
Me hods: He e, we e alua e he impac o p io speci ica ions in a sh inkage-based Bayesian a iable selec ion
me hod which is based on a mix u e o uni o m p io s applied o gene ic ma ke e ec s ha we p esen ed in a
p e ious s udy. Unlike mos o he sh inkage app oaches, he use o a mix u e o uni o m p io s p o ides a cohe en
amewo k o in e ence based on Bayes ac o s. To e alua e he obus ness o gene ic associa ion unde a ying p io
speci ica ions, Bayes ac o s a e compa ed as signals o posi i e ma ke associa ion, whe eas genomic es ima ed
b eeding alues a e conside ed o genomic selec ion. The impac o speci ic p io speci ica ions is educed by
calcula ion o combined es ima es om mul iple speci ica ions. A Gibbs sample is used o pe o m Ma ko chain
Mon e Ca lo es ima ion (MCMC) and a gene alized expec a ion-maximiza ion algo i hm as a as e al e na i e o
maximum a pos e io i poin es ima ion. The pe o mance o he me hod is e alua ed by using wo publicly a ailable
da a examples: he simula ed QTLMAS XII da a se and a eal da a se om a popula ion o pigs.
Resul s: Combined es ima es o Bayes ac o s we e e y success ul in iden i ying quan i a i e ai loci, and he
anking o Bayes ac o s was ai ly s able among ma ke s wi h posi i e signals o associa ion unde a ying p io
assump ions, bu hei magni udes a ied conside ably. Genomic es ima ed b eeding alues using he mix u e o
uni o m p io s compa ed well o o he app oaches o bo h da a se s and loss o accu acy wi h he gene alized
expec a ion-maximiza ion algo i hm was small as compa ed o ha wi h MCMC.
Conclusions: Since no e o - ee me hod o speci y p io s is a ailable o complex biological phenomena, explo ing a
wide a ie y o p io speci ica ions and combining esul s p o ides some solu ion o his p oblem. Fo his pu pose,
he mix u e o uni o m p io s app oach is especially sui able, because i comp ises a wide and lexible amily o
dis ibu ions and compu a ionally in ensi e es ima ion can be ca ied ou in a easonable amoun o ime.
*Co espondence: mjs@ ol .helsinki. i
1Depa men o Ma hema ics and S a is ics, P.O. Box 68, Uni e si y o Helsinki,
Helsinki, FIN-00014, Finland
2Depa men o Ma hema ical Sciences/S a is ics, P.O. Box 3000, Uni e si y o
Oulu, Oulu, FIN-90014, Finland
Full lis o au ho in o ma ion is a ailable a he end o he a icle
© 2013 Knü 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.
Knü e al. Gene ics Selec ion E olu ion 2013, 45:24 Page 2 o 16
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Backg ound
Gene ic associa ion s udies, quan i a i e ai loci (QTL)
mapping and genomic p edic ion ely on inc easingly
dense DNA in o ma ion such as single nucleo ide poly-
mo phisms (SNP). The inc easing abundance o ma ke
da a ampli ies one o he essen ial s a is ical p oblems
in such s udies: he numbe o po en ial explana o y
a iables ep esen ed by single ma ke s is o en la ge
han he numbe o obse a ions in he sample s ud-
ied, and some egula iza ion is equi ed o ensu e he
iden i iabili y o he ma ke e ec s. Sui able s a is i-
cal models can accomplish his egula iza ion by a i-
able (i.e. ma ke ) selec ion, sh inkage o ma ke e ec s
owa ds ze o o a combina ion o hese wo s a egies
[1-4].
Many a iable selec ion and sh inkage echniques based
on Bayesian modelling and Ma ko chain Mon e Ca lo
(MCMC) algo i hms ha e been p oposed o gene ic asso-
cia ion s udies, QTL mapping and genomic p edic ion
(see [5,6]). They di e in he se -up o he s a is ical
model and in hei p io speci ica ions. P obably he
mos popula al e na i es a e e e sible jump MCMC
[7-9], s ochas ic sea ch a iable selec ion (SSVS) [10,11]
and locus-indica o models [12]. To a oid some o he
complica ions in model selec ion, sa u a ed models ha e
been p oposed in which gene ic e ec s om all pos-
sible explana o y ma ke s a e collec ed simul aneously
in o he model and hei iden i iabili y is inc eased by
p io assump ions ha esul in sh inkage o e ec sizes
owa ds ze o [1,4,13]. Such a sh inkage-inducing me hod
leads o a solu ion in which la ge e ec s end o occu only
a a he ew posi ions along he genome in he pos e io
dis ibu ion.
In a p e ious s udy, we p esen ed a new class o
sh inkage-inducing p io s: a mix u e o disc e e uni o m
dis ibu ions (MU), and compa ed i o o he me hods in
he con ex o QTL de ec ion [14]. Compa ed o me h-
ods commonly used in genomic p edic ion, he main
di e ences and simila i ies a e he ollowing: MU is a
sh inkage-based me hod like BayesA [1] and Bayesian
LASSO [13,15], bu i is iche in he a ie y o uning-
pa ame e s. This may be bad om a uning poin o iew,
bu he hype -pa ame e combina ions in he p io spec-
i ica ion po en ially co e s a wide spec um o di e en
scena ios conce ning he gene ic a chi ec u e o he ai ,
he i abili y, ma ke spacing o s uc u e o linkage dise-
quilib ium (LD) in he da a. Like BayesB [1] and SSVS
[10,11], MU includes a hype -pa ame e o he p io
p obabili y o no ma ke associa ion, bu unlike BayesB
and SSVS, he p io o MU does no include any indi-
ca o a iables. The e o e, use o such sepa a e indica o
a iables is a oided in he es ima ion algo i hms o MU,
which o he wise could nega i ely a ec he speed and
he mixing p ope ies du ing MCMC simula ion o cause
mul imodali y p oblems in maximum a pos e io i es ima-
ion (see [16]).
Bayesian sh inkage me hods a e common ools in
genomic p edic ion, bu igo ous decision making in
he con ex o QTL de ec ion ia such models is no
well es ablished [17]. He e, we shall examine in mo e
de ail he p ope ies o MU, ocusing in pa icula on
how obus he esul s a e in he analysis o he well-
s udied QTLMAS XII da a se wi h igh ly linked ma ke s
[18,19]. In addi ion, we es he p edic ion abili y o
genomic selec ion pu poses in a eal da a se on a pop-
ula ion o pigs [20]. As sugges ed in [14], MU appea s
o be sensi i e o p io pa ame e s. In his s udy, we
esume he issue o p io sensi i i y and we ex end he
analysis. As a po en ial solu ion o he p io sensi i i y
issue, we de ine a ini e se o p io speci ica ions and
use ”poo -man’s” model a e aging o e hese by gi ing
equal p obabili y/weigh o each p io se ing. We com-
pa e hese consensus es ima es o he p esumably less
obus ones om single p io speci ica ions. MU com-
p ises a wide and lexible amily o p io dis ibu ions,
because i is con olled by h ee hype -pa ame e s ins ead
o wo o one as in mos o he sh inkage app oaches
wi hou indica o s in he model. Fu he mo e, he p io
assump ions in MU p o ide a cohe en amewo k o
o mal hypo hesis es ing and calcula ion o Bayes ac-
o s, which is lacking in mos o he sh inkage-based a i-
able selec ion me hods [17]. As ano he excep ion wi h
a cohe en amewo k, a decision ule based on Bayes
ac o s has been p oposed o he ex ended Bayesian
LASSO [21].
Fo MCMC simula ion o he pos e io dis ibu ion, we
ha e implemen ed a Gibbs sample , o which we p o-
ide he ully condi ional dis ibu ions in Addi ional ile
1 and he C code as an ex ension module o he so -
wa e package R [22] in Addi ional ile 2. As a as e
al e na i e o MCMC es ima ion, we ha e cons uc ed a
gene alized expec a ion-maximiza ion (GEM) algo i hm
o maximum a pos e io i (MAP) poin es ima ion [23],
o which we p o ide he es ima ion de ails and he C
implemen a ion in Addi ional ile 3.
Me hods
Da a model and Bayesian hie a chical se -up
Conside a popula ion-based sample o Nindi idu-
als wi h pheno ype measu emen s Yj(j=1, ...,N).
Suppose each indi idual has been sco ed a Mma k-
e s and he geno ype obse a ion o an indi idual a
ma ke m(m=1, ...,M)isdeno edbyxjm. Assum-
ing bi-allelic ma ke s such as SNP (single nucleo ide
polymo phisms) and only addi i ely ac ing gene e ec s,
geno ype obse a ions a e coded as −1, 0 and 1 co e-
sponding o he h ee possible geno ypes, say AA,Aa
and aa.
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The pheno ype o indi idual jis modelled by he ollow-
ing eg ession equa ion
Yj=α+
M

m=1
βmxjm +j.(1)
He e, αis he in e cep common o all indi iduals in
he popula ion. Fu he mo e, each βmholds he addi-
i e e ec o ma ke m,andj he e o e m o he
indi idual. A comple e desc ip ion o he dis ibu ional
assump ion made o speci y he likelihood as well as i s
ma hema ical o mula a e included as suppo ing in o -
ma ion [see Addi ional ile 1]. Cons an a iances a e
assumed o αand {βm}in hei espec i e p io spec-
i ica ions, whe eas a common andom a iance σ2is
assumed o he e o e ms. Condi ional on σ2, mu ual
independence is assumed among he o he pa ame e s
(α,{βm},{j}). I app op ia e, he eg ession can ead-
ily be ex ended o include a polygenic componen wi h
kinship-based a iance-co a iance s uc u e o accoun
o in ini esimal ma ke e ec s and/o backg ound QTL.
P io speci ica ions o sh inkage-based a iable selec ion
As ypical in his ype o Bayesian a iable selec ion
app oaches, es ic i e sh inkage p io s a e assigned o
he e ec size pa ame e s o egula ise he model, o
a oid o e i ing and o ensu e he iden i iabili y o gene ic
ma ke e ec s.In he ollowing,wedesc ibesuchan
app oach, which p o ides a mechanism o sh ink spu ious
e ec sizes owa ds 0. We use a mix u e o h ee dis inc
uni o m dis ibu ions (MU), he pe o mance o which
has been p e iously e alua ed using wo well-documen ed
eal da a se s and compa ing i o wo o he Bayesian
a iable selec ion app oaches [14]. Since we used he so -
wa e package OpenBUGS [24] in ou p e ious s udy o
pe o m MCMC simula ion, ou epo was es ic ed o
sampleswi hmuch ewe indi idualsandma ke s hanin
his s udy. He e, we o e come his d awback by a Gibbs
sample implemen a ion o MCMC simula ion o he
pos e io dis ibu ion and a GEM algo i hm o as max-
imum a pos e io i poin es ima ion in he low-le el C
p og amming language.
Bo h ypes o algo i hms a e based on he ully condi-
ional uni a ia e pos e io dis ibu ions and single pa am-
e e s a e upda ed one a a ime; whe eas he Gibbs sample
i e a es o e andom d aws om hese dis ibu ions,
GEM only i e a es o e he ully condi ional expec ed
alues be o e eaching con e gence in a - possibly local -
maximum o he pa ame e space. Fo a de ailed discus-
sion on GEM and i s a ini y wi h s anda d EM and ela ed
algo i hms see [25].
The assump ions o he p io dis ibu ion a e com-
ple ely speci ied in he suppo ing in o ma ion [see Addi-
ional ile 1]. In Addi ional ile 1, we also de i e he
uni a ia e ully condi ional pos e io dis ibu ions needed
o a single-si e Gibbs sample and he ully condi ional
expec ed alues o GEM. The C codes o bo h algo-
i hms a e p o ided in he suppo ing in o ma ion [see
Addi ional iles 2 and 3].
In MU, each e ec size, βm, is assigned a p io dis ibu-
ion wi h p obabili y densi y unc ion
p(βm)=p0·1
2bI(−b,b)(βm)
+1−p0
2·1
l−bI[−l,−b](βm)+I[b,l](βm),
(2)
whe e IA(x)is he indica o unc ion o a se A,i.e.i s alue
is 1 i x∈Aand 0 o he wise; u he mo e, p0∈(0, 1)is
he p io p obabili y ha βmob ains a alue close o 0 in
he in e al (−b,b), wi h he bo de alue se o b>0, and
1−p0is consequen ly he p io p obabili y ha βmlies
u he away om 0, ei he in [ −l,−b]o in[b,l], wi h he
e ec size limi se o l>b. I he h ee hype -pa ame e s
p0,band la e app op ia ely chosen, his densi y has a na -
ow peak a ound ze o and is la on he es o i s suppo .
Thus, his densi y is a s ep unc ion, esembling a spike
and a slab [26]. The slab is some imes also e e ed o as a
smea (e.g. [27]).
The mix u e o h ee uni o m dis ibu ions is speci ied
by alloca ing a majo amoun o p obabili y mass, p0,on
asmallin e al(−b,b) ha co e s 0 and he emaining
p obabili y mass, 1 −p0, on wo in e als ha lie sym-
me ically a ei he side away om 0. Dis ibu ing he
p obabili y mass in his way e lec s he p io pe cep ion
ha a ma ke chosen a bi a ily om a la ge se is unlikely
o explain a subs an ial po ion o he pheno ypic a ia-
ion. In o he wo ds, mos ma ke e ec s a e expec ed o
be so close o 0 ha hei con ibu ions can be conside ed
negligible.
Biological expe knowledge and p ac ical consid-
e a ions should de e mine he choice o he h ee
hype -pa ame e s. Conside ing he con ibu ion o he
pheno ypic a ia ion o e ec sizes lying wi hin he spike
(|βm|<b) as negligible, yields a c i e ion o disc imina e
be ween associa ed and non-associa ed ma ke s. How-
e e , o he aspec s such as sample size and coa seness
o measu emen a ec he choice o b,becauseasmall
sample size and imp ecise da a educe he chances o
iden i y small ma ke e ec s. I |βm|≥bis used as he
c i e ion o QTL iden i ica ion, he p io belie conce n-
ing he o al numbe o associa ed ma ke s can be di ec ly
exp essed ia he choice o p0; he numbe o ma ke s
wi h |βm|≥bhas ap io ia binomial dis ibu ion wi h
mean M(1−p0)due o he independence assumed
among {βm}.Thehype -pa ame e l es ic s he absolu e
e ec size o a ma ke o a ce ain uppe limi , which is
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di icul o quan i y ap io i, because he gene ic a chi ec-
u e o he ai and speci ically he dis ibu ion o e ec
sizes a e no known. Howe e , empi ical s udies indica e
ha e ec sizes o mo e han a ew pheno ypic s anda d
de ia ions seem unlikely (see [28-30]).
In he con ex o eg ession models o genomic p e-
dic ion, a ough guideline has been sugges ed o choos-
ing hype -pa ame e s in he p io dis ibu ion o gene ic
e ec s based on a connec ion be ween he p io a iance
o SNP e ec s and he expec ed he i abili y o he ai
(c .[6]).Fo MU, he a ianceo hee ec o asingleSNP
can be easily ob ained om Equa ion (2) and in eg a ion
yields
Va (βm)=1
3b2+l(l+b)(1−p0).
Gianola e al. [31] de i ed ha
Va (βm)=VA
2M
m=1 m(1− m)
unde idealized condi ions (Ha dy-Weinbe g equilib ium,
linkage equilib ium be ween QTLs, and QTL posi ions
coinciding wi h ma ke posi ions). He e, VAis he addi i e
gene ic a iance and m he allele equency a ma ke m.
Unde hese condi ions, he na ow-sense he i abili y, i.e.
h2=VA/VPwi h VPbeing he pheno ypic a iance, can
be exp essed as
h2=2Va (βm)M
m=1 m(1− m)
VP
.(3)
As poin ed ou by de los Campos e al. [6], i he geno ypes
a each ma ke a e s anda dized o ha e a mean o 0 and
a a iance o 1 ins ead o using -1, 0, and 1 as geno ype
codes, he ela ionship jus men ioned becomes
h2=Va (βm)M
VP
.(4)
No e ha he alues o h2a e no es ic ed o he in e -
al (0, 1)bu me ely o (0, ∞). He e, i is no ewo hy ha
al e ing he geno ype codes ia s anda diza ion a ec s he
in e p e a ion o he e ec size es ima es, since βmsdono
ep esen addi i e gene ic e ec s on he pheno ype scale
in his case.
Tools o in e ence
As in ou p e ious s udy, we calcula ed he Bayes ac o o
he hypo hesis ha he absolu e alue o he ma ke e ec
exceeds a ce ain h eshold alue o assess he s eng h
o he associa ion be ween he pheno ype and a single
ma ke m. As in any sh inkage-inducing app oach, choos-
ing his h eshold is a bi a y o needs o be con olled
bype mu a iono hepheno ype[4].In hecaseo MU,
howe e , he choice o bas he h eshold esul s in a
amewo k which is cohe en wi h he p io assump ions
conce ning he e ec size βm, namely ha he con ibu-
ion o ma ke s wi h e ec sizes in he in e al (−b,b)
a e negligible. By de ining an indica o a iable Sm=
I[b,l](|βm|), he pos e io p obabili y o he hypo hesis can
be exp essed as P(Sm=1|da a).Toob ain heBayes ac-
o o he wo compe ing hypo heses H1:Sm=1agains
H0:Sm=0, he pos e io odds is di ided by i s p io
odds [32,33]:
BFm=P(Sm=1|da a)
1−P(Sm=1|da a)P(Sm=1)
1−P(Sm=1),
whe e he p io p obabili y P(Sm=1)=1−p0is eadily
a ailable om he p io speci ica ion o βmin MU.
Kass and Ra e y [32] ha e sugges ed he ollowing ca -
ego ies o classi y he s eng h o e idence p o ided by
wice he na u al loga i hm o he Bayes ac o , 2ln(BFm),
as a sligh modi ica ion o he ca ego ies p esen ed by
Je eys [34]: e idence in a ou o he hypo hesis is con-
side ed e y s ong o alues >10, s ong o alues in
(6, 10], posi i e o alues in (2, 6], and no wo h mo e
han a ba e men ion o alues in (0, 2], espec i ely.
As men ioned abo e, he choice o a h eshold o
he e ec size βmis gene ally p oblema ic in sh inkage
app oaches, whe eas he p io speci ica ion o MU en ails
a jus i ica ion o a speci ic h eshold in MU. Unless indi-
ca o a iables a e in eg a ed in o he likelihood o he
model (e.g. as in [35]), mos sh inkage app oaches do no
p o ide an unequi ocal ame o hypo heses necessa y
o he Bayes ac o . A no able excep ion is he ex ended
Bayesian LASSO [21], whe e he p io dis ibu ions o
locus-speci ic a iances depend on egula izing sh ink-
age pa ame e s, which can be es ed o QTL p esence ia
Bayes ac o s.
Besides he choice o a h eshold o βm, ano he con-
cep ual p oblem may a ise in sh inkage app oaches in
which imp ope p io s o he e ec sizes a e used, such
as he model p oposed in [36] as a modi ica ion o he
app oach in [4]; al hough he pos e io p obabili y P(Sm=
1|da a)and consequen ly he pos e io odds may exis
also o imp ope p io s, hep io oddsisno a ail-
able o he complemen a y hypo heses b<|βm| s.
|βm|≤b, because he in eg al o e he p io dis ibu ion
co esponding o he o me hypo hesis does no exis .
We assessed he sensi i i y o single analyses by com-
pa ing esul s unde a ying p io speci ica ions, and o
MCMC addi ionally unde iden ical p io speci ica ions o
de ec con e gence o mixing p oblems. In addi ion, we
combined Bayes ac o in o ma ion om di e en anal-
yses o inc ease he obus ness in de ec ing associa ion
signals.
We also e alua ed he p edic i e abili ies o ou model
by compa ison o genomic es ima ed b eeding alues
(GEBV) ei he wi h he ue b eeding alues (TBV), as
a ailable in simula ed da a se s, o wi h he pheno ype
Knü e al. Gene ics Selec ion E olu ion 2013, 45:24 Page 5 o 16
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measu emen s di ec ly, as a ailable in eal da a se s. The
GEBV o indi idual iis
GEBVi=
M

m=1
βmxim,
whe e 
βmis he pos e io mean o βmin hecaseo
MCMC o he MAP poin es ima e in he case o he
GEM algo i hm, and (xim)is he ec o o geno ype codes
o he indi idual. Fo c oss- alida ion o ou esul s,
we employed he as e GEM algo i hm. Also he e, we
compa ed es ima es om single p io speci ica ions wi h
combined es ima es om mul iple ones. A mo e de ailed
desc ip ion o hese p ocedu es is gi en in he ollowing
sec ions.
Analysis o he simula ed QTLMAS XII da a
This simula ed da a se was o iginally dis ibu ed as a pa
o he 12 h Eu opean wo kshop on QTL mapping and
ma ke assis ed selec ion (QTLMAS XII) held in Uppsala,
Sweden, on 15–16 May 2008. De ailed in o ma ion on he
publicly a ailable da a [37] has been p esen ed by C ooks
e al. [18] and Lund e al. [19].
The simula ion o he pheno ype in ol ed a o al o 50
bi-allelic QTLs wi h addi i e e ec s. C ooks e al. [18]
classi ied 15 o hese as majo QTL (deno ed by M1-M15),
because hey yield P- alues o less han 0.05 a e Bon e -
oni co ec ion in a mul iple linea eg ession including
all geno ypes o ue QTLs. The whole da a se a ail-
able o QTL de ec ion consis s o 4665 indi iduals om
a pedig ee o consecu i e gene a ions. We excluded he
165 indi iduals o he i s gene a ion om ou analy-
sis, because hey do no o m ull-sib amilies o size 10
like he 4500 indi iduals in he subsequen gene a ions.
The ounde s o each gene a ion we e 15 males and 150
emales. In he i s gene a ion, all indi iduals we e used
as pa en s, whe eas in he second and hi d gene a ion,
hey we e andomly sampled. Each male pa en was ma ed
o 10 emales, each p oducing 10 ull-sib o sp ing. Thus,
he pedig ee ac ually has a ull-sib and hal -sib s uc u e.
Howe e , we did no ake in o accoun he amilial esem-
blance be ween hal -sibs o be ween pa en s and o sp ing
om consecu i e gene a ions in ou s a is ical model.
Fo simplici y, we me ely conside ed polygenic amily
e ec s (uk) o ull-sib amilies and ex ended he eg es-
sion in Equa ion (1) o
Ykj =α+
M

m=1
βmxkjm +uk+kj,
o indi idual j(j=1, ...,Nk) om amilyk(k=
1, ...,K). The polygenic e ms ukwe e assumed condi-
ionally independen andom e ec s wi h a mean o 0 and
a common andom a iance σ2
u.
Ou esul s a e based on N=4500 indi iduals in K=
450 ull-sib amilies, each o size Nk=10. The ma ke
da a consis s o 6000 comple ely geno yped SNP equidis-
an ly spaced by 0.1 cM spanning six ch omosomes wi h
1000 ma ke s each. We emo ed he 106 ma ke s wi h
mino allele equency o less han 0.01, yielding M=
5894 ma ke s o analysis o he comple e genome.
Associa ion mapping
We an MCMC simula ions o ou di e en se s o p io
speci ica ions (see de ails in Table 1). Ou i s goal was o
e alua e he powe o MU o de ec QTL and he alse pos-
i i e e o a e in his da a se wi h igh ly-linked ma ke s
and o compa e he indings wi h he esul s om he six
associa ion s udies epo ed in [18]. Secondly, we aimed a
assessing he obus ness o ou esul s in se e al MCMC
uns unde iden ical and a ying p io speci ica ions. Fo
each se o p io speci ica ions, we s a ed wo MCMC
chains om di e en s a ing alues. Thus, he esul s
a e based on a o al o eigh chains (ma ked by A-H). In
each un, we simula ed 220 000 Gibbs i e a ions, o which
he i s 20 000 we e disca ded as bu n-in. This bu n-
in size was de e mined based on in o mal con e gence
checks. We applied hinning o sa e disk space and only
s o ed e e y 20 h i e a ion. Thus, each o he eigh uns
yielded 10 000 MCMC samples o he analysis o he join
pos e io dis ibu ion. The MCMC simula ion o a sin-
gle chain ook 6 - 6.5 hou s on a compu e wi h a 3 GHz
dual co e p ocesso and a physical memo y o 2 GB. All
simula ions sha ed he ollowing p io speci ica ions: he
uppe limi o he e ec size pa ame e s βmwas se o
l=sd(Y)=2.10, he p io a iance o he common in e -
cep α o c=106, and he shape and a e pa ame e s
(su, u,s, ) we e all se o 0.01 in he in e se-gamma dis-
ibu ions used as p io s o he a iance componen s σ2
u
and σ2[see Addi ional ile 1 o he pa ame isa ion o he
in e se-gamma dis ibu ion]. Fo an in e se-gamma dis-
ibu ion wi h shape pa ame e sand a e pa ame e ,i s
mean has he alue
s−1,i s>1, and i s a iance has he
alue 2
(s−1)2(s−2),i s>2. Thus, he mean and a iance
do no exis o ou choice o shape and a e pa ame e s
because o a hea y igh ail. Howe e , he mode exis s,
wi h a alue o
s+1=1
101 .Wi hbo h and sdec easing
owa ds 0, he in e se-gamma dis ibu ion app oaches he
nonin o ma i e scale-in a ian , bu imp ope p io wi h
densi y ∝1/σ 2.
Genomic p edic ion
In addi ion o he ou gene a ions used o QTL de ec-
ion, he QTLMAS XII da a spans o e h ee mo e gene -
a ions, p o iding a alida ion se o genomic p edic ion
models. Each o hese gene a ions holds 400 indi iduals
wi h comple e geno ype in o ma ion and TBV.

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Table 1 Compa ison o he p io speci ica ions in he eigh MCMC chains A-H used o analyse he QTLMAS XII da a,
pos e io es ima es o model pa ame e s and summa y s a is ics
P io speci ica ion Pos e io mean (sd) o
Chain p0b(a)NQα102σ2
uσ2h2
M(b)NQ
A 0.99 0.01 58.9 2.0 (0.6) 1.7 (1.2) 3.0 (0.1) 0.32 (0.02) 23.0 (2.5)
B 0.99 0.01 58.9 2.6 (0.7) 1.7 (1.2) 3.0 (0.1) 0.32 (0.02) 22.9 (2.6)
C 0.99 0.001 58.9 2.3 (0.5) 3.0 (1.9) 3.0 (0.1) 0.30 (0.02) 31.5 (2.5)
D 0.99 0.001 58.9 2.6 (0.5) 3.0 (1.9) 3.0 (0.1) 0.29 (0.02) 31.1 (2.5)
E 0.999 0.01 5.9 2.1 (0.5) 1.9 (1.3) 3.0 (0.1) 0.31 (0.02) 15.3 (1.3)
F 0.999 0.01 5.9 2.8 (0.5) 2.1 (1.4) 3.0 (0.1) 0.31 (0.02) 14.3 (1.3)
G 0.999 0.001 5.9 1.9 (0.4) 3.9 (2.3) 3.1 (0.1) 0.28 (0.02) 21.5 (1.4)
H 0.999 0.001 5.9 2.0 (0.7) 3.7 (2.2) 3.1 (0.1) 0.28 (0.02) 22.6 (1.8)
(a)gi eninuni so pheno ypics anda dde ia ions(sd(Y)=2.10).
(b)The ue o e all he i abili y o he ai is 0.30 [19].
Hype -pa ame e p0de ines he p io p obabili y ha he e ec size lies in he in e al o he spike, (−b,b).NQis a summa y s a is ic o he numbe o QTL (see ex
o de ails), α he common in e cep in he eg ession, σ2
u he a iance componen o he polygenic e ms, σ2 he esidual a iance, and h2
M he pa o he he i abili y
due o ma ke e ec s.
To assess he p edic i e abili ies o ou model, we i s
calcula ed GEBV o he alida ion indi iduals, using he
pos e io means o he e ec sizes, βm, om he MCMC
chains. Fo simplici y, he es ima ed amily e ec s, uk,
e lec ing pedig ee in o ma ion wi hin he aining gene -
a ions, we e no aken in o accoun , because he polygenic
e ec was negligible in ou analysis (see Resul s sec ion),
as well as in a p e ious s udy [25]. Fu he mo e, he am-
ily e ec s we e es ima ed o ull-sib amilies wi hin he
aining gene a ions and could hus no be applied o he
indi iduals in he alida ion gene a ions.
We e alua ed hese GEBV o single p io speci ica ions
and hei a e ages ac oss he ou p io speci ica ions con-
side ed. As in [19], we assessed he p edic i e abili y o
he GEBV in he alida ion indi iduals by h ee measu es:
he accu acy was es ima ed as he Pea son co ela ion
be ween GEBV and TBV; in addi ion, he Spea man ank
co ela ion was calcula ed be ween GEBV and TBV o
he 10% o he indi iduals wi h he la ges TBV; inally,
he bias o GEBV was es ima ed as he coe icien o
eg ession o TBV on GEBV.
We also ob ained GEBV om he GEM algo i hm and
assessed hei p edic i e abili y as jus desc ibed. Again
o simplici y, we excluded he amily e ec s, uk, om he
model. Ins ead o using he o iginal pheno ype and geno-
ype in o ma ion, we s anda dized he pheno ype and he
geno ype codes a each SNP o ha e a sample mean o 0
and a a iance o 1 in he aining se . The GEBV we e
hen es ima ed as abo e and ansla ed back o he o igi-
nal scale. The GEM algo i hm o one p io speci ica ion
equi ed3 o14secondsand19 o125i e a ions ocon-
e ge on he same compu e as men ioned abo e (wi h a
3 GHz p ocesso and 2 GB memo y). Con e gence was
decla ed when he sum o de ia ions be ween cu en and
upda ed pa ame e alues was smalle han (M+2)×10−7,
whe e M+2=5896 is he numbe o pa ame e s in he
model.
As TBV a e only a ailable in simula ed da a se s, we
also applied a c oss- alida ion (CV) app oach as a me hod
o assess p edic i e abili y o he model in eal da a se s.
He e, we used only he 4500 indi iduals in he h ee ain-
ing gene a ions. Speci ically, we used wo di e en 10- old
CV s a egies: (I) we andomized he da a in o 10 dis-
inc alida ion se s, each holding 45 ull-sib amilies, i.e.
all membe s o a amily belonged o he same alida ion
se ; (II) each o he 10 ull-sibs o a amily was andomly
assigned o a di e en alida ion se . To p edic GEBV o
he indi iduals o a single alida ion se , he o he nine se s
we e combined o o m he aining se . We di ided he
co ela ion be ween GEBV and pheno ype by he squa e
oo o he i abili y h=√0.30 [19] o con e i o an es i-
ma e o he accu acy o he GEBV. The bias o GEBV was
es ima ed as he coe icien o eg essing pheno ype on
GEBV.
Analysis o he eal da a
To es he p edic i e abili y o ou me hod in eal da a,
we analysed a pig da a se made a ailable by Pig Imp o e-
men Company (a Genus company) o he scien i ic
communi y [20]. He e, we used one o he i e pheno-
ypes p o ided (T5), which was eco ded o 3184 geno-
yped indi iduals and o which a he i abili y o 0.62 was
epo ed in [20]. Be o e analysis, he ai was s anda d-
ized o ha e a sample mean o 0 and a s anda d de ia ion
o 1.
A o al o 52 843 SNP we e con ained in he geno ype
da a made public. The o iginal geno ype codes we e 0, 1,
and 2 o he h ee SNP geno ypes, espec i ely, and o
missing geno ypes (<1%), a non-in ege be ween 0 and 2
had been impu ed (see [20] o de ails). Fo ou analysis,
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geno ype codes we e s anda dized o ha e a mean o 0
and a s anda d de ia ion o 1 a each SNP. He e, we used
ou subse s o hese SNP: (i) a andom se o 10 000
SNP om he en i e SNP da a; (ii) a andom pick o
1000 SNP om he se in (i); (iii) a subse o 10 000 SNP,
each wi h a mino allele equency >0.05 and il e ed
om he en i e SNP da a by su e independence sc eening
(SIS) o he ma ginal co ela ions be ween he pheno-
ype and SNP [38]; (i ) a subse o 1000 SNP, also each
wi h a mino allele equency >0.05 and il e ed om
he en i e SNP da a by SIS; his was a subse o he se
in (iii). No e ha he se o 10 000 SNP il e ed by SIS
is iden ical o he one used in [25]. We epo esul s
including p edic ion accu acies o all ou se s o SNP
(i)-(i ).
As he esul s ob ained om o he Bayesian app oaches
we e shown o be nea ly una ec ed by he inclusion o
pedig ee in o ma ion in his da a se [25], we chose no o
include a polygenic componen in his pa o he analysis.
Fo pa ame e es ima ion, we applied he GEM algo i hm
and conside ed nume ous combina ions o he hype -
pa ame e s p0and b, which anged om 0.9 o 0.9999 and
om 0.0001 o 0.036, espec i ely. The hype -pa ame e l
waskep cons an a 2.
The accu acy o GEBV was es ima ed by hei co ela-
ion wi h pheno ypic alues di ided by he squa e oo o
he epo ed he i abili y, i.e. √0.62. The b eeding alue o
an indi idual was p edic ed ia 10- old c oss- alida ion,
in which each indi idual was andomly assigned o one
o 10 subse s. Each o hese subse s was used once as
he alida ion se , wi h he o he nine subse s o ming
he aining se . By using he same subse s as in [25],
ou esul s a e di ec ly compa able o he ones ob ained
in ha s udy. We also ob ained an es ima e o he bias
o GEBV as he coe icien o eg ession o he pheno-
ype on GEBV. I is impo an o no e ha , simila o
[25], he p e-selec ion by SIS was done using all indi id-
uals, i.e. i was in luenced no only by aining bu also
by alida ion indi iduals, and may ha e caused he sub-
sequen c oss- alida ion p ocedu e o o e -es ima e he
accu acies.
Resul s
QTL de ec ion in he QTLMAS XII da a
Compa ison o common model pa ame e s
We begin wi h an o e iew o he pos e io es ima ion
o he model pa ame e s, wi h he excep ion o ma ke -
speci ic pa ame e s and compa e esul s ob ained om
he eigh MCMC chains A-H. Table 1 shows he a ying
p io speci ica ions o he MCMC chains and pos e-
io esul s o model pa ame e s and summa y s a is-
ics. The alues o he bo de pa ame e , b,a egi en
in uni s o pheno ypic s anda d de ia ions (sd(Y)=
2.10). We de ined a summa y s a is ic o he numbe o
QTL based on he ma ke -speci ic indica o a iables by
NQ=
M

m=1
Sm, and o he he i abili y due o ma ke
e ec s by h2
M=1−(σ2+2σ2
u)/ a (Y),whe e hesam-
ple a iance a (Y)wasusedasanapp oxima iono he
pheno ypic a iance, igno ing he ela edness be ween he
indi iduals s udied. He e, he a iance componen σ2
uo
he polygenic e ec s was mul iplied by a ac o 2, because
he coe icien o he addi i e gene ic co a iance be ween
ull sibs is 1/2(seee.g.chap e 7in[39]).
Fo he common in e cep , somewha highe de ia ions
o he pos e io esul s we e obse ed be ween chains
wi h iden ical p io speci ica ions when he bo de alue
o he e ec sizes was se o b=0.01 (chains A s. B
and E s. F) han when i was se o b=0.001 (chains
C s. D and G s. H). Thus, a leas o hese pa ame-
e s, he p io speci ica ion b=0.001 yielded mo e obus
esul s.
All chains p oduced i ually iden ical es ima es o he
esidual a iance σ2. The poin es ima es o he be ween-
amily a iance σ2
uwe e o abou wo o de s o magni-
ude smalle han σ2. This indica es ha he polygenic
e ec s, uk, abso bed a he li le pheno ypic a ia ion in
he simul aneous analysis o all ch omosomes, which is
consis en wi h he esul s epo ed by Lund e al. [19].
Since he gene ic a ia ion in he da a was explained
almos comple ely by he ma ke e ec s, li le in o ma-
ion would be los i he polygenic e ms we e excluded
om he model. In he analysis o only one ch omo-
some, he polygenic e ms played a mo e in luen ial ole
( esul s no shown), since hey can abso b gene ic e ec s
om he es o he genome (c . [40]). Al hough he es i-
ma es o σ2
uwe e small when analysing he comple e
genome, we obse e di e ences be ween p io speci ica-
ions: mo e pheno ypic a ia ion was explained by he
polygenic e ec s when b=0.001, i.e. in chains C, D, G
and H, since σ2
uob ained la ge pos e io means in hese
chains han in he o he s. This also explains he sligh ly
highe es ima es o h2
M o b=0.01. He e, we should
no e ha he ue he i abili y o he ai o he ull pedi-
g ee da a is 0.30 [19], which closely coincides wi h ou
es ima es, which anged om 0.27 o 0.32.
Es ima es o he summa y s a is ic NQ o he numbe
o QTL we e, as expec ed, highe o he chains wi h p0=
0.99, i.e. wi h a smalle p io p obabili y o ma ke exclu-
sion. He e we no e ha he p io mean o NQis M·(1−p0).
Thus o p0=0.99, he pos e io mean alues be ween
24 and 33 we e lowe han he p io mean o 60. In con-
as , he p io mean o NQwas 6 o p0=0.999, bu he
pos e io means we e la ge wi h alues anging om 14
o 22. In his sense, he in ui ion ha he p io speci ica-
ions wi h p0=0.999 a e mo e conse a i e is con i med.
We also obse ed ha he chains wi h b=0.01 p oduced
lowe pos e io means o NQ o ixed p0.This esul
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is in ui i e also, since ma ke indica o s a e expec ed o
each he alue 1 mo e easily, when he in e al (−b,b)is
sho ened.
Ma ke -speci ic esul s
Two o ou main goals we e (1) o assess how well MU
iden i ies ue QTL in his da a se and (2) o e alua e
he isk o alse posi i e QTL de ec ion when apply-
ing he Bayes ac o as he measu e o he e idence in
a ou o ma ke associa ion. In Table 2, he 20 ma k-
e s wi h he s onges signals in ou analysis a e lis ed.
He e, we used he ollowing c i e ion o ank he s eng hs
o associa ion om all M=5894 ma ke s: o each
ma ke , we calcula ed he Bayes ac o o he hypo h-
esis Sm=1(seeabo e,Tools o in e ence)ineach
o he eigh MCMC chains A-H. Nex , we anked he
Bayes ac o s wi hin each chain and calcula ed a ma ke -
speci ic mean ank ac oss chains as a measu e o sum-
ma ize in o ma ion om he eigh chains. This was done
o inc ease he obus ness in assessing he s eng h o
e idence by making he esul s less dependen on he spe-
ci ic choices o he hype -pa ame e s in single MCMC
chains.
Fo each o hese 20 ma ke s, hei posi ion in he
genome, mino allele equency and dis ance o he clos-
es ue majo QTL a e gi en in Table 2 (c . Table one
o [18]). The mino allele equencies o he ue QTL
we e added as a e e ence. The able also p o ides he
pos e io means o 2 ln(BFm)a e aged ac oss chains as a
consensus measu e o e idence, he minimal and maxi-
mal means ac oss he chains, and he absolu e alues o
he e ec sizes (|βQ|) o he ue majo QTL as epo ed
in[18].He eweshouldno e ha ,in hecaseo asin-
gle alue o an e ec size, i is su icien o epo only
he absolu e alue, since he sign o he alue will depend
on he geno ype coding o he da a se . O cou se, ou
es ima es also depend on he geno ype coding. Ne e he-
less, we epo he signed pos e io means o he e ec
sizes, Epos (βm), om ou analysis, because he minima
and maxima om he eigh MCMC chains could ha e
opposi e signs – al hough his did no happen o he
20 ma ke s epo ed. Finally, he pos e io means o he
pe cen age o pheno ypic a iance explained a e gi en
in Table 2. They we e calcula ed by Epos (%PVE)=
2MAFm(1−MAFm)Epos β2
m/ a (Y). He e, MAFmis
he mino allele equency o ma ke m,Epos β2
m he
pos e io mean o β2
m,and a (Y)is as de ined abo e. No e
ha Epos (%PVE)a e es ima es o single ma ke s and
simply summing hem up does no yield an es ima e o
he en i e p opo ion o a iance explained by ma ke s, as
co a iances due o LD be ween ma ke s a e missed in his
sum. Howe e , he p opo ion o a iance accoun ed o
by he eg ession on ma ke s is cap u ed in ou es ima es
h2
M(see Table 1).
Iden i ica ion o ue QTL by Bayes ac o s and alse
posi i es
Twel e o he 15 majo ue QTL we e loca ed wi hin 5
cM om he ma ke s epo ed in Table 2. In he compa -
a i e s udy o six associa ion analyses, C ooks e al. [18]
conside ed a QTL o be iden i ied co ec ly i a posi i e
signal was epo ed wi hin 5 cM om he QTL. The mos
success ul s udy by Ledu e al. [41] de ec ed 11 ue majo
QTL (see Table ou in [18]). No s udy compa ed in [18]
iden i ied he ue majo QTL M7, whe eas we ound a
ma ke wi h a signal o associa ion wi hin 2.01 cM o ha
QTL. The only s udy iden i ying M9 was Ledu e al. [41],
who ound an associa ion wi h exac ly he same ma ke
as we did, namely a 60.1 cM on ch omosome 3. Ano he
QTL, M14 a 5.15 cM, was iden i ied by only one s udy:
Bink and an Eeuwijk [42] de ec ed a signal a 2.0 cM, bu
he ma ke we iden i ied a 4.2 cM is somewha close o
his QTL.
Th ee ue majo QTL, namely M5, M10 and M11, a e
absen om Table 2. M5 is e y close o M4, a 2.59 cM
om M4 a posi ion 30.00 cM on ch omosome 2. Each
o he six analyses compa ed in [18] iden i ied ei he M4
o M5 only. M10 a posi ion 3.2 cM on ch omosome 4
was iden i ied by all six s udies and explained 4% o he
pheno ypic a iance. I is he e o e qui e in iguing ha
ou esul s ega ding M10 con as so ma kedly. M11 was
iden i ied only by Cle eland and Deeb [43].
In he lis o he 20 ma ke s wi h he s onges sig-
nals in ou analysis, wo ma ke s we e mo e han 5 cM
om a majo ue QTL and would ha e been conside ed
alse posi i es in [18]: one o hem, a posi ion 54.1 cM
on ch omosome 3, was 5.9 cM om M9 (a 60.00 cM),
and he o he , a 85.9 cM on ch omosome 4, was loca ed
abou midway be ween M12 (a 76.06 cM) and M13
(a 96.49 cM).
Up o now, we ha e conside ed an a bi a y num-
be , namely 20, o ma ke s showing he s onges signals
o associa ion ac oss di e en MCMC chains. In many
empi ical s udies, a decision making ool is used o clas-
si y ma ke s in o wo g oups: ma ke s wi h ”signi ican ”
and ”non-signi ican ” QTL signals. Fo his pu pose, one
can apply a h eshold o , say, 10 o he a e age o 2 ln(BFm)
ac oss he chains when mul iple chains a e conside ed.
Six een o he ma ke s shown in Table 2 ul il his c i e-
ion and ou do no . In addi ion o he h ee ue majo
QTL men ioned abo e (M5, M10, M11), M14 would also
emain uniden i ied i his c i e ion was used. Mo eo e ,
he ma ke s a 54.1 cM on ch omosome 3 and a 85.9 cM
on ch omosome 4 would s ill be alse posi i es, wi h bo h
Bayes ac o s exceeding he h eshold.
Th ee o he six analyses compa ed in [18] p oduced no
alse posi i e signals. To achie e his le el o ype I e o ,
he h eshold has o be se o 12 in ou analysis. This would
esul in missing wo addi ional QTL (M7 and M8) and
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Table 2 The 20 ma ke s wi h he s onges signals o associa ion ac oss chains in he analysis o he QTLMAS XII da a
Ma ke Closes ue 2ln(BFm)|βQ|Epos (βm)%PVEQEpos (%PVEm)
majo QTL(a)
Ch Pos MAF MAF Name Dis a g min max a g min max a g min max
1 19.5 0.28 0.28 M1 0.50 30 28 32 0.62 0.60 0.59 0.61 3.5 3.4 3.2 3.5
1 40.1 0.09 0.07 M2 -0.10 15 9 21 0.56 -0.35 -0.46 -0.17 0.9 0.6 0.3 0.8
1 77.7 0.28 0.29 M3 -0.47 28 22 32 0.37 0.43 0.41 0.46 1.3 1.7 1.6 1.9
2 26.9 0.44 0.44 M4 0.51 12 10 15 0.35 0.22 0.15 0.28 1.4 0.9 0.5 1.3
2 28.2 0.24 0.44 M4 -0.79 12 7 18 0.35 0.20 0.09 0.33 1.4 0.6 0.3 1.2
2 48.2 0.38 0.40 M6 0.42 25 14 32 0.37 -0.41 -0.45 -0.36 1.5 1.8 1.5 2.2
2 72.9 0.11 0.18 M7 2.01 11 8 15 0.50 0.15 0.04 0.24 1.6 0.2 0.1 0.4
3 13.2 0.33 0.40 M8 1.71 11 7 18 0.30 0.14 0.03 0.28 1.0 0.4 0.1 0.9
3 14.8 0.39 0.40 M8 0.11 8 6 12 0.30 -0.04 -0.08 -0.01 1.0 0.1 0.0 0.3
3 54.1 0.27 0.07 M9 5.90 11 6 16 0.68 0.12 0.01 0.21 1.3 0.3 0.0 0.5
3 60.1 0.16 0.07 M9 -0.10 24 20 29 0.68 -0.39 -0.41 -0.37 1.3 1.0 0.9 1.1
4 75.7 0.05 0.41 M12 0.36 26 12 32 0.58 -0.72 -0.78 -0.58 3.7 1.1 0.9 1.3
4 76.4 0.46 0.41 M12 -0.34 30 28 32 0.58 0.64 0.61 0.67 3.7 4.6 4.2 5.1
4 85.9 0.18 0.41 M12 -9.84 10 9 12 0.58 0.12 0.02 0.19 3.7 0.3 0.0 0.5
4 96.4 0.27 0.19 M13 0.09 13 11 14 0.29 -0.19 -0.27 -0.06 0.6 0.5 0.1 0.8
4 96.6 0.18 0.19 M13 -0.11 9 4 16 0.29 0.11 0.02 0.28 0.6 0.3 0.0 0.7
4 98.3 0.23 0.19 M13 -1.81 9 5 13 0.29 -0.08 -0.20 -0.02 0.6 0.2 0.0 0.4
5 4.2 0.19 0.21 M14 0.95 8 6 11 0.18 -0.06 -0.15 -0.01 0.2 0.1 0.0 0.3
5 93.4 0.36 0.26 M15 0.10 30 28 32 0.75 -0.71 -0.73 -0.68 5.0 5.3 4.9 5.6
5 94.5 0.09 0.26 M15 -1.00 22 15 32 0.75 -0.50 -0.53 -0.48 5.0 1.0 0.9 1.1
(a)The h ee ue majo QTLs missing a e:
M5 on ch . 2 a pos. 30.00 (MAF=0.21, |βQ|=0.33, %PVEQ=0.8),
M10 on ch . 4 a pos. 3.21 (MAF=0.39, |βQ|=0.61, %PVEQ=4.0),
M11 on ch . 4 a pos. 36.93 (MAF=0.24, |βQ|=0.34, %PVEQ=1.0).
Ch = ch omosome, Pos = posi ion in cM om he s a o he ch omosome, MAF = mino allele equency, Dis = di ec ed dis ance in cM o a ma ke o he closes
ue majo QTL, 2ln(BFm)= pos e io mean o he 2×log- ans o med Bayes ac o in a o o ma ke associa ion, |βQ|and Epos (βm)= ueabsolu e alueandsigned
pos e io mean o he addi i e e ec size, espec i ely, %PVEQand Epos (%PVEm)= ue alue and pos e io mean o he pe cen age o a iance explained,
espec i ely.T ue alues a e aken om Table one in [18].
he o al numbe o de ec ed QTL would dec ease o nine.
One s udy (wi h no alse posi i es) de ec ed mo e QTL,
namely ha o Ledu e al. [41], wi h 11 QTL. Howe e ,
his s udy also exploi ed haplo ype in o ma ion.
Robus ness o ma ke -speci ic esul s
As shown in Table 2, he Bayes ac o s a ied a he li le
ac oss chains o some ma ke s and a lo o o he s: e.g.
he minimal and maximal 2ln- ans o med Bayes ac o s
we e 28 and 32, espec i ely, o he ma ke a 19.5 cM on
ch omosome 1, bu we e 4 and 16 o he ma ke a 96.6
cM on ch omosome 4. Thus, he la e ma ke showed
e y s ong e idence in one chain bu ”only” posi i e e i-
dence in ano he one, acco ding o he classi ica ion by
Kass and Ra e y [32].
To quan i y he obus ness among he eigh MCMC
chains, we calcula ed pai wise Spea man’s ank co -
ela ion coe icien s ρbe ween he chains o he 20
Bayes ac o s epo ed in Table 2 (see he uppe igh
iangle in Table 3). When compa ing chains wi h iden i-
cal p io speci ica ions, he s onges pai wise ag eemen
was obse ed be ween chains A and B (p0=0.99 and
b=0.01), wi h a co ela ion o 0.99, and he weak-
es ag eemen be ween chains C and D (p0=0.999
and b=0.001), wi h a co ela ion equal o 0.84. Fo
chains wi h di e en p io speci ica ions, he co ela ion
coe icien ob ained i s lowes alue, 0.67, be ween chains
C and E, which di e in bo h p0and b.
We also epo he a ios o he 2 ×log- ans o med
Bayes ac o s a e aged ac oss he 20 ma ke s o pai s o
chains in he lowe le iangle o Table 3. These mean
a ios gi e an indica ion o he di e ences in magni ude
o he Bayes ac o s be ween he chains. On a e age,
chains A, B and C yielded he la ges Bayes ac o s o
abou he same magni ude. The la ges di e ences in
Bayes ac o s we e obse ed be ween chains A and H and
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doi:10.1186/1297-9686-45-24
Ci e his a icle as: Knü e al.:Impac o p io speci ica ions in a
sh inkage-inducing Bayesian model o quan i a i e ai mapping and
genomic p edic ion. Gene ics Selec ion E olu ion 2013 45:24.
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