RESEARCH Open Access
Allele coding in genomic e alua ion
Ismo S andén
1*
and Ole F Ch is ensen
2
Abs ac
Backg ound: Genomic da a a e used in animal b eeding o assis gene ic e alua ion. Se e al models o es ima e
genomic b eeding alues ha e been s udied. In gene al, wo app oaches ha e been used. One app oach es ima es
he ma ke e ec s i s and hen, genomic b eeding alues a e ob ained by summing ma ke e ec s. In he second
app oach, genomic b eeding alues a e es ima ed di ec ly using an equi alen model wi h a genomic ela ionship
ma ix. Allele coding is he me hod chosen o assign alues o he eg ession coe icien s in he s a is ical model. A
common allele coding is ze o o he homozygous geno ype o he i s allele, one o he he e ozygo e, and wo
o he homozygous geno ype o he o he allele. Ano he common allele coding changes hese eg ession
coe icien s by sub ac ing a alue om each ma ke such ha he mean o eg ession coe icien s is ze o wi hin
each ma ke . We call his cen e ed allele coding. This s udy conside ed e ec s o di e en allele coding me hods
on in e ence. Bo h ma ke -based and equi alen models we e conside ed, and es ic ed maximum likelihood and
Bayesian me hods we e used in in e ence.
Resul s: Theo e ical de i a ions showed ha pa ame e es ima es and es ima ed ma ke e ec s in ma ke -based
models a e he same i espec i e o he allele coding, p o ided ha he model has a ixed gene al mean. Fo he
equi alen models, he same esul s hold, e en hough di e en allele coding me hods lead o di e en genomic
ela ionship ma ices. Calcula ed genomic b eeding alues a e independen o allele coding when he es ima e o
he gene al mean is included in o he alues. Reliabili ies o es ima ed genomic b eeding alues calcula ed using
elemen s o he in e se o he coe icien ma ix depend on he allele coding because di e en allele coding
me hods imply di e en models. Finally, allele coding a ec s he mixing o Ma ko chain Mon e Ca lo algo i hms,
wi h he cen e ed coding being he bes .
Conclusions: Di e en allele coding me hods lead o he same in e ence in he ma ke -based and equi alen
models when a ixed gene al mean is included in he model. Howe e , eliabili ies o genomic b eeding alues a e
a ec ed by he allele coding me hod used. The cen e ed coding has some nume ical ad an ages when Ma ko
chain Mon e Ca lo me hods a e used.
Backg ound
The e has been g owing in e es in he use o ma ke -
based models [1] in ecen yea s. In s udies using hese
models, desc ip ions o he e ec o allele coding sys em
on in e ence and compu a ions a e o en ague o missing.
By allele coding, we e e o he coe icien s in he ma ke
ma ix o ma ke -based models. Coe icien s, commonly
used o allele coding o a ma ke is 0 when he indi idual
is homozygous o he i s allele, 1 when he indi idual is
he e ozygous, and 2 when he indi idual is homozygous
o he second allele. Depending on which o he alleles
has been chosen as he i s allele, he coe icien s a e
di e en . Thus, his allele coding me hod does no gi e
unique eg ession coe icien s.
The e a e o he allele coding me hods such as he one
ha use coe icien s -1, 0, and 1 ins ead o 0, 1, and 2,
espec i ely. Di e en allele coding me hods a ec coe i-
cien s in he s a is ical models bu hey do no seem o
change he amoun o in o ma ion o s a is ical in e ence.
Hence, one would expec ha he use o di e en allele
coding me hods would lead o he same in e ence. How-
e e , allele coding can be o i al impo ance in compu a-
ions. Fi s , con e gence o i e a i e me hods such as
Ma ko chain Mon e Ca lo (McMC) me hods o en used
in Bayesian in e ence can be assumed o be a ec ed by he
allele coding me hod used because di e en allele coding
me hods change he co ela ion s uc u e be ween ma ke
* Co espondence: [email p o ec ed]i
Full lis o au ho in o ma ion is a ailable a he end o he a icle
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Selec ion
E olu ion
© 2011 S andén and Ch is ensen; 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.
e ec s. Second, equi alen models ha e become popula in
animal b eeding [2,3]. An impo an concep in hese
me hods is he genomic ela ionship ma ix. Di e ences in
allele coding will yield di e en genomic ela ionship
ma ices [4]. Thus, some elemen s o he in e se o
he coe icien ma ix can be di e en and, consequen ly,
eliabili ies may be di e en .
We in es iga ed e ec s o di e en allele coding me h-
ods using heo e ical de i a ions and a p ac ical example
when es ic ed maximum likelihood (REML) o Baye-
sian in e ence is used. E ec s on pa ame e es ima es,
eliabili ies, and McMC compu a ions we e s udied. We
conside ed ma ke models and hei equi alen b eeding
alue models.
Me hods
Genomic ma ke model
Le us conside a uni a ia e linea mixed e ec s model
o genomic ma ke e ec es ima ion, e.g. [1],
y
=1μ+Zg +e
,
(1)
whe e yis n× 1 ec o o obse a ions, μis he gen-
e al mean, Zis a n×mma ix con aining a column o
each ma ke locus, gis a m× 1 ec o o andom SNP
ma ke e ec s, and eis a andom esidual ec o . The e
can be o he ixed o andom e ec s in he model bu
hei inclusion does no change he ollowing
de i a ions.
Allele coding me hods
The e a e se e al al e na i es o coding he coe icien s
in he Zma ix. Fou allele coding sys ems a e consid-
e ed. A simple ans o ma ion can be made om one
allele coding sys em o ano he . Ou basic allele coding
sys em coun s he numbe o copies o one o he
alleles. Depending on which o he alleles is coun ed,
he ma ix can be di e en . In he allele coding sys em
012, he numbe o copies o he less equen allele is
coun ed. Thus, he coe icien is 0 i he indi idual is
homozygous o he mo e equen allele, 1 i i is he -
e ozygous, o 2 i i is homozygous o he less equen
allele. In his case, he Zma ix o he basic allele cod-
ing sys em 012 is deno ed by Z
0
.
A gene al o m o he allele coding ans o ma ion
om he basic allele coding sys em is Z0−1n 
m
whe e
m
is a m× 1 ec o . This allows many ypes o allele
coding me hods. No e ha he ans o ma ion keeps dis-
ances be ween allele codes wi hin a ma ke he same.
So, he coding 0,1,2 can be changed o - 1,0,1 o 0.5,
1.5, 2.5 by his ans o ma ion, bu no o -10,0,10.
We de ine he cen e ed allele coding sys em as Z
c
=
Z
0
-P
c
, whe e each column o he ma ix P
c
con ains
he a e age allele coun o he co esponding ma ke
column. Thus, summing alues in each column will gi e
a ec o o ze os, i.e., 1
n
(Z0−Pc
)
is a ec o o ze os.
Fo he cen e ed allele coding sys em, we ha e
m=1
n
Z
01
n
, i.e., Zc=Z0−1
n
1n1
nZ
0
.No e ha
m
/2
gi es he allele equencies o he ma ke s in he da a.
The allele coding ans o ma ion allows shi s in he
allele codes. The 101 allele coding sys em is such ha -
1 is assigned o he geno ype homozygous o he mo e
equen allele, 0 o he he e ozygous indi idual, and 1
o he indi idual homozygous o he less equen allele.
Fo he 101 allele coding sys em, we ha e
m
=1
m
.The
101 allele coding sys em is equal o he cen e ed allele
coding sys em when all allele equencies a e equal o
0.5.
In he ollowing, he de i a ions will use he gene al
allele coding ans o ma ion. The ma ix Z
0
is unique.
Howe e , in gene al he decision on which o he alleles
o coun is a bi a y. Le he 210 allele coding sys em be
such ha he mo e equen allele is coun ed. Then he
Zma ix o he 210 allele coding sys em can be calcu-
la ed om he 012 coding ma ix by 21n1
m
−Z
0
whe e
1
n
is n× 1 ec o o ones. The 210 allele coding sys em
is he opposi e o he 012 allele coding sys em bu
esul s in his pape apply o he 210 allele coding sys-
em as well, wi h some modi ica ions men ioned
sepa a ely.
Di e en allele coding me hods imply di e en models
(1) and di e en models may lead o di e en pa ame e
es ima es. Howe e , he in e ence conside ed in his
pape is no a ec ed by allele coding, as we demons a e
below.
In e ence in ma ke -based model
Va iance componen es ima ion by es ic ed maximum
likelihood (REML), p edic ion o andom e ec s, and
Bayesian in e ence a e all based on he likelihood a e
ma ginaliza ion o he ixed e ec s, i.e., he ixed e ec s
ha e been in eg a ed ou . Bayesian in e ence equi es
e en mo e ma ginaliza ion. In o de o show ha in e -
ence is no a ec ed by allele coding, i is su icien o
show ha he likelihood a e in eg a ing ou he gene al
mean is he same i espec i e o allele coding. The ol-
lowing de i a ion makes no assump ions on he p io
densi ies o ma ke e ec s. Thus, he esul s apply o
many models, including BLUP, BayesA and BayesB in
[1].
The ma ginal likelihood o he mixed e ec s model is
p(y|g,θ)=
∞
∫
−
∞
p(y|μ,g,θ)dμ
,
whe e p(y|μ,g,θ) is he condi ional densi y o y,
o en a Gaussian densi y, and θcon ains all pa ame e s
in he dis ibu ion o e, o en only he esidual a iance
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pa ame e σ
2
e
. Using he ans o ma ion esul (7) in
Appendix A and a change o in eg a ion a iable
μ0=μ− 
mg
, we can w i e
p(y|g,θ)=
∞
∫
−∞
p0(y|μ− 
mg,g,θ)d
μ
=
∞
∫
−∞
p0(y|μ0,g,θ)dμ0
=p0
(
y|g,θ
)
,
whe e p
0
deno es he 012 allele coding sys em. Hence,
he ma ginal likelihood does no depend on allele cod-
ing, a p ope y used in he ollowing de i a ions.
The REML-likelihood is de ined when gand ea e
mul i a ia e Gaussian dis ibu ed, and equals
L
(
θ,η
)
=∫p
(
y|g,θ
)
p
(
g|η
)
dg
,
whe e hcon ains all pa ame e s in he dis ibu ion o g,
commonly only he gene ic ma ke a iance pa ame e
σ
2
g
. This likelihood is independen o allele coding, and,
hence, REML pa ame e es ima ion is independen o
allele coding. No e ha maximum likelihood es ima ion
is based on L(μ,θ,h)=∫p(y|μ,g,θ)p(g|h)dgand is
a ec ed by allele coding because, in his case, he gene al
mean is no in eg a ed ou .
BLUP es ima ion o ma ke e ec s gassumes ha he
a iance pa ame e s (θ,h) a e known. The condi ional
dis ibu ion p(g|y,θ,h)=p(y|g,θ)p(g|h)/L(θ,h)is
independen o allele coding. Hence, BLUP ĝand asso-
cia ed unce ain ies do no depend on he allele coding.
In Bayesian in e ence, he join pos e io a e in e-
g a ing ou μis
p(g,θ,η|y)=∫p(μ,g,θ,η|y)dμ
=p(y|g,θ)p(g|η)p(θ,η)

p(y),
whe e p(θ,h) is he join p io o he pa ame e s,
and he denomina o p(y) is he in eg al o he nume a-
o . All e ms in he nume a o a e independen o allele
coding, and by ma ginaliza ion p(y) sa is ies he same.
Hence, p(g,θ,h|y) does no depend on allele coding.
The gene al in e cep μis, howe e , no independen
o allele coding. Fo simplici y o he a gumen , we
assume ha pa ame e s (θ,h)a eknown,andomi
showing hese alues. Acco ding o he ans o ma ion
esul (8) in Appendix A and a change o in eg a ion
a iable μ0=μ− 
mg
, he condi ional expec a ion o
he gene al mean is
ˆμ=∫μ∫p(μ,g|y)dgdμ
=∫∫μp0(μ− 
mg,g|y)dμdg
=∫∫(μ0+ 
mg)p0(μ0,g|y)dμ0dg
=∫μ0p0(μ0|y)dμ0+∫ 
mgp0(g|y)d
g
=ˆμ0+ 
m
ˆ
g,
(2)
whe e p
0
deno es densi y o he 012 allele coding,
and ˆ
μ0
is he condi ional expec a ion o he gene al
mean when using he 012 allele coding. Thus, he gen-
e al mean es ima e is di e en by allele coding when

m
ˆ
g
is no ze o. When gand ea e mul i a ia e Gaus-
sian dis ibu ed, he condi ional expec a ions ĝand ˆ
μ
equal he BLUP and BLUE es ima es, espec i ely.
Finally, he in e ence is indi e en o he allele being
coun ed. This is demons a ed by s udying he cen e ed
coding sys em and assuming ha allele in he i s ma -
ke is coun ed in he opposi e way, i.e., he i s column
in Zis minus he i s column in Z
c
,o z
1
=-z
c1
.We
see ha Z
g
=Zc
˜
g
whe e he en ies in
˜
g
a e equal o he
en ies in g, excep o he i s en y which equals
minus he i s en y in g.Sincegand ˜
g
ha e he same
dis ibu ion, hese wo models a e equi alen .
Genomic b eeding alues
Es ima ing b eeding alues
In b eeding alue e alua ion, he main in e es is in es i-
ma ion o genomic b eeding alues o he geno yped
animals. In o he wo ds, es ima ion o ˆ
a=Zˆ
g
whe e ĝ
a esolu ions o hema ke e ec sbyama ke -based
model like model (1). Because he ma ke e ec solu-
ions a e he same o di e en allele coding sys ems,
he es ima ed genomic b eeding alues a e di e en due
o di e ences in he coe icien ma ix Z.
Allele coding does no , howe e , change ela i e di -
e ences be ween he es ima ed genomic b eeding
alues, because Zˆ
g−Z0ˆ
g=−1n( 
m
ˆ
g
)
shows ha hey
a e jus shi ed by a cons an . Le us de ine comple e
genomic b eeding alues as ˆ
ad=1nˆμ+Zˆ
g
. Subs i u ing
Z=Z0−1n 
m
and using equa ion (2) we ob ain
ˆ
ad=1nˆμ+(Z0−1n 
m)ˆ
g
=1n(ˆμ− 
mˆ
g)+Z0ˆ
g
=1nˆμ0+Z0ˆ
g
.
Consequen ly, he es ima ed comple e b eeding alues
ˆ
a
d
a e he same i espec i e o allele coding.
Equi alen model and allele coding
Assume ha he ma ke e ec s ha e a Gaussian dis i-
bu ion g∼N(0,Imσ2
g)
whe e I
m
is an m×miden i y
ma ix. The b eeding alues a=Zg can be calcula ed
di ec ly wi hou es ima ing ĝby he model [2,3]
y
=1nμ+a+e
,
whe e he b eeding alues ha e p io densi y o
a∼N(0,ZZ’σ2
g)
.O en heco a iancema ixo he
b eeding alues is scaled by a alue such as
p=2
m

i
=1
pi(1 −pi
)
whe e p
i
is he allele equency o
ma ke i. Then, he b eeding alues ha e a p io densi y
o a∼N(0,Gσ
2
a)
whe e he genomic ela ionship ma ix
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is G=ZZ’ 1
p
and gene ic a iance is σ2
a=σ2
g
p
. Assuming
ha he esidual dis ibu ion is e~N(0,R), hen he
mixed model equa ions o he equi alen model a e

1
nR−
1
1n1nR−
1
R−11nR−1+G−1/σ2
aˆμ
ˆ
a

=

1
nR−1y
R−1y

.
(3)
The b eeding alue solu ions â om hese mixed
model equa ions (3) a e he same as he genomic b eed-
ing alues calcula ed by ˆ
a=Zˆ
g
whe e ˆ
g
a e ma ke
e ec s es ima ed by he ma ke -based model (1).
The e o e, he conclusion o he ma ke -based models
abou ela i e di e ences be ween genomic b eeding
alues being una ec ed by allele coding is also ue o he
equi alen model, al hough di e en allele coding me hods
lead o di e en genomic ela ionship ma ices. Simila ly,
a iance componen es ima ion by REML and Bayesian
me hods a e una ec ed by he allele coding due o equi a-
lence o models wi h he ma ke -based models.
The mixed model equa ions in (3) a e no well-de ined
when he genomic ela ionship ma ix Gis singula .
Howe e , mixed model equa ions no equi ing an
in e ible Gma ix do exis ; see page 48 in [5]. The
genomic ela ionship ma ix Gcan be singula o se -
e al easons. Fo example, he e can be iden ical wins
o clones ha ha e he same geno ypes. In addi ion, o
he cen e ed allele coding sys em he genomic ela ion-
ship ma ix is Gc=ZcZ
c
.Thelas owo
Zc=(I−
1
n
1n1
n)Z
0
is equal o he sum o all he o he
ows. Hence, Z
c
is no o ull ank, and G
c
is singula .
P edic ion e o a iances and eliabili ies
Gaussian models a e o en used in p ac ical genomic
e alua ion o animals. In hese models, eliabili ies o
es ima ed b eeding alues âa e calcula ed using ele-
men s o he in e se o he mixed model equa ions such
as (3). Reliabili y o â
i
is
2
i=1−PEVi
σ2
a
Gii
,
(4)
whe e PEV
i
is he p edic ion e o a iance, i.e., Va (a
i
|
y), o animal i, and G
ii
is he diagonal elemen o animal i
in he genomic ela ionship ma ix G; e.g. [6], p. 51 in [7].
The p edic ion e o a iance o animal iis he diagonal
elemen o he in e se o he coe icien ma ix o mixed
model equa ions (3) o animal i. Al e na i ely,
PEV = Va (a|y)
=Va (Zg|y)
=ZVa (g|y)Z
’
=ZCgZ’
,
(5)
whe e C
g
is he genomic ma ke e ec subma ix in he
in e se o he coe icien ma ix o he mixed model
equa ion o ma ke -based model (1) (see Appendix B).
The subma ix C
g
=Va (g|y) is he same i espec i e o
he allele coding me hod used as shown in he chap e
on in e ence on ma ke -based models. Because he coe -
icien ma ix Zis di e en depending on allele coding,
PEV is also di e en depending on allele coding. Conse-
quen ly, he eliabili y o âdepends on allele coding.
Mo e gene ally, o any o he models conside ed in
his pape , a=Zg whe e p(g|y) is independen o allele
coding and Zdepends on allele coding. The e o e, he
dis ibu ion p(a|y) and, in pa icula , he a iance-co -
a iance ma ix Va (a|y) and eliabili ies o âdepend
on allele coding.
Thecomple eb eeding aluedis ibu ionp(a
d
|y)
does no depend on allele coding, unlike PEV associa ed
wi h â. The p oo is based on he demons a ion ha
when applying any unc ion , he expec a ions a e inde-
penden o he allele coding sys em,
E[ (ad)|y]
=∫∫ (1nμ+Zg)p(μ,g|y)dμdg
=∫∫ (1n(μ− 
mg)+Z0g)
×p0(μ− 
mg,g|y)dμdg
=∫∫ (1nμ0+Z0g)p0(μ0,g|y)dμ0d
g
=E
0[
(
ad
)
|
y
],
whe e E
0
is he expec a ion when using he basic allele
coding me hod. The e o e, he a iance-co a iance
ma ix Va (a
d
|y) and all highe o de momen s o he
dis ibu ion a e independen o allele coding. Howe e ,
he esul does no p o ide ac ual o mulas o he
momen s.
A closed o m o mula o he a iance-co a iance
ma ix is de i ed o a Gaussian model. Assume
g∼N(0,Imσ2
g)
and e~N(0,R). Fo his model, in
Appendix B we ob ain ha
Va (ad|y)= 1n1
n
+(In− 1n1
n
R−1)ZCgZ’(In− R−11n1
n
)
,
whe e =1/(1
n
R−
1
1n
)
and
C
g=[Z’(R−1− R−11n1
nR−1)Z+Im/σ2
g
]−
1
.Asdemon-
s a ed ea lie in his sec ion, his a iance-co a iance
ma ix is independen o allele coding. When
R
=Inσ
2
e
,
he a iance-co a iance ma ix simpli ies o
Va (ad|y)
=1n1
nσ2
e/n+Zc(Z
cZc/σ2
e+Im/σ2
g
)−1Z
c
,
(6)
whe e Z
c
is based on he cen e ed allele coding
me hod. The diagonal elemen s in (6) a e di e en om
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he PEV
i
s in (4) because hey con ain unce ain y abou
he unknown mean μas well.
Fo he comple e b eeding alues a
d
,weha eshown
abo e ha p edic ion e o a iances a e independen o
allelecodingandweha ep o ideda o mula o he
Gaussian model. Reliabili ies o â
d
, howe e , can no be
de ined in a meaning ul way. Subs i u ing he diagonal
elemen s om (6) o he PEV
i
s in (4) is no app op ia e
since he denomina o in (4) is Va (a
i
)no Va (a
d
)
ii
.
The denomina o in he eliabili y o mula should con-
ain he ma ginal (uncondi ional) a iance Va (a
d
)
ii
=∫
Va (μ+a
i
|μ)dμ, bu his a iance is in ini e.
McMC compu a ions
Theo e ical con e gence a e
The con e gence and mixing o an McMC algo i hm
depend on he pa ame iza ion o he model and on he
algo i hm used. Theo e ical esul s abou he geome ic
a e o con e gence o he s a iona y dis ibu ion o
Gibbs sampling algo i hms a e shown in [8], and as men-
ioned in [9] his a e also desc ibes he mixing o he
algo i hm. Below we show speci ic esul s abou he con-
e gence a e o Gibbs sampling algo i hms o simu-
la ing om [μ,g|y] in he ma ke -based model (1),
whe e gis Gaussian g∼N(0,Imσ2
g)
and eis Gaussian
e∼N(0,Inσ2
e)
. These esul s p o ide some ideas abou
mo e gene al models and algo i hms whe e heo e ical
esul s canno be ob ained.
Sec ion 2.2 in [8] con ains esul s abou a ious
Gibbs-sampling schemes o simula ing om a mul i-
a ia e dis ibu ion. He e we apply hese esul s (see
Appendix C o de ails) o wo ypes o Gibbs upda ing
schemes. The i s scheme i e a es be ween upda ing μ
and a block o all componen s in g, and will be called
he block upda ing scheme he eina e . The second
scheme upda es μ,g
1
,...,g
m
sequen ially one a a ime,
and will be called he single si e upda ing scheme.
Fo he block scheme, he con e gence a e is
ρ1=n
σ2
e
¯zC−1
g¯z
,
whe e ¯z=Z’1n
/n
is a m×1 ec o ,and
C
g=Z’Z/σ2
e+Im/σ
2
g
.
Fo he single si e scheme, he con e gence a e is
ρ2=ρl (L−1
g
(D−1¯z¯zn/σ2
e−Ug)
,
whe e L
g
is he ma ix con aining he lowe iangle
and he diagonal o D
-1
C
g
,Dis he diagonal o C
g
,and
U
g
is he ma ix con aining he uppe iangle o D
-1
C
g
.
This single si e Gibbs sampling algo i hm is he s ochas-
ic coun e pa o he Gauss-Seidel algo i hm o sol ing
he mixed model equa ions, and he con e gence esul s
a e simila , see [10].
When he cen e ed allele coding me hod
Zc=(In−1n1
n
/n)Z0is used, ¯
z
is a ec o o ze os
and, hence,
1
= 0. The cen e ed allele coding me hod
b eaks dependency be ween he gene al mean and
gene ic ma ke e ec s, as seen om he a iance-co -
a iance ma ix (de i ed in Appendix B o a mo e gen-
e al si ua ion)
Va (μ,g|y)
=σ2
en0
0(Z
cZcσ2
e+Im

σ2
g)
−1
.
Consequen ly, abso p ion o he gene al mean is done
wi hou needing o compu e abso p ion explici ly. No e
ha , in gene al, his holds only when he esidual a -
iance-co a iance ma ix is Iσ
2
e
. Fo he block McMC
scheme, he con e gence and mixing o he algo i hms
a e o he same o de as o non-McMC algo i hms ha
simula e di ec ly om he dis ibu ion o in e es . Fo
he single si e McMC scheme, he cen e ed allele coding
me hod s ill b eaks he dependence be ween he gene al
mean and ma ke e ec s, bu as
2
>0 illus a es, he
indi idual ma ke e ec s g
1
, ..., g
m
a e no independen
and he McMC samples a e au oco ela ed.
Da a and me hods
Da a
Da a o he XII
h
QTLMAS wo kshop [11] we e used o
illus a e he heo y. The simula ed da a had ou gen-
e a ions. In each gene a ion, 15 si es and 150 dams
we e selec ed andomly o p oduce he nex gene a ion.
Each si e was ma ed o 10 dams and each ma ing p o-
duced 10 p ogeny. Thus, he base gene a ion had 165
indi iduals, and he subsequen h ee gene a ions, 1500
indi iduals each. In o al, he analyzed da a had 4665
animals wi h pheno ypes. The simula ed ai had a he -
i abili y o 0.30. The da a had 6000 equally spaced SNP
ma ke s on six ch omosomes. We dele ed ma ke s ha
had a mino allele equency less han 1% among he
pheno yped indi iduals and his educed he numbe o
ma ke s o 5896.
The 012 allele coding me hod was used o make he
base da a se . So, he leas equen allele was coun ed.
In addi ion, 210, 101, and cen e ed allele coding da a
se s we e analyzed.
Va iance componen analysis
The ma ke -based model (1) wi h common gene ic a -
iance was used o analyze he da a: e∼N(0,Iσ2
e)
and
g∼N(0,Iσ
2
g)
.McMCcompu a ionsbyasinglesi e
upda ing Gibbs sample we e used o calcula e pos e io
mean es ima es o he loca ion (μ,g) and dispe sion
(σ2
g
,σ2
e
)
pa ame e s. The leng h o he McMC chain was
100000 i e a ions, o which he bu n-in pe iod o 10000
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was omi ed. E e y en h sample was sa ed, gi ing 9000
sa ed samples. E ec i e sample sizes we e calcula ed o
all pa ame e s using he ini ial mono one sequence
app oach [12]. The app oach es ima es he numbe o
independen samples om he pos bu n-in samples.
Theo e ical con e gence a es we e calcula ed o all he
allele coding me hods in he single si e and block upda ing
schemes. He e, he con e gence a e also desc ibes he
mixing o he McMC chain and was measu ed by he co -
ela ion be ween successi e McMC samples. Because o
he Ma ko p ope y, he k-lag co ela ion is
k
, whe e is
he con e gence a e and kis he lag o dis ance be ween
samples. In o de o compa e he heo e ical con e gence
a e o he obse ed e ec i e sample size, heo e ical mix-
ing was calcula ed ela i e o he 012 allele coding sys em
as ollows. Le
012
be he con e gence a e om he 012
allele coding sys em, and he con e gence a es om
ano he allele coding sys em. Mixing o he o he allele
coding sys em is equal o mixing o he 012 allele coding
sys em when e e y k
h
sample is aken om he McMC
samples and
012
=
k
. Thus, he ela i e mixing is k= log
(
012
)/log( ).
Pa ame e es ima ion by REML o bo h he ma ke -
based model and he equi alen model and o all ou
allele coding me hods was done using so wa e DMU
[13]. Bo h AI-REML and EM-REML we e in es iga ed.
Fo he equi alen model and he cen e ed allele coding
me hod, he singula genomic ela ionship ma ix was
modi ied by mul iplying he diagonals by 1.001 in o de
o be able in e he ma ix. The e ec o allele coding
on con e gence was in es iga ed, and i was checked
ha he pa ame e es ima es we e he same.
Reliabili ies o genomic b eeding alues
Reliabili ies we e calcula ed by di e en allele coding
me hods using (4) and elemen s o he in e se o he
mixed model equa ions. The a iance componen s we e
hose es ima ed by he cen e ed allele coding me hod
using he single si e McMC app oach. P edic ion e o
a iances we e calcula ed by bo h he ma ke -based
model equa ions (5) and equi alen model equa ions (3).
Mean, minimum, maximum and s anda d de ia ions o
eliabili ies we e calcula ed o all allele coding me hods.
Also, co ela ions be ween eliabili ies om all allele
coding me hods we e calcula ed.
Resul s and Discussion
Pos e io mean es ima es o ma ke e ec s and a iance
componen s we e almos equal be ween di e en allele
coding me hods (Table 1). Co ela ions o es ima es o
he ma ke e ec s be ween allele coding me hods we e
highe han 99.98%. Only he gene al mean (μ)hada
di e en es ima e, as expec ed. The es ima ed a iance
componen s ag eed well wi h hose used o simula e he
da a. Addi i e gene ic a iance was ˆσ2
a=ˆσ2
g
p=1.55
6
,
whe e p=2
m

i
=1
pi(1 −pi) = 2323.0
0
,andp
i
a e he
obse ed allele equencies in he e e ence da a. Thus,
he he i abili y es ima e was ˆ
h2=ˆσ2
a
/( ˆσ2
a
+ˆσ2
e
)=0.3
4
,
compa ed o he simula ed alue o 0.30.
E ec i e sample sizes di e ed depending on allele
coding me hod. The cen e ed allele coding me hod had
he bes mixing, and he 210 allele coding me hod had
he wo s (Table 2). In pa icula , he inc ease in e ec-
i e sample size was la ges o he gene al mean. Fo
he cen e ed allele coding me hod, he gene al mean
was independen om he ma ke e ec g, which led o
excellen mixing o his pa ame e . In gene al, he ma -
ke e ec s showed excellen mixing. Wi h all allele cod-
ing me hods, e ec i e sample sizes we e a leas 5500
o all ma ke e ec s, and on a e age we e equal o
abou 8800.
Theo e ical con e gence a es (Table 3) displayed he
same esul s as he e ec i e sample sizes discussed
abo e. No e ha ou Gibbs sample used single si e
upda es o all pa ame e s. Fo he single si e upda ing
algo i hm, he 210 allele coding sys em was p edic ed o
need 5.64 imes mo e i e a ions han he 012 allele cod-
ing sys em. The numbe o e ec i e samples o he
gene al mean pa ame e (μ) was 5.11 imes bigge o
he 012 han o he 210 allele coding sys em. These ig-
u es we e 0.48 and 0.48 o he 101 allele coding sys em,
and 0.070 and 0.0051 o he cen e ed allele coding sys-
em. Theo e ical con e gence a es o he block Gibbs
sample showed he same pa e n as o he single si e
upda e (Table 3).
Su p isingly, he block Gibbs sample was p edic ed o
be wo se han he single si e Gibbs sample o all allele
coding sys ems excep o he cen e ed allele coding sys-
em. Howe e , i is well known in he li e a u e ha
block-upda ing schemes may some imes be wo se han
Table 1 Pos e io means o selec ed pa ame e s by allele
coding
Allele coding
Pa ame e 012 210 101 cen e ed
μ1.698 0.801 1.083 1.359
σ2
g
6.700 × 10
-4
6.703 × 10
-4
6.691 × 10
-4
6.698 × 10
-4
σ2
e
2.996 2.996 2.996 2.996
Table 2 E ec i e sample sizes in McMC compu a ions by
allele coding
Allele coding
Pa ame e 012 210 101 cen e ed
μ46 9 96 8961
σ2
g
723 330 1001 1701
σ2
e
7814 6861 7661 7720
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single si e upda ing schemes, o examples see [8]. The
excellen con e gence a e o he block Gibbs sample
wi h cen e ed allele coding was expec ed because, in
his case, he Gibbs sample is equal o Mon e Ca lo
sampling.
Table 4 shows con e gence o REML in pa ame e es i-
ma ion. Fo he ma ke -based model, he con e gence
was independen o he allele coding sys em, whe eas o
he equi alen model, he con e gence was as es o he
cen e ed coding sys em and slowes o he 210 coding
sys em, al hough he di e ences we e small. The pa a-
me e es ima es ob ained (Table 5) we e he same, wi h
he excep ion o σ
2
e
o he cen e ed coding sys em. The
di e ence is due o he need o make he genomic ela-
ionship ma ix G
c
o be ull ank by mul iplica ion o he
diagonals by 1.001. In summa y, REML pa ame e es i-
ma ion is only sligh ly a ec ed by allele coding.
Reliabili ies we e a ec ed depending on he allele coding
me hod used. Di e ences we e la ge (Table 6). A e age
eliabili ies anged om 0.37 wi h he 210 allele coding
me hod o 0.80 wi h he cen e ed allele coding me hod.
The cen e ed allele coding me hod ga e highe eliabili ies
han achie ed by any o he o he allele coding me hods.
Reliabili ies calcula ed by di e en allele coding me hods
we e also di e en as judged by he co ela ion o each
o he (Table 7). Reliabili ies calcula ed by he ma ke -
based model and he equi alen model app oaches we e
equal wi hin he nume ical ounding e o .
The obse ed la ge di e ences in eliabili ies using di -
e en allele coding me hods can be explained by di e -
ences in es ima ion unce ain y. Di e en allele coding
sys ems ha e di e en Zma ices. Conside i s he 012
and 210 allele coding me hods. The 012 allele coding sys-
em has a 0 coe icien when he indi idual is homozygous
o he mo e equen allele while he 210 allele coding
sys em has a coe icien o 2 ins ead. In he ma ke -based
model, unce ain y o he in e se o he coe icien ma ix
is he same i espec i e o allele coding me hod. Reliabili y
is calcula ed by mul iplying he ma ke unce ain y by he
Zma ix (5). Consequen ly, unce ain y is less in he 012
allele coding sys em han in he 210 allele coding sys em
because he mo e equen homozygous allele mul iplies
he ma ke solu ion and unce ain y by ze o. Thus, homo-
zygous geno ypes o he mo e equen allele do no
inc ease unce ain y when es ima ing genomic b eeding
alues. Thus, he 012 allele coding sys em will yield highe
eliabili ies han he 210 allele coding sys em, as was
obse ed. This a gumen can be gene alized as ollows. In
he genomic model conside ed, unce ain y o a geno ype
in es ima ing genomic b eeding alue is alued ela i e o
a chosen base geno ype. The u he away an obse ed
geno ype is om he base geno ype, he la ge he coe i-
cien in absolu e alue in he Zma ix and he highe he
unce ain y in genomic b eeding alue. In he 012 allele
coding sys em, he base geno ype is homozygous o he
mo e equen allele, while in he 210 allele coding, i is
homozygous o he less equen allele. In he 101 allele
coding sys em he base geno ype is he he e ozygo e. The
highe he numbe o he e ozygous indi iduals is in he
da a he smalle will he unce ain y be, i.e., he highe he
eliabili y will be o he 101 allele coding sys em. Fo he
cen e ed allele coding sys em he base geno ype is he
a e age geno ype in he da a. Thus, o his allele coding
sys em he base popula ion is oughly he popula ion we
wo k wi h [14], and i has he smalles a e age dis ance o
obse ed geno ypes om he base geno ype. In p ac ice,
his can be expec ed o lead o he highes eliabili ies.
Di e en allele coding sys ems ha e di e en model
design ma ices Z, and, hence, imply di e en models.
Thus, eliabili ies om di e en allele coding sys ems a e
in ac om di e en s a is ical models. Compa ison o
eliabili ies om di e en models is meaningless. How-
e e , he di e en allele coding sys ems lead o he same
pa ame e es ima es. I he co ec allele coding me hod,
i.e., s a is ical model, is known, i should be used. Because
he ue model is unknown and compa ison o eliabili ies
by allele coding me hod is meaningless, some p inciples
mus be used o decide on which allele coding me hod
should be used. These p inciples will no gua an ee he
use o a co ec model o co ec eliabili ies. One such
p inciple should be consis ency o eliabili ies be ween
e alua ions. The cen e ed allele coding me hod changes
model om one e alua ion o he nex because mo e ma -
ke da a accumula e. Hence, acco ding o he consis ency
p inciple, i canno be ecommended o compu a e
Table 3 P edic ed absolu e and ela i e con e gence
a es in McMC compu a ions by allele coding
Allele coding
012 210 101 cen e ed
single si e
2
0.9974795 0.9995523 0.9947515 0.9647670
ela i e 1.00 5.64 0.48 0.070
block
1
0.9995429 0.9998860 0.9989434 0.00
ela i e 1.00 4.01 0.43 -
Table 4 Numbe o i e a ions in REML by allele coding
and model
Model Allele coding AI-REML EM-REML
Ma ke 012 9 45
Ma ke 210 9 45
Ma ke 101 9 45
Ma ke cen e ed 9 45
G 012 7 34
G 210 9 44
G 101 7 31
G cen e ed 7 28
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eliabili ies. Likewise, he base geno ype in he 012 and 210
allele coding me hods depend also on he obse ed allele
equencies, i.e., ma ke da a.
The cen e ed allele coding me hod is simila o ha
in oduced in [4] whe e he allele equencies we e om
an unselec ed base popula ion. I was used in o de o
“gi e mo e c edi o a e alleles han o common alleles
when calcula ing genomic ela ionships”.Asshown,
in e ence is he same i espec i e o he allele coding
me hod when a ixed gene al mean is in he model.
Howe e , eliabili ies a e a ec ed as shown. The use o
base popula ion allele equencies in he cen e ed allele
coding me hod would emo e he abo e men ioned p o-
blem o inconsis ency be ween e alua ions, bu es ima -
ing hese allele equencies is elusi e. Recen ly, [15]
p esen ed a me hod o adjus ing he G
c
ela ionship
ma ix o become a ela ionship ma ix ela i e o he
base popula ion, he eby a oiding he es ima ion o base
popula ion allele equencies.
The esul s in his pape a e based on he assump ion
ha pheno ypes and geno ypes a e a ailable o all animals
in he analysis. This assump ion may o en no be sa is ied.
Models based on an ex ension o he genomic ela ionship
ma ix o include also non-geno yped animals ha e been
p esen ed by [16-18]. The esul s in he p esen pape
abou pa ame e es ima es and es ima ed b eeding alues
no depending on allele coding do no ca y o e o he
models wi h an ex ended genomic ela ionship ma ix.
Conclusions
We showed ha , in heo y, di e en allele coding me h-
ods led o he same in e ence in ma ke -based models
when he model has a ixed gene al mean e ec . P ac i-
cal analyses led o he same conclusions. Also in heo y,
he cen e ed allele coding me hod was expec ed o gi e
be e mixing p ope ies when Ma ko chain Mon e
Ca lo me hods we e used. This was also obse ed in
p ac ice. When an equi alen b eeding alue model was
used, di e en allele coding me hods p o ed o lead o
he same in e ence as in he ma ke -based model. How-
e e , eliabili ies o b eeding alues depend on he cho-
sen allele coding sys em because di e en allele coding
me hods change he amoun o unce ain y in he es i-
ma ed b eeding alues.
Appendix A
In he ollowing, we conside he e ec o allele coding
me hod on he densi ies p(y|μ,g)andp(μ,g,|y). Fo
simplici y o p esen a ion, he pa ame e s in he dis i-
bu ion o gand ea e omi ed. Le p
0
deno e densi y o
012 allele coding. Because he loca ion pa ame e s μand
ma ke e ec s g ela e o he obse a ions yonly
h ough 1
n
μ+Zg, we i s s udy his e m. By subs i u -
ing Z=Z0−1n 
m
in o he e m, we ha e
1nμ+Zg =1nμ+Z0g−1n 
m
g
=1n(μ− 
m
g)+Z0g.
So, when di e en allele coding sys ems a e used, he
densi ies ha e equali y by
p(y|μ,g)=p0(y|μ− 
m
g,g)
.
(7)
By changing he in eg a ion a iable μ0=μ− 
mg
,we
ob ain ∫p(y|μ,g)dμ=∫p
0
(y|μ
0
,g)dμ
0
and, hence,
p(y)=∫∫p(y|μ,g)p(g)dμdg=p
0
(y). F om hese
esul s, we see ha
p(μ,g|y)=p(y|μ,g)p(g)/p(y)
=p0(y|μ− 
mg,g)p0(g)/p0(y
)
=p0(μ− 
m
g,g|y).
(8)
Table 6 Summa y s a is ics o genomic b eeding alue
eliabili ies by allele coding
Allele coding min mean max s d
012 0.41 0.49 0.59 0.022
210 0.30 0.37 0.42 0.017
101 0.55 0.62 0.73 0.024
cen e ed 0.72 0.80 0.95 0.026
Table 7 Co ela ions be ween genomic b eeding alue
eliabili ies by allele coding
Allele coding 210 101 cen e ed
012 -0.22 0.32 0.43
210 0.82 0.59
101 0.91
Table 5 REML es ima es by allele coding
Allele coding
Model Pa ame e 012 210 101 cen e ed
Ma ke σ2
g
6.623 × 10
-4
6.623 × 10
-4
6.623 × 10
-4
6.623 × 10
-4
Ma ke σ2
e
2.993 2.993 2.993 2.993
Gσ2
a
1.540 1.540 1.540 1.540
Gσ
2
e
2.993 2.993 2.993 2.992
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The esul s (7) and (8) a e ai ly gene al in e ms o
dis ibu ional assump ions. The only equi emen s a e
ha p(y|μ,g)dependsonμand gonly h ough 1
n
μ+
Zg, ha an imp ope uni o m p io is used o μ,and
ha p(y) is ini e. The la e equi emen is o assu e he
pos e io dis ibu ion becomes a p ope dis ibu ion,
and his has o be p o en o a model o be alid when
an imp ope p io is used.
When e~N(0,R), i is no di icul o show ha
∫p(y|μ,g)dμ∝exp(−1
2
(y−Zg)M(y−Zg)
)
wi h M=R−1−R−11n1
n
R−1/(1
n
R−11n). The e o e, ∫
p(y|μ,g)dμ<c
0
whe e he cons an c
0
is independen
o g. Thus, p(y)=∫∫p(y|μ,g)dμp(g)dg<c
0
∫p(g)dg=
c
0
is ini e, i espec i e o dis ibu ion o he ma ke
e ec s g.
Appendix B
We conside a Gaussian dis ibu ion model whe e
g∼N(0,Imσ
2
g)
and e~N(0,R). Consequen ly, he dis-
ibu ion [μ,g|y] is a mul i a ia e Gaussian dis ibu-
ion. In he ollowing, we de i e he a iance-co a iance
ma ix o his dis ibu ion. The condi ional densi y is
p(μ,g|y)∝p(y|g,μ)p(g)
∝exp(−1
2(y−1nμ−Zg)R−1(y−1nμ−Zg)
−1
2g’g/σ2
g
)
∝exp

−1
2[μ−ˆμg’ −ˆ
g]Q

μ−ˆμ
g−ˆ
g

,
whe e
Q=[Va (μ,g|y)]−1=

−1¯z
¯z Cg

Wi h =1/(1
n
R−
1
1n
)
,¯
z
=Z’R−11
n
and
C
g=Z’R−1Z+Imσ−
2
g
. The ma ix Qis he coe icien
ma ix in he mixed model equa ions and he in e se o
his ma ix is
Va (μ,g|y)=⎡
⎣
+ 2z
Cgz − z
Cg
− Cgz Cg⎤
⎦,
whe e
C
g=(C
g
− ¯z ¯z
)−
1
. No e ha he subma ix
C
g
=Va (g|y) is independen o allele coding, because,
asshownin hemain ex ,p(g|y) does no depend on
allele coding.
Va iance-co a iance ma ix o he comple e b eeding
alue is
Va (ad|y)
=1nZ + 2¯z
Cg¯z − ¯z
Cg
− Cg¯z Cg1
n
Z
= 1n1
n+ 21n¯z
Cg¯z 1
n
− ZCg¯z 1
n− 1n¯zCgZ+ZCgZ
= 1n1
n
+(In− 1n1
n
R−1)ZCgZ’(In− R−11n1
n
)
.
Appendix C
The esul s in [8] s a e ha he con e gence a e o a
Gibbs sample is equal o he la ges modulus eigen a-
lue o a ce ain ma ix Bwhe e he eigen alues can be
complex numbe s. As men ioned in [9] his con e gence
a e is also a a measu e o co ela ion be ween succes-
si e McMC samples, i.e., mixing o he algo i hm. The
close he con e gence a e is o ze o he less co ela ed
a e he successi e samples.
The Bma ix is cons uc ed as ollows. Le Qbe he
in e se o he a iance-co a iance ma ix o he a ge
mul i a ia e no mal dis ibu ion, in ou case he coe i-
cien ma ix in he mixed model equa ions. Assume ha
a Gibbs sampling scheme is used whe e he a iables
a e g ouped in o sblocks and Qis spli acco dingly in o
sblocks. Fi s , de ine
A=I−diag(Q−1
11
,...,Q−1
ss
)Q
.
Le Lbe he block lowe iangula ma ix wi h blocks
in he lowe diagonal being hose o A, and le U=A-
L.Thus,Uis a s ic ly uppe iangle ma ix wi h ze os
in he diagonal. Then he ma ix o in e es is
B=
(
I−L
)
−
1
U
.
We conside he genomic ma ke model (1) whe e gis
Gaussian g∼N(0,Imσ2
g)
and eis Gaussian
e∼N(0,Inσ2
e)
. The condi ional dis ibu ion [μ,g|y]is
a mul i a ia e no mal dis ibu ion wi h some mean ec-
o and a a iance-co a iance ma ix wi h in e se
Q=[Va (μ,g|y)]−1=

n/σ2
e1
nZ/σ2
e
Z’1n/σ2
eCg
,
whe e Cg=Z’Z/σ2
e+Im/σ
2
g
; see Appendix B. We con-
side wo McMC upda ing schemes. The i s scheme
i e a es wo blocks: μand g. The second scheme
upda es successi ely all pa ame e s: μ,g
1
, ..., g
m
. Fo he
i s McMC upda ing scheme we ha e
S andén and Ch is ensen Gene ics Selec ion E olu ion 2011, 43:25
h p://www.gsejou nal.o g/con en /43/1/25
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