O iginal
a icle
Mixed
e ec s
linea
models
wi h
-dis ibu ions
o
quan i a i e
gene ic
analysis:
a
Bayesian
app oach
Ismo
S andén
Daniel
Gianola
a
Depa men
o
Animal
Sciences,
Uni e si y
o
Wisconsin,
Madison,
WI
53706,
USA
b
Animal
P oduc ion
Resea ch,
Ag icul u al
Resea ch
Cen e -
MTT,
31600
Jokioinen,
Finland
(Recei ed
21
July
1998;
accep ed
27
No embe
1998)
Abs ac -
A
Bayesian
app oach
o
in e ences
abou
pa ame e s
o
mixed
e ec s
linea
models
wi h
-dis ibu ions
is
p esen ed,
wi h
emphasis
on
quan i a i e
gene ic
applica ions.
The
implemen a ion
is
ia
he
Gibbs
sample .
Da a
om
a
simula ed
mul iple
o ula ion
and
emb yo
ans e
scheme
in
dai y
ca le
b eeding
wi h
non-
andom
p e e en ial
ea men
o
some
cows
is
used
o
illus a e
he
p ocedu es.
Ex ensions
o
he
model
a e
discussed.
©
In a/Else ie ,
Pa is
mixed
e ec s
models
/
Bayesian
in e ence
/
obus
es ima ion
/
Gibbs
sampling
/
S uden ’s
-dis ibu ion
Résumé -
Modèles
linéai es
mix es
a ec
dis ibu ions
de
S uden
en
géné ique
quan i a i e :
app oche
bayésienne.
On
p ésen e
une
app oche
bayésienne
en
ue
de
l’in é ence
conce nan
les
pa amè es
de
modèles
linéai es
mix es
a ec
des
dis ibu ions
de
S uden ,
en
me an
l’accen
su
les
applica ions
en
géné ique
quan i a i e.
L’applica ion
s’e ec ue
g âce
à
l’échan illonnage
de
Gibbs.
Des
données
p o enan
d’un
schéma
de
sélec ion
simulé
u ilisan
le
ans e
emb yonnai e
chez
les
bo ins
lai ie s
en
p ésence
d’un
ai emen
p é é en iel
de
quelques
aches
son
u ilisées
pou
illus e
les
p océdu es.
Les
ex ensions
du
modèle
son
discu ées.
©
In a/Else ie ,
Pa is
modèle
mix e
/
in é ence
bayésienne
/
es ima ion
obus e
/
échan illonnage
de
Gibbs
/
dis ibu ion
de
S uden
*
Co espondence
and
ep in s
E-mail:
[email p o ec ed]
1.
INTRODUCTION
Mixed
e ec s
linea
models
a e
used
widely
in
animal
and
plan
b eeding
and
in
e olu iona y
gene ics
[27].
Thei
applica ion
o
animal
b eeding
was
pionee ed
by
Hende son
[17,
19-21],
p ima ily
om
he
poin
o
iew
o
making
in e ences
abou
candida es
o
gene ic
selec ion
by
bes
linea
unbiased
p edic ion
(BLUP).
Because
BLUP
elies
on
knowledge
o
he
dispe sion
s uc u e,
es ima ion
o
a iance
and
co a iance
componen s
is
cen al
in
p ac ical
implemen a ion
[14,
18,
29,
32].
Typically,
he
dispe sion
s uc u e
is
es ima ed
using
a
likelihood-based
me hod
and,
hen,
in e ences
p oceed
as
i
hese
es ima es
we e
he
ue
alues
(e.g.
[8]).
Al hough
no mali y
is
no
equi ed
by
BLUP,
i
is
p ecisely
when
no mali y
holds
ha
i
can
be
iewed
as
an
app oxima ion
o
he
bes
p edic o
[4,
8,
12,
19].
Mo e
ecen ly,
Bayesian
me hods
ha e
been
ad oca ed
o
he
analysis
o
quan i a i e
gene ic
da a wi h
mixed
linea
models
[8,
9,
34,
39,
40],
and
he
Bayesian
solu ions
sugges ed
employ
Gaussian
sampling
models
as
well
as
no mal
p io s
o
he
andom
e ec s.
I
is
o
p ac ical
in e es ,
he e o e,
o
s udy
s a is ical
models
ha
a e
less
sensi i e
han
Gaussian
ones
o
depa u es
om
assump ions.
Fo
example,
i
is
known
in
dai y
ca le
b eeding
ha
mo e
aluable
cows
ecei e
p e e en ial
ea men ,
and
o
he
ex en
ha
such
ea men
canno
be
accommoda ed
in
he
model,
his
leads
o
bias
in
he
p edic ion
o
b eeding
alues
[23,
24].
Ano he
sou ce
o
bias
in
in e ences
is
an
inco ec
speci ica ion
o
he
inhe i ance
mechanism
in
he
model.
I
is
o en
pos ula ed
ha
he
geno ypic
alue
o
a
quan i a i e
ai
is
he
esul
o
he
addi i e
ac ion
o
alleles
a
a
p ac ically
in ini e
numbe
o
unlinked
loci
and,
hus,
no mali y
esul s
[4].
This
assump ion
is
e u ed
in
an
ob ious
manne
when
inb eeding
dep ession
is
obse ed,
o
when
unknown
genes
o
majo
e ec
a e
seg ega ing.
Howe e ,
in
he
absence
o
clea ly
con adic o y
e idence,
no mali y
is
a
p ac ical
assump ion
o
make,
as
hen
he
machine y
o
mixed
e ec s
linea
models
can
be
exploi ed.
An
appealing
al e na i e
is
o
i
linea
models
wi h
obus
dis ibu ions
o
he
e o s
and
o
he
andom
e ec s.
One
o
such
dis ibu ions
is
S uden ’s
,
bo h
in
i s
uni a ia e
and
mul i a ia e
o ms.
Se e al
au ho s
[2,
7,
26,
37, 38,
41,
42]
ha e
s udied
linea
and
non-linea
eg ession
p oblems
wi h
S uden ’s
-dis ibu ions,
bu
he e
is
a
sca ci y
o
li e a u e
on
andom
e ec s
models.
Wes
[41]
desc ibed
a
one-way
andom
e ec s
layou
wi h
-dis ibu ed
e o s
and
a
hea y
ailed
p io
o
he
andom
e ec s.
Assuming
ha
he
a io
be ween
esidual
a iance
and
he
a iance
o
he
andom
e ec s
was
known,
he
showed
ha
his
model
could
discoun
e ec s
o
ou lie s
on
in e ences.
Pinhei o
e
al.
[30]
desc ibed
a
obus
e sion
o
he
Gaussian
mixed
e ec s
model
o
Lai d
and
Wa e
[25]
and
used
maximum
likelihood.
They
hypo hesized
ha
he
dis ibu ion o
he
esiduals
had
he
same
deg ees
o
eedom
as
ha
o
he
andom
e ec s,
and,
also,
ha
andom
e ec s
we e
independen ly
dis ibu ed.
The
i s
assump ion
is
un ealis ic
as
i
is
ha d
o
accep
why
wo
di e en
andom
p ocesses
( he
dis ibu ions
o
andom
e ec s
and
o
he
esiduals)
should
be
go e ned
by
he
same
deg ees
o
eedom
pa ame e .
The
second
assump ion
is
no
enable
in
gene ics
because
andom
gene ic
e ec s
o
ela i es
may
be
co ela ed.
In
quan i a i e
gene ics
he
andom
e ec s
o
unc ions
he eo
a e
o
cen al
in e es .
Fo
example,
in
animal
b eeding
p og ams
he
objec i e
is
o
inc ease
a
linea
o
non-linea
me i
unc ion
o
gene ic
alues
which,
ideally,
akes
in o
accoun
he
economics
o
p oduc ion
[16,
28,
33].
He e,
i
would
seem
na u al
o
conside
he
condi ional
dis ibu ion o
he
andom
e ec s
gi en
he
da a,
o
d aw
in e ences.
The e
a e
wo
di icul ies
wi h
his
sugges ion.
Fi s ,
i
is
no
always
possible
o
cons uc
his
condi ional
dis ibu ion.
Fo
example,
i
he
andom
e ec s
and
he
e o s
ha e
independen
-dis ibu ions,
he
condi ional
dis ibu ion
o
in e es
is
unknown.
Second,
his
condi ional
dis ibu ion
would
no
inco po a e
he
unce ain y
abou
he
pa ame e s,
a
well-known
p oblem
in
animal
b eeding,
which
does
no
ha e
a
simple
equen is
o
likelihood-based
solu ion
(e.g.
[10,
15]).
I ,
on
he
o he
hand,
he
pa ame e s
( he
ixed
e ec s
and
he
a iance
componen s)
a e
o
p ima y
in e es ,
he
me hod
o
maximum
likelihood
has
some
impo an
d awbacks.
In e ences
a e
alid
asymp o ically
only,
unde
egula i y
condi ions,
and
ini e
sample
esul s
o
mixed
e ec s
models
a e
no
a ailable,
which
is
pa icula ly
ue
o
a
model
wi h
-dis ibu ions.
In
addi ion,
some
gene ic
models
impose
cons ain s
such
ha
he
pa ame e
space
depends
on
he
pa ame e s
hemsel es,
so
i
would
be
nai e
o
apply
a
egula
asymp o ic
heo y.
Fo
example,
wi h
a
pa e nal
hal -sib
amily
s uc u e
[6],
he
a iance
be ween
amilies
is
bounded
be ween
0
and
one- hi d
o
he
a iance
wi hin
amilies.
Mo eo e ,
maximum
likelihood
es ima ion
in
he
mul i-pa ame e
case
has
he
no o ious
de iciency
o
no
accoun ing
well
o
nuisance
pa ame e s
[3,
8,
13].
A
Bayesian
app oach
o
d awing
in e ences
abou
ixed
and
andom
e ec s,
and
abou
a iance
componen s
o
mixed
linea
models
wi h
-dis ibu ed
an-
dom
and
esidual
e ms
is
desc ibed
he e.
Sec ion
2
p esen s
he
p obabili y
model,
emphasizing
a
s uc u e
sui able
o
analysis
o
quan i a i e
gene ic
da a.
Sec ion
3
gi es
a
Ma ko
chain
Mon e
Ca lo
implemen a ion.
A
Bayesian
analysis
o
a
simula ed
animal
b eeding
da a
se
is
p esen ed
in
sec ion
4.
Po en-
ial
applica ions
and
sugges ions
o
addi ional
esea ch
a e
in
he
concluding
sec ion
o
he
pape .
2.
THE
UNIVARIATE
MIXED
EFFECTS
LINEAR
MODEL
2.1.
Sampling
model
and
likelihood
unc ion
Conside
he
uni a ia e
linea
model
whe e
y
is
an
n
x
1
ec o
o
obse a ions;
X
is
a
known,
ull
ank,
incidence
ma ix
o
o de
n
x
p
o
’ ixed’
e ec s;
b
is
a
p
x
1
ec o
o
unknown
’ ixed’
e ec s;
Z
is
a
known
incidence
ma ix
o
o de n
x
q
o
addi i e
gene ic
e -
ec s;
u
is
a
q x
1
ec o
o
unknown
addi i e
gene ic
e ec s
( andom)
and
e
is
an
n
x
1
ec o
o
andom
esidual
e ec s.
Al hough
only
a
single
se
o
an-
dom
e ec s
is
conside ed,
he
model
and
subsequen
esul s
can
be ex ended
in
a
s aigh o wa d
manne .
I
is
assumed
ha
u
and
e
a e
dis ibu ed
inde-
penden ly.
Suppose
he
da a
ec o
can
be
pa i ioned
acco ding
o
’clus e s’
induced
by
a
common
ac o ,
such
as
he d
o
he d-yea
season
o
cal ing
in
a
ca le
b eeding
con ex .
The
model
can
hen
be
p esen ed
as:
whe e
m
is
he
numbe
o
’clus e s’
(e.g.
he ds).
He e
yi
is
he
da a
ec o
o
clus e
i (i
=
1, 2, ... ,
m),
Xi
and
Zi
and
a e
he
co esponding
incidence
ma ices
and
ei
is
he
esidual
ec o
pe aining
o
yi.
Obse a ions
in
each
clus e
will
be
modeled
using
a
mul i a ia e
-
dis ibu ion
such
ha ,
gi en
b and
u,
da a
in
he
same
he d
a e
unco ela ed
bu
no
independen ,
whe eas
eco ds
in
di e en
clus e s
a e
(condi ionally)
independen .
Le
yi !b, u, 62 N
ni
(X
ib
+
Zi
u,
1,,, o, e
2,
e
),
whe e
ni
is
he
num-
be
o
obse a ions
in
clus e
i (i
= 1, 2, ... ,
m),
o ’
is
a
scale
pa ame e
and
.
is
he
deg ees
o
eedom.
I
ni
=
1 o
all
i,
he
sampling
model
becomes
uni a ia e
.
The
condi ional
densi y
o
all
obse a ions,
gi en
he
pa ame e s,
is
Al hough
he
m
dis ibu ions
ha e
he
same
,
and
Qe
pa ame e s,
hese
a e
no
iden ical.
In
pa icula ,
no e
ha
E(y2 !b, u, Qe, e)
Xib
+
Zi
u,
and
Va (y2!b, u, Qe, e)
=
hz !./( ! - 2),
i =
1,2,...,m,
so
he
mean
ec o
is
peculia
o
each
clus e .
Homoscedas ici y
is
assumed,
bu
his
es ic ion
can
be
li ed
wi hou
di icul y.
When
each
clus e
con ains
a
single
obse a ion,
he
e o
dis ibu ion
is
he
independen
-model
o Lange
e
al.
[26];
hen,
he
obse a ions
a e
condi ionally
independen .
When
all
obse a ions
a e
pu
in
a
single
clus e ,
he
mul i a ia e
-model
o
Zellne
[42]
esul s;
in
his
case,
he
deg ees
o
eedom
canno
be
es ima ed.
Each
o
he
m
e ms
in
equa ion
(3)
can
be
ob ained
om
he
mix u e
o
he
no mal
dis ibu ion:
wi h
he
mixing
p ocess
being:
whe e
x
e
is
a
chi-squa ed
andom
a iable
on
Ve
deg ees
o
eedom
[26,
38,
41,42].
2.2.
Bayesian
s uc u e
Fo mally,
bo h
b and
u
a e
loca ion
pa ame e s
o
he
condi ional
dis ibu-
ion
in
equa ion
(3).
The
dis inc ion
be ween
’ ixed’
and
’ andom’
is
equen is ,
bu
om
a
Bayesian
pe spec i e
i
co esponds
o
a
si ua ion
whe e
he e
is
a
di e en ial
amoun
o
p io
in o ma ion
on
b and
u
[8,
13].
In
pa icula ,
he
Bayesian
coun e pa
o
a
’ ixed’
e ec
is
ob ained
by
assigning
a
la
p io
o
b,
so
ha
he
p io
densi y
o
his
ec o
would
be:
in
Rp.
This
dis ibu ion
is
imp ope ,
bu
lowe
and
uppe
limi s
can
be
assigned
o
each
o
he
elemen s
o
b,
as
in
So ensen
e
al.
[34],
o
make
i
p ope .
The
p io
dis ibu ion
o
addi i e
gene ic
alues
u
will
be
aken
o
be
a
mul i a ia e
-dis ibu ion,
and
independen
o
ha
o
b.
F om
a
quan i a i e
gene ics
poin
o
iew
his
can
be
in e p e ed
as
an
addi i e,
mul i a ia e
no mal
model
(as
in
[4]),
bu
wi h
a
andomly
a ying
addi i e
gene ic
a iance.
Because
he
mul i a ia e
-dis ibu ion
has
hicke
ails
han
he
no mal,
he
p oposed
model
is
expec ed
o
be
somewha
bu e ed
agains
depa u es
om
he
assump ions
made
in
an
addi i e
gene ic
e ec s
model,
so
’gene ic
ou lie s’
s emming
om
nonaddi i i y
o
om
majo
genes
become,
pe haps,
less
in luen ial
in
he
o e all
analysis.
All
p ope ies
o
he
mul i a ia e
no mal
dis ibu ion
a e
p ese ed,
e.g.
any
ec o
o
scala
alued
linea
combina ion
o
addi i e
gene ic
alues
has
a
mul i a ia e
-dis ibu ion,
he
ma ginal
dis ibu ions
o
all
e ms
in
u
a e
,
and
all
condi ional
dis ibu ions
a e
as
well.
In
pa icula ,
i
he
addi i e
gene ic
alues
o
pa en s
and
he
seg ega ion
esidual
o
an
o sp ing
a e
join ly
dis ibu ed
as
mul i a ia e
,
he
addi i e
gene ic
alue
o
he
o sp ing
has
a
uni a ia e
-dis ibu ion
wi h
he
same
deg ees
o
eedom.
This
implies
ha
he
coances y
p ope ies
o
he
usual
Gaussian
model
a e
p ese ed.
We
hen
ake
as
p io
dis ibu ion:
wi h
densi y
Abo e,
q
is
he
numbe
o
indi iduals
included
in
u
(some
o
which
may
no
ha e
da a),
A
is
a
known
ma ix
o
addi i e
ela ionships,
au
is
a
scale
pa ame e
and
u
is
he
deg ees
o eedom
pa ame e .
Hence,
Va (ulo, 2,
!)
=
A< ! /( !2),
which
educes
o
he
a iance-co a iance
ma ix
o
addi i e
gene ic
alues
o
a
Gaussian
model
when
u
-!
oo.
The
scale
pa ame e s
o 2
and
o 2
a e
aken
o
ha e
independen
scaled
in e ed
chi-squa e
dis ibu ions,
wi h
densi ies:

espec i ely,
o
a
>
0
and
o
>
0.
He e,
T’e
(
Tu
)
is
a
s ic ly
posi i e
’deg ee
o
belie ’
pa ame e ,
and
Te
(T
u)
can
be
hough
o
as
a
p io
alue
o
he
scale
pa ame e .
These
dis ibu ions
ha e
ini e
means
and
a iances
whene e
he
T
pa ame e s
a e
la ge
han
2
and
4,
espec i ely.
In
animal
b eeding
esea ch,
i
is
common
p ac ice
o
assign
imp ope
la
p io s
o
he
a iance
componen s
o
a
Gaussian
linea
model
[8,
9,
39].
I
uni o m
p io s
a e
o
be
used,
i
is
ad isable
o
es ic
he
ange
o
alues
hey
can
ake,
o
a oid
imp op ie y
(o en
di icul
o
ecognize,
see
[22]).
He e,
one
can
ake
Typically,
he
lowe
bounds
a e
se
o
ze o,
whe eas
he
uppe
bounds
can
be
elici ed
om
mechanis ic
conside a ions,
o
se
up
a bi a ily.
P io
dis ibu ions
o
he
deg ees
o
eedom
can
be
disc e e
as
in
Albe
and
Chib
[1]
and
Besag
e
al.
[2],
o
con inuous
as
in
Geweke
[7],
wi h
he
join
p io
densi y
aken
as
p( e, !)
=
p(
e
)p(
u
).
In
he
disc e e
se ing,
le
j,
j
=
1, 2, ... , d
e,
and
wk,
k
=
1, 2, ... , d!,
be
se s
o
s a es
o
he
esidual
and
gene ic
alues
deg ees
o
eedom,
espec i ely.
The
independen
p io
dis ibu ions
a e:
Because
a
mul i a ia e
-dis ibu ion
is
assigned
o
he
whole
ec o
u,
he e
is
no
in o ma ion
con ained
in
he
da a
abou
,,.
The e o e,
equa ion
(11)
is
eco e ed
in
he
pos e io
analysis.
The e
a e
a
leas
wo
possibili ies
he e:
1)
o
assign
a bi a y
alues
o
Vu
and
examine
how
a ia ion
in
hese
alues
a ec s
in e ences,
o
2)
o
c ea e
clus e s
o
gene ic
alues
by,
e.g.
hal -sib
o
ull-sib
amilies,
and
hen
assume
ha
clus e s
a e
mu ually
independen
bu
wi h
common
deg ees
o
eedom.
He e
he
.
pa ame e
would
be
es imable,
bu
a
he
expense
o
igno ing
gene ic
ela ionships
o he
han
hose
om
hal -sib
o
ull-sib
s uc u es.
Al e na i e
(2)
may
be
sui able
o
dai y
ca le
b eeding
(whe e
mos
o
he
ela ionships
a e
due
o
si es)
o
humans
(whe e
mos
amilies
a e
nuclea ).
A
hi d
al e na i e
would
be
o
use
(2),
hen
ind
he
mode
o
he
pos e io
dis ibu ion
o
u,
and
hen
use
(1)
as
i
his
mode
we e
he
ue
alue.
In
he
ollowing
de i a ion,
we
adop
op ion
(1).
The
join
p io
densi y
o
all
unknowns
is
hen:
wi h
ob ious
modi ica ions
i
equa ions
(8)
and
(9)
a e
used
ins ead
o
equa ions
(6)
and
(7).
The
join
pos e io
densi y
is
ound
by
combining
likelihood
equa ion
(3)
and
app op ia e
p io s
in
equa ions
(4)-(11),
o
ob ain:
whe e
b
E
!p,
u
E
K,j,o!
>
0, Q!
>
0
and
e
E
j
, j
=
1, 2, ... ,
,de}
i
a
disc e e
p io
is
employed.
The
hype -pa ame e s
a e
7e
,
TM!
Te
, T
u
and
u
because
we
assume
his
las
one
o
be
known.
He ea e ,
we
supp ess
he
dependency
on
he
hype -pa ame e s
in
he
no a ion.
3.
THE
GIBBS
SAMPLING
SCHEME
A
Ma ko
chain
Mon e
Ca lo
me hod
such
as
Gibbs
sampling
is
acili a ed
using
an
augmen ed
pos e io
dis ibu ion
ha
esul s
om
mix u e
models.
The
-dis ibu ion
wi hin
each
clus e
in
equa ion
(3)
is
iewed
as
s emming
om
he
mix u e
p ocesses
no ed
ea lie .
Likewise,
he
-dis ibu ion
in
equa-
ion
(5)
can
be
a i ed
a
by
mixing
he
ulA,
o 2,s2 -
N(O,
AU2
/82
)
p ocess
wi h
s2
wu N
X2.
I ,,.
The
augmen ed
join
pos e io
densi y
is
m
whe e
s,
=
(se
l
, ... , sP
m
)
and
N
=
! n2.
In eg a ion
o
equa ion
(14)
wi h
!
i=i
espec
o
Se
and
s2
yields
equa ion
(13),
so
hese
pos e io s
a e
’equi alen ’.
The e
is
a
connec ion
he e
wi h
he
he e ogeneous
a iance
models
o
animal
b eeding
gi en,
e.g.
in
Gianola
e
al.
[11]
and
in
San
C is obal
e
al.
[31].
These
au ho s
pa i ioned
b eeding
alues
and
esiduals
in o
clus e s
as
well,
each
clus e
ha ing
a
speci ic
a iance
ha
a ied
a
andom
acco ding
o
a
scale
in e ed
chi-squa e
dis ibu ion
wi h
known
pa ame e s.
The
ull
condi ional
dis ibu ions
equi ed
o
ins umen
a
Gibbs
sample
a e
de i ed
om
equa ion
(14).
Resul s
gi en
in
Wang
e
al.
[40]
a e
used.
Deno e
C
=
!c2!!,
i, j
=
1, 2,...,p + q, ,
and
=
{
j,
i,j
=
1, 2,...,p +
q o
be
he
coe icien
ma ix
and
igh -hand
side
o
Hende son’s
mixed
model
equa ions,
espec i ely,
whe e
p +
is
he
numbe
o
unknowns
( ixed
and
andom
e ec s),
gi en
he
dispe sion
componen s
Se
,
s l
and
he
scale
pa ame e s
Qe
and
o
2 u
The
mixed model
equa ions
a e:
bes
linea
unbiased
es ima o
(BLUE)
o
b,
and
u
is
he
bes
linea
unbiased
p edic o
(BLUP)
o
u.
Collec
he
ixed
and
andom
e ec s
in o
a’ =
(b’,
u’)
_
(a,,
a2
, ... ,
ap
+
q) .
Le
a’
i =
(a
l
, a
2
, ... ,
ai-1,
a
i+l
,
... ,
ap
+
q).
The
condi ional
pos e io
dis ibu-
ion
o
each
o
he
elemen s
o
a
is
/
P+9
B
whe e
4i
=
c-
1
! -
y! c,j’aj L
i, j
=
1,
2, ... ,
p
+
q.
This
ex ends
o
blocks
B
j-1
/
B
j$i
/
o
elemen s
o
a
in
a
na u al
way.
I
ai
is
a
sub- ec o
o
a,
he
condi ional
dis ibu ion o
ai
gi en
e e y hing
else,
is
mul i a ia e
no mal
wi h
mean
ai
=
Ci
il
i
-
E Cija
i
o
app op ia e
de ini ions
o
C
ij
,
i
and
a!
as
B
j-1
/
B
j54,
/
ma ices
and
ec o s.
The
condi ional
pos e io
densi y
o
each
o
he
se
is
in
he
o m
o
a
gamma
densi y
whe e
s, _,
is
s,
wi hou
S;
i’
Equi alen ly,
,
&dquo;e
e
/
Simila ly,
he
condi ional
pos e io
densi y
o
s!
also
has
he
gamma
densi y
o m
The
condi ional
pos e io
dis ibu ion
o
Qe
is
a
scaled
in e ed
chi-squa e
dis ibu ion
wi h
o m
I
a
bounded
uni o m
dis ibu ion
is
used
as
p io
o
Q
e,
i s
condi ional
pos e io
is
he
unca ed
dis ibu ion:
The
condi ional
pos e io
densi y
o
ou
is:
dis ibu ions,
han
hose
ob ained
wi h
a
mixed
e ec s
Gaussian
linea
model,
he
cu en
pa adigm
in
quan i a i e
gene ics
[19,
21,
27].
Ou
app oach
was
illus a ed
wi h
simula ed
da a
om
a
dai y
ca le
b eeding
scheme,
whe e
cows
we e
subjec
o
ai ly
p e alen
and
s ong
p e e en ial
ea men .
A
uni a ia e
-model
o
he
e o s
led
o
mo e
accu a e
in e ences
abou
addi i e
gene ic
a iance
han
ei he
a
he d-clus e ed
-model
o
a
Gaussian
sampling
p ocess.
The
pos e io
dis ibu ions
o
b eeding
alues
o
some
example
animals
we e
sha pe
in
he
uni a ia e
-model.
Ou
model
and
implemen a ion
can
be
ex ended
in
se e al
espec s.
Fo
example,
i
he
deg ees
o
eedom
o
he
dis ibu ion
o
gene ic
alues
needs
o
be
assessed,
i
is
possible
o
clus e
he
gene ic
alues
in o
’independen ’
amilies
and
p oceed
as
o
he
esidual
a iance.
Howe e ,
such
clus e ing
would
lead
o
a
loss
o
accu acy
in
he
speci ica ion
o
he
gene ic
a iance-co a iance
s uc u e,
because
ela ionships
be ween
indi iduals
in
di e en
clus e s
would
no
be
aken
in o
accoun .
Ano he
ex ension
would
be
o
ake
he
deg ees
o
eedom
as
con inuous
and
use
a
ejec ion
algo i hm
o
a
Me opolis-
Has ings
walk
o
d aw
samples,
combined
wi h
he
Gibbs
sample
o
he
es
o
he
pa ame e s
o
he
model.
Addi ional
andom
e ec s,
such
as
pe manen
en i onmen al
e ec s
a ec ing
all
eco ds
o
a
cow,
can
be
inco po a ed,
e.g.
by
aking
a
uni a ia e
-dis ibu ion
as
p io .
Residuals
can
be
clus e ed
in
di e en
manne s.
Fo
example,
clus e ing
e o s
by
si e
o
ull-sib
amilies
may
cope
wi h
inadequa e
gene ic
assump ions,
e.g.
unknown
majo
genes
may
be
seg ega ing.
In
addi ion,
i
is
possible
o
allow
o
he e ogeneous
a iance
in
he
model
wi hou
majo
di icul y.
O he
esidual
dis ibu ions
such
as
he
logis ic
o
he
slash
may
be
conside ed
as
well.
A
p esen ,
i is
no
ye
possible
o
apply
hese
me hods
o
he
la ge
da a
se s
used
o
ou ine
gene ic
e alua ion
in
he
dai y
ca le
b eeding
indus y,
whe e
he
models
can
ha e
millions
o
indi idual
b eeding
alues.
Hence,
i
i
is
es ablished
ha
models
based
on
he
-dis ibu ion
imp o e
gene ic
e alua ions,
compu a ionally
simple
o
as e
me hods
should
be
de eloped.
One
possibili y
would
be
o
employ
Laplacian
app oxima ions
o
assess
he
mode
o
he
join
dis ibu ion
o
he
dispe sion
pa ame e s
and
hen
use
some
o m
o
condi ional
analysis
o
ob ain
poin
p edic o s
o
b eeding
alues.
In
a
model
wi h
-
dis ibu ed
andom
e ec s,
modal
es ima es
o
ixed
and
andom
e ec s,
gi en
he
scale
pa ame e s
and
he
deg ees
o
eedom
can
be
ound
using
an
i e a i e
p ocedu e
[35].
This
equi es
sol ing
eweigh ed
mixed model
equa ions
se e al
imes,
as
in
h eshold
models.
App oxima e
solu ions
a e
a
couple
o
i e a ions
may
be
adequa e
o
p ac ical
pu poses.
This
Laplacian-i e a ion
app oach
would
be
coun e pa
o
he
s anda d
REML-BLUP
analysis
in
linea
models
unde
Gaussian
assump ions.
Finally,
we
would
like
o
obse e
some
in e p e a i e
di e ences
be ween
he
Gaussian
and
he
-model
om
a
quan i a i e
gene ic
poin
o
iew.
He i abili y
is
he
eg ession
o
geno ype
on
pheno ype,
which
is
o’!/(<!+o’!)
in
a
Gaussian
model.
In
he
-models
discussed,
his
eg ession
is
wi h
he
nume a o
(and
he
app op ia e
pa
o
he
denomina o )
being
equal
o
o,2i
gene ic
alues
a e
assumed
o
be
Gaussian,
ins ead
o
-dis ibu ed,
as

was
he
case
in
ou
analysis.
The
pos e io
dis ibu ion o
he i abili y
can
be
es ima ed
by
o ming
a
sample
alue
o
h2
om
he
co esponding
d aws
o
ol 2,
!,
o 2and
e.
Fo
ou
simula ion,
pos e io
mean
es ima es
o
he i abili y
we e
0.47
and
0.45
o
he
Gaussian
and
he d-clus e ed
-models,
espec i ely,
and
0.19
o
he
uni a ia e
-model,
his
being
closes
o
he
inpu
alue.
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J.H.,
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S.,
Bayesian
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o
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J.,
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K.A.,
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C.R.,
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C.R.,
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A.E.,
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