The genetic model A general genetic model for the partition of genetic variance (particularly epistatic variance) in a random mating population was first given by Cockerham [7 , 13 ] and extended to multiple alleles by Kempthorne [8 , 9 ], following the basic genetic model formulated by Fisher [6 ]. The model for two loci A and B with multiple alleles was expressed as follows
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MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqabeWabaaabaGaem4raC0aa0baaSqaaiabdQgaQjabdYgaSbqaaiabdMgaPjabdUgaRbaakiabg2da9GGaciab=X7aTHGaaiab+TcaRiab=f7aHnaaCaaaleqabaacbiGae0xAaKgaaOGaey4kaSIae8xSde2aaSbaaSqaaiabdQgaQbqabaGccqGHRaWkcqWF0oazdaqhaaWcbaGaemOAaOgabaGaemyAaKgaaOGaey4kaSIae8NSdi2aaWbaaSqabeaacqWGRbWAaaGccqGHRaWkcqWFYoGydaWgaaWcbaGaemiBaWgabeaakiabgUcaRiabeo7aNnaaDaaaleaacqWGSbaBaeaacqWGRbWAaaGccqGHRaWkcqGGOaakcqWFXoqydaahaaWcbeqaaiabdMgaPbaakiab=j7aInaaCaaaleqabaGaem4AaSgaaOGaeiykaKcabaGaey4kaSIaeiikaGIae8xSde2aaWbaaSqabeaacqWGPbqAaaGccqWFYoGydaWgaaWcbaGae0hBaWgabeaakiabcMcaPiabgUcaRiabcIcaOiab=f7aHnaaBaaaleaacqqFQbGAaeqaaOGae8NSdi2aaWbaaSqabeaacqWGRbWAaaGccqGGPaqkcqGHRaWkcqGGOaakcqWFXoqydaWgaaWcbaGaemOAaOgabeaakiab=j7aInaaBaaaleaacqWGSbaBaeqaaOGaeiykaKIaey4kaSIaeiikaGIae8xSde2aaWbaaSqabeaacqWGPbqAaaGccqaHZoWzdaqhaaWcbaGaemiBaWgabaGaem4AaSgaaOGaeiykaKcabaGaey4kaSIaeiikaGIae8xSde2aaSbaaSqaaiabdQgaQbqabaGccqaHZoWzdaqhaaWcbaGaemiBaWgabaGaem4AaSgaaOGaeiykaKIaey4kaSIaeiikaGIae8hTdq2aa0baaSqaaiabdQgaQbqaaiabdMgaPbaakiab=j7aInaaCaaaleqabaGaem4AaSgaaOGaeiykaKIaey4kaSIaeiikaGIae8hTdq2aa0baaSqaaiabdQgaQbqaaiabdMgaPbaakiab=j7aInaaBaaaleaacqWGSbaBaeqaaOGaeiykaKIaey4kaSIaeiikaGIae8hTdq2aa0baaSqaaiabdQgaQbqaaiabdMgaPbaakiabeo7aNnaaDaaaleaacqWGSbaBaeaacqWGRbWAaaGccqGGPaqkaaGaaCzcaiaaxMaadaqadaqaaiabigdaXaGaayjkaiaawMcaaaaa@AE28@
where the genotypic value
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is the expected phenotype of an individual carrying alleles A
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As an ANOVA model, it is known that not all the parameters in model (1) are estimable. A number of constraint conditions on these parameters are therefore needed. Let p i , q k denote allelic frequencies for alleles on paternal gametes, and p
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MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqaaeWabaaabaWaaabuaeaacqWGWbaCdaahaaWcbeqaaiabdMgaPbaaiiGakiab=f7aHnaaCaaaleqabaGaemyAaKgaaOGaeyypa0JaeGimaaJaeiilaWIaeeiiaaYaaabuaeaacqWGWbaCdaahaaWcbeqaaiabdMgaPbaakiab=r7aKnaaDaaaleaacqWGQbGAaeaacqWGPbqAaaGccqGH9aqpcqaIWaamcqGGSaalcqqGGaaidaaeqbqaaiabdchaWnaaBaaaleaacqWGQbGAaeqaaOGae8hTdq2aa0baaSqaaiabdQgaQbqaaiabdMgaPbaakiabg2da9iabicdaWiabcYcaSaWcbaGaemOAaOgabeqdcqGHris5aaWcbaGaemyAaKgabeqdcqGHris5aaWcbaGaemyAaKgabeqdcqGHris5aaGcbaWaaabuaeaacqWGWbaCdaahaaWcbeqaaiabdMgaPbaakiabcIcaOiab=f7aHnaaCaaaleqabaGaemyAaKgaaOGae8NSdi2aaWbaaSqabeaacqWGRbWAaaGccqGGPaqkcqGH9aqpcqaIWaamcqGGSaalaSqaaiabdMgaPbqab0GaeyyeIuoakiabbccaGmaaqafabaGaemiCaa3aaWbaaSqabeaacqWGRbWAaaGccqGGOaakcqWFXoqydaahaaWcbeqaaiabdMgaPbaakiab=j7aInaaCaaaleqabaGaem4AaSgaaOGaeiykaKIaeyypa0JaeGimaaJaeiilaWcaleaacqWGRbWAaeqaniabggHiLdaakeaadaaeqbqaaiabdchaWnaaBaaaleaacqWGQbGAaeqaaOGaeiikaGIae8hTdq2aa0baaSqaaiabdQgaQbqaaiabdMgaPbaakiab=j7aInaaCaaaleqabaGaem4AaSgaaOGaeiykaKIaeyypa0JaeGimaaJaeiilaWIaeeiiaaYaaabuaeaacqWGXbqCdaahaaWcbeqaaiabdUgaRbaakiabcIcaOiab=r7aKnaaDaaaleaacqWGQbGAaeaacqWGPbqAaaGccqWFYoGydaahaaWcbeqaaiabdUgaRbaakiabcMcaPiabg2da9iabicdaWiabcYcaSaWcbaGaem4AaSgabeqdcqGHris5aaWcbaGaemOAaOgabeqdcqGHris5aOGaeeiiaaIaeS47IWKaeS47IWeaaiaaxMaacaWLjaWaaeWaaeaacqaIYaGmaiaawIcacaGLPaaaaaa@A8EF@
Under the assumption of random mating and linkage equilibrium and allowing for different allelic frequencies in paternal and maternal gametes, the mean and genetic effects can be expressed as follows based on the least squares principle:
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MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakqaabeqaaGGaciab=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j7aInaaCaaaleqabaGaem4AaSgaaOGaeiykaKIaeyypa0Jaem4raC0aa0baaSqaaiabdQgaQjabc6caUaqaaiabc6caUiabdUgaRbaakiabgkHiTiabdEeahnaaDaaaleaacqWGQbGAcqGGUaGlaeaacqGGUaGlcqGGUaGlaaGccqGHsislcqWGhbWrdaqhaaWcbaGaeiOla4IaeiOla4cabaGaeiOla4Iaem4AaSgaaOGaey4kaSIaem4raC0aa0baaSqaaiabc6caUiabc6caUaqaaiabc6caUiabc6caUaaakiabcYcaSaqaaiabcIcaOiab=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@CBD7@
where
G
..
..
=
∑
i
,
j
,
k
,
l
p
i
p
j
q
k
q
l
G
j
l
i
k
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGhbWrdaqhaaWcbaGaeiOla4IaeiOla4cabaGaeiOla4IaeiOla4caaOGaeyypa0ZaaabeaeaacqWGWbaCdaahaaWcbeqaaiabdMgaPbaakiabdchaWnaaBaaaleaacqWGQbGAaeqaaOGaemyCae3aaWbaaSqabeaacqWGRbWAaaGccqWGXbqCdaWgaaWcbaGaemiBaWgabeaakiabdEeahnaaDaaaleaacqWGQbGAcqWGSbaBaeaacqWGPbqAcqWGRbWAaaaabaGaemyAaKMaeiilaWIaemOAaOMaeiilaWIaem4AaSMaeiilaWIaemiBaWgabeqdcqGHris5aaaa@4F36@
,
G
..
i
.
=
∑
j
,
k
,
l
p
j
q
k
q
l
G
j
l
i
k
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGhbWrdaqhaaWcbaGaeiOla4IaeiOla4cabaGaemyAaKMaeiOla4caaOGaeyypa0ZaaabeaeaacqWGWbaCdaWgaaWcbaGaemOAaOgabeaakiabdghaXnaaCaaaleqabaGaem4AaSgaaOGaemyCae3aaSbaaSqaaiabdYgaSbqabaGccqWGhbWrdaqhaaWcbaGaemOAaOMaemiBaWgabaGaemyAaKMaem4AaSgaaaqaaiabdQgaQjabcYcaSiabdUgaRjabcYcaSiabdYgaSbqab0GaeyyeIuoaaaa@4A77@
, and so on. The total genetic variance is
V
G
=
∑
i
,
j
,
k
,
l
p
i
p
j
p
k
p
l
(
G
j
l
i
k
−
μ
)
2
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGwbGvdaWgaaWcbaGaem4raCeabeaakiabg2da9maaqababaGaemiCaa3aaWbaaSqabeaacqWGPbqAaaGccqWGWbaCdaWgaaWcbaGaemOAaOgabeaakiabdchaWnaaCaaaleqabaGaem4AaSgaaOGaemiCaa3aaSbaaSqaaiabdYgaSbqabaGccqGGOaakcqWGhbWrdaqhaaWcbaGaemOAaOMaemiBaWgabaGaemyAaKMaem4AaSgaaOGaeyOeI0ccciGae8hVd0MaeiykaKYaaWbaaSqabeaacqaIYaGmaaaabaGaemyAaKMaeiilaWIaemOAaOMaeiilaWIaem4AaSMaeiilaWIaemiBaWgabeqdcqGHris5aaaa@525B@
, and has an orthogonal partition under random mating and linkage equilibrium
V
G
=
V
A
1
+
V
A
2
+
V
D
1
+
V
D
2
+
V
A
1
A
2
+
V
A
1
D
2
(
4
)
+
V
D
1
A
2
+
V
D
1
D
2
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@60B6@
with
V
A
1
=
∑
i
p
i
(
α
i
)
2
+
∑
j
p
j
(
α
j
)
2
V
D
1
=
∑
i
,
j
p
i
p
j
(
δ
j
i
)
2
V
A
2
=
∑
k
q
k
(
β
k
)
2
+
∑
l
q
l
(
β
l
)
2
V
D
2
=
∑
k
,
l
q
k
q
l
(
γ
l
k
)
2
V
A
1
A
2
=
∑
i
,
k
p
i
q
k
(
α
i
β
k
)
2
+
∑
j
,
l
p
j
q
l
(
α
j
β
l
)
2
+
∑
i
,
l
p
i
q
l
(
α
i
β
l
)
2
+
∑
j
,
k
p
j
q
k
(
α
j
β
k
)
2
V
A
1
D
2
=
∑
i
,
k
,
l
p
i
q
k
q
l
(
α
i
γ
l
k
)
2
+
∑
j
,
k
,
l
p
j
q
k
q
l
(
α
j
γ
l
k
)
2
V
D
1
A
2
=
∑
i
,
j
,
k
p
i
p
j
q
k
(
δ
j
i
β
k
)
2
+
∑
i
,
j
,
l
p
i
p
j
q
l
(
δ
j
i
β
l
)
2
V
D
1
D
2
=
∑
i
,
j
,
k
,
l
p
i
p
j
q
k
q
l
(
δ
j
i
γ
l
k
)
2
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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f7aHnaaCaaaleqabaGaemyAaKgaaOGae8NSdi2aaWbaaSqabeaacqWGRbWAaaGccqGGPaqkdaahaaWcbeqaaiabikdaYaaaaeaacqWGPbqAcqGGSaalcqWGRbWAaeqaniabggHiLdGccqGHRaWkdaaeqbqaaiabdchaWnaaBaaaleaacqWGQbGAaeqaaOGaemyCae3aaSbaaSqaaiabdYgaSbqabaGccqGGOaakcqWFXoqydaWgaaWcbaGaemOAaOgabeaakiab=j7aInaaBaaaleaacqGFSbaBaeqaaOGaeiykaKYaaWbaaSqabeaacqaIYaGmaaaabaGaemOAaOMaeiilaWIaemiBaWgabeqdcqGHris5aaGcbaGaaCzcaiabgUcaRmaaqafabaGaemiCaa3aaWbaaSqabeaacqWGPbqAaaGccqWGXbqCdaWgaaWcbaGaemiBaWgabeaakiabcIcaOiab=f7aHnaaCaaaleqabaGaemyAaKgaaOGae8NSdi2aaSbaaSqaaiab+XgaSbqabaGccqGGPaqkdaahaaWcbeqaaiabikdaYaaaaeaacqWGPbqAcqGGSaalcqWGSbaBaeqaniabggHiLdGccqGHRaWkdaaeqbqaaiabdchaWnaaBaaaleaacqWGQbGAaeqaaOGaemyCae3aaWbaaSqabeaacqWGRbWAaaGccqGGOaakcqWFXoqydaWgaaWcbaGae4NAaOgabeaakiab=j7aInaaCaaaleqabaGae43AaSgaaOGaeiykaKYaaWbaaSqabeaacqaIYaGmaaaabaGaemOAaOMaeiilaWIaem4AaSgabeqdcqGHris5aaGcbaGaemOvay1aaSbaaSqaaiabdgeabnaaBaaameaacqaIXaqmaeqaaSGaemiraq0aaSbaaWqaaiabikdaYaqabaaaleqaaOGaeyypa0ZaaabuaeaacqWGWbaCdaahaaWcbeqaaiabdMgaPbaakiabdghaXnaaCaaaleqabaGaem4AaSgaaOGaemyCae3aaSbaaSqaaiabdYgaSbqabaGccqGGOaakcqWFXoqydaahaaWcbeqaaiabdMgaPbaakiabeo7aNnaaDaaaleaacqGFSbaBaeaacqGFRbWAaaGccqGGPaqkdaahaaWcbeqaaiabikdaYaaaaeaacqWGPbqAcqGGSaalcqWGRbWAcqGGSaalcqWGSbaBaeqaniabggHiLdGccqGHRaWkdaaeqbqaaiabdchaWnaaBaaaleaacqWGQbGAaeqaaOGaemyCae3aaWbaaSqabeaacqWGRbWAaaGccqWGXbqCdaWgaaWcbaGaemiBaWgabeaakiabcIcaOiab=f7aHnaaBaaaleaacqGFQbGAaeqaaOGaeq4SdC2aa0baaSqaaiab+XgaSbqaaiab+TgaRbaakiabcMcaPmaaCaaaleqabaGaeGOmaidaaaqaaiabdQgaQjabcYcaSiabdUgaRjabcYcaSiabdYgaSbqab0GaeyyeIuoaaOqaaiabdAfawnaaBaaaleaacqWGebardaWgaaadbaGaeGymaedabeaaliabdgeabnaaBaaameaacqaIYaGmaeqaaaWcbeaakiabg2da9maaqafabaGaemiCaa3aaWbaaSqabeaacqWGPbqAaaGccqWGWbaCdaWgaaWcbaGaemOAaOgabeaakiabdghaXnaaCaaaleqabaGaem4AaSgaaOGaeiikaGIae8hTdq2aa0baaSqaaiabdQgaQbqaaiabdMgaPbaakiab=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@9DD5@
Using indicator variables, we can represent model (1) in another form. Assume that the two loci A and B have alleles A
i
, i = 1, 2, ..., n 1 ; and B
k
, i = 1, 2, ..., n 2 , respectively. We define the following indicator variables to represent the segregation of alleles in a population.
z
M
i
(
1
)
=
{
1
,
for
A
i
allele from paternal gamete
0
,
otherwise
.
z
F
j
(
1
)
=
{
1
,
for
A
j
allele from maternal gamete
0
,
otherwise
.
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@B74B@
for i , j = 1, 2, ..., n 1 at locus A , and
z
M
k
(
2
)
=
{
1
,
for
B
k
allele from paternal gamete
0
,
otherwise
.
z
F
l
(
2
)
=
{
1
,
for
B
l
allele from maternal gamete
0
,
otherwise
.
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakqaabeqaaiabdQha6naaDaaaleaacqWGnbqtdaWgaaadbaGaem4AaSgabeaaaSqaaiabcIcaOiabikdaYiabcMcaPaaakiabg2da9maaceaabaqbaeaabiGaaaqaaiabigdaXiabcYcaSaqaaiabbAgaMjabb+gaVjabbkhaYjabbccaGGqaciab=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@B766@
for k , l = 1, 2, ..., n 2 at locus B . In terms of these indicator variables, we have the following.
• Hardy-Weinberg equilibrium (HWE) implies that {
z
M
i
(
1
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG6bGEdaqhaaWcbaGaemyta00aaSbaaWqaaiabdMgaPbqabaaaleaacqGGOaakcqaIXaqmcqGGPaqkaaaaaa@33AE@
, i = 1, 2, ..., n 1 } are independent of {
z
F
j
(
1
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG6bGEdaqhaaWcbaGaemOray0aaSbaaWqaaiabdQgaQbqabaaaleaacqGGOaakcqaIXaqmcqGGPaqkaaaaaa@33A2@
, j = 1, 2, ..., n 1 }, and {
z
M
k
(
2
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG6bGEdaqhaaWcbaGaemyta00aaSbaaWqaaiabdUgaRbqabaaaleaacqGGOaakcqaIYaGmcqGGPaqkaaaaaa@33B4@
, k = 1, 2, ..., n 2 } are independent of {
z
F
l
(
2
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG6bGEdaqhaaWcbaGaemOray0aaSbaaWqaaiabdYgaSbqabaaaleaacqGGOaakcqaIYaGmcqGGPaqkaaaaaa@33A8@
, l = 1, 2, ..., n 2 }.
• Linkage equilibrium (LE) implies that {
z
M
i
(
1
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG6bGEdaqhaaWcbaGaemyta00aaSbaaWqaaiabdMgaPbqabaaaleaacqGGOaakcqaIXaqmcqGGPaqkaaaaaa@33AE@
, i = 1, 2, ..., n 1 } are independent of {
z
M
k
(
2
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG6bGEdaqhaaWcbaGaemyta00aaSbaaWqaaiabdUgaRbqabaaaleaacqGGOaakcqaIYaGmcqGGPaqkaaaaaa@33B4@
, k = 1, 2, ..., n 2 }, and {
z
F
j
(
1
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG6bGEdaqhaaWcbaGaemOray0aaSbaaWqaaiabdQgaQbqabaaaleaacqGGOaakcqaIXaqmcqGGPaqkaaaaaa@33A2@
, j = 1, 2, ..., n 1 } are independent of {
z
F
l
(
2
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG6bGEdaqhaaWcbaGaemOray0aaSbaaWqaaiabdYgaSbqabaaaleaacqGGOaakcqaIYaGmcqGGPaqkaaaaaa@33A8@
, l = 1, 2, ..., n 2 }.
• There is another type of disequilibrium; i.e., the so-called genotypic disequilibrium [14 ] for two alleles on different gametes and at different loci. So, the genotypic equilibrium (GE) here means that {
z
M
i
(
1
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG6bGEdaqhaaWcbaGaemyta00aaSbaaWqaaiabdMgaPbqabaaaleaacqGGOaakcqaIXaqmcqGGPaqkaaaaaa@33AE@
, i = 1, 2,..., n 1 } are independent of {
z
F
l
(
2
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG6bGEdaqhaaWcbaGaemOray0aaSbaaWqaaiabdYgaSbqabaaaleaacqGGOaakcqaIYaGmcqGGPaqkaaaaaa@33A8@
, l = 1, 2, ..., n 2 }, and {
z
F
j
(
1
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG6bGEdaqhaaWcbaGaemOray0aaSbaaWqaaiabdQgaQbqabaaaleaacqGGOaakcqaIXaqmcqGGPaqkaaaaaa@33A2@
, j = 1, 2, ..., n 1 } are independent of {
z
M
k
(
2
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG6bGEdaqhaaWcbaGaemyta00aaSbaaWqaaiabdUgaRbqabaaaleaacqGGOaakcqaIYaGmcqGGPaqkaaaaaa@33B4@
, k = 1, 2, ..., n 2 }.
It is known that under random mating we have both HWE and GE, which together are called gametic phase equilibrium. Now, let G denote the genotypic value of a progeny drawn randomly from the current population. Based on Cockerham model, G can be expressed as
G
=
μ
+
∑
i
=
1
n
1
α
i
z
M
i
(
1
)
+
∑
j
=
1
n
1
α
j
z
F
j
(
1
)
+
∑
i
,
j
δ
j
i
z
M
i
(
1
)
z
F
j
(
1
)
+
∑
k
=
1
n
2
β
k
z
M
k
(
2
)
+
∑
l
=
1
n
2
β
l
z
F
l
(
2
)
+
∑
k
,
l
γ
l
k
z
M
k
(
2
)
z
F
l
(
2
)
+
[
∑
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k
(
α
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)
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i
(
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2
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∑
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k
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2
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∑
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MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakqaabeqaaiabdEeahjabg2da9GGaciab=X7aTjabgUcaRmaaqahabaGae8xSde2aaWbaaSqabeaacqWGPbqAaaGccqWG6bGEdaqhaaWcbaGaemyta00aaSbaaWqaaiabdMgaPbqabaaaleaacqGGOaakcqaIXaqmcqGGPaqkaaaabaGaemyAaKMaeyypa0JaeGymaedabaGaemOBa42aaSbaaWqaaiabigdaXaqabaaaniabggHiLdGccqGHRaWkdaaeWbqaaiab=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f7aHnaaCaaaleqabaGaemyAaKgaaOGae8NSdi2aaSbaaSqaaiab+XgaSbqabaGccqGGPaqkcqWG6bGEdaqhaaWcbaGaemyta00aaSbaaWqaaiabdMgaPbqabaaaleaacqGGOaakcqaIXaqmcqGGPaqkaaGccqWG6bGEdaqhaaWcbaGaemOray0aaSbaaWqaaiabdYgaSbqabaaaleaacqGGOaakcqaIYaGmcqGGPaqkaaaabaGaemyAaKMaeiilaWIaemiBaWgabeqdcqGHris5aaGcbaGaey4kaSYaaabuaeaacqGGOaakcqWFXoqydaWgaaWcbaGae4NAaOgabeaakiab=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f7aHnaaCaaaleqabaGaemyAaKgaaOGaeq4SdC2aa0baaSqaaiabdYgaSbqaaiabdUgaRbaakiabcMcaPiabdQha6naaDaaaleaacqWGnbqtdaWgaaadbaGaemyAaKgabeaaaSqaaiabcIcaOiabigdaXiabcMcaPaaakiabdQha6naaDaaaleaacqWGnbqtdaWgaaadbaGaem4AaSgabeaaaSqaaiabcIcaOiabikdaYiabcMcaPaaakiabdQha6naaDaaaleaacqWGgbGrdaWgaaadbaGaemiBaWgabeaaaSqaaiabcIcaOiabikdaYiabcMcaPaaaaeaacqWGPbqAcqGGSaalcqWGRbWAcqGGSaalcqWGSbaBaeqaniabggHiLdGccqGHRaWkdaaeqbqaaiabcIcaOiab=f7aHnaaBaaaleaacqGFQbGAaeqaaOGaeq4SdC2aa0baaSqaaiabdYgaSbqaaiabdUgaRbaakiabcMcaPiabdQha6naaDaaaleaacqWGgbGrdaWgaaadbaGaemOAaOgabeaaaSqaaiabcIcaOiabigdaXiabcMcaPaaakiabdQha6naaDaaaleaacqWGnbqtdaWgaaadbaGaem4AaSgabeaaaSqaaiabcIcaOiabikdaYiabcMcaPaaakiabdQha6naaDaaaleaacqWGgbGrdaWgaaadbaGaemiBaWgabeaaaSqaaiabcIcaOiabikdaYiabcMcaPaaaaeaacqWGQbGAcqGGSaalcqWGRbWAcqGGSaalcqWGSbaBaeqaniabggHiLdGccqGGDbqxaeaacqGHRaWkcqGGBbWwdaaeqbqaaiabcIcaOiab=r7aKnaaDaaaleaacqWGQbGAaeaacqWGPbqAaaGccqWFYoGydaahaaWcbeqaaiabdUgaRbaakiabcMcaPiabdQha6naaDaaaleaacqWGnbqtdaWgaaadbaGaemyAaKgabeaaaSqaaiabcIcaOiabigdaXiabcMcaPaaakiabdQha6naaDaaaleaacqWGgbGrdaWgaaadbaGaemOAaOgabeaaaSqaaiabcIcaOiabigdaXiabcMcaPaaakiabdQha6naaDaaaleaacqWGnbqtdaWgaaadbaGaem4AaSgabeaaaSqaaiabcIcaOiabikdaYiabcMcaPaaaaeaacqWGPbqAcqGGSaalcqWGQbGAcqGGSaalcqWGRbWAaeqaniabggHiLdGccqGHRaWkdaaeqbqaaiabcIcaOiab=r7aKnaaDaaaleaacqWGQbGAaeaacqWGPbqAaaGccqWFYoGydaWgaaWcbaGae4hBaWgabeaakiabcMcaPiabdQha6naaDaaaleaacqWGnbqtdaWgaaadbaGaemyAaKgabeaaaSqaaiabcIcaOiabigdaXiabcMcaPaaakiabdQha6naaDaaaleaacqWGgbGrdaWgaaadbaGaemOAaOgabeaaaSqaaiabcIcaOiabigdaXiabcMcaPaaakiabdQha6naaDaaaleaacqWGgbGrdaWgaaadbaGaemiBaWgabeaaaSqaaiabcIcaOiabikdaYiabcMcaPaaaaeaacqWGPbqAcqGGSaalcqWGQbGAcqGGSaalcqWGSbaBaeqaniabggHiLdGccqGGDbqxaeaacqGHRaWkdaaeqbqaaiabcIcaOiab=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@FD6E@
This is simply a different presentation of Cockerham model with the same constraint conditions applied on the coefficient parameters. For a given individual with genotype A
i
B
k
/A
j
B
l
, G will take the same value of
G
j
l
i
k
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGhbWrdaqhaaWcbaGaemOAaOMaemiBaWgabaGaemyAaKMaem4AaSgaaaaa@3368@
as before. However, this expression is helpful for us to understand some details about each component of genetic effects. We can see this more clearly in the examination of some reduced models later.
In general, the genetic effects can be defined separately for alleles that are paternally and maternally transmitted to account for possible biological differences. As a fully parameterized model for
G
j
l
i
k
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGhbWrdaqhaaWcbaGaemOAaOMaemiBaWgabaGaemyAaKMaem4AaSgaaaaa@3368@
, model (3) may give
G
j
l
i
k
≠
G
j
k
i
l
≠
G
i
l
j
k
≠
G
i
k
j
l
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGhbWrdaqhaaWcbaGaemOAaOMaemiBaWgabaGaemyAaKMaem4AaSgaaOGaeyiyIKRaem4raC0aa0baaSqaaiabdQgaQjabdUgaRbqaaiabdMgaPjabdYgaSbaakiabgcMi5kabdEeahnaaDaaaleaacqWGPbqAcqWGSbaBaeaacqWGQbGAcqWGRbWAaaGccqGHGjsUcqWGhbWrdaqhaaWcbaGaemyAaKMaem4AaSgabaGaemOAaOMaemiBaWgaaaaa@4D0F@
depending on how genetic effects are defined. If locus A has n 1 alleles, and locus B has n 2 alleles, there are N =
n
1
2
n
2
2
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGUbGBdaqhaaWcbaGaeGymaedabaGaeGOmaidaaOGaemOBa42aa0baaSqaaiabikdaYaqaaiabikdaYaaaaaa@33A0@
possible phased genotypes in total with the partition of the degrees of freedom given in Table 1 .
Table 1 Partition of degrees of freedom for two loci with number of alleles n 1 and n 2 (a general case) If we assume that the union of paternal gamete A
i
B
k
with maternal gamete A
j
B
l
have the same mean effect as that of paternal gamete A
j
B
l
with maternal gamete A
i
B
k
(i.e.,
G
j
l
i
k
=
G
i
k
j
l
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGhbWrdaqhaaWcbaGaemOAaOMaemiBaWgabaGaemyAaKMaem4AaSgaaOGaeyypa0Jaem4raC0aa0baaSqaaiabdMgaPjabdUgaRbqaaiabdQgaQjabdYgaSbaaaaa@3B34@
), the coupling and repulsion heterozygotes have the same genotypic value (i.e.,
G
j
l
i
k
=
G
j
k
i
l
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGhbWrdaqhaaWcbaGaemOAaOMaemiBaWgabaGaemyAaKMaem4AaSgaaOGaeyypa0Jaem4raC0aa0baaSqaaiabdQgaQjabdUgaRbqaaiabdMgaPjabdYgaSbaaaaa@3B34@
), and paternal and maternal gametes have the same gametic frequency distribution, we do not need to distinguish paternal and maternal effects. In this case, the two loci can be regarded as 2 factors and each factor has
n
i
2
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGUbGBdaqhaaWcbaGaemyAaKgabaGaeGOmaidaaaaa@308B@
(i = 1, 2) levels produced by the allelic combinations of n
i
alleles (cf. [7 ]). The total number of genotypes is N = n 1 (n 1 + 1)n 2 (n 2 + 1)/4 and the partition of degrees of freedom is shown in Table 2 . Since in this case, α i = α
i
, β k = β
k
, ..., and so on, the model can also be expressed as follows
G
=
μ
+
∑
i
=
1
n
1
α
i
(
z
M
i
(
1
)
+
z
F
i
(
1
)
)
+
∑
i
,
j
δ
j
i
z
M
i
(
1
)
z
F
j
(
1
)
+
∑
k
=
1
n
2
β
k
(
z
M
k
(
2
)
+
z
F
k
(
2
)
)
+
∑
k
,
l
γ
l
k
z
M
k
(
2
)
z
F
l
(
2
)
+
∑
i
,
k
(
α
i
β
k
)
(
z
M
i
(
1
)
+
z
F
i
(
1
)
)
(
z
M
k
(
2
)
+
z
F
k
(
2
)
)
+
∑
i
,
k
,
l
(
α
i
γ
l
k
)
(
z
M
i
(
1
)
+
z
F
i
(
1
)
)
z
M
k
(
2
)
z
F
l
(
2
)
+
∑
i
,
j
,
k
(
δ
j
i
β
k
)
z
M
i
(
1
)
z
F
j
(
1
)
(
z
M
k
(
2
)
+
z
F
k
(
2
)
)
+
∑
i
,
j
,
k
,
l
(
δ
j
i
γ
l
k
)
z
M
i
(
1
)
z
F
j
(
1
)
z
M
k
(
2
)
z
F
l
(
2
)
)
(
6
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakqaabeqaaiabdEeahjabg2da9GGaciab=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j7aInaaBaaaleaacqWGRbWAaeqaaOGaeiykaKcaleaacqWGPbqAcqGGSaalcqWGQbGAcqGGSaalcqWGRbWAaeqaniabggHiLdGccqWG6bGEdaqhaaWcbaGaemyta00aaSbaaWqaaiabdMgaPbqabaaaleaacqGGOaakcqaIXaqmcqGGPaqkaaGccqWG6bGEdaqhaaWcbaGaemOray0aaSbaaWqaaiabdQgaQbqabaaaleaacqGGOaakcqaIXaqmcqGGPaqkaaGccqGGOaakcqWG6bGEdaqhaaWcbaGaemyta00aaSbaaWqaaiabdUgaRbqabaaaleaacqGGOaakcqaIYaGmcqGGPaqkaaGccqGHRaWkcqWG6bGEdaqhaaWcbaGaemOray0aaSbaaWqaaiabdUgaRbqabaaaleaacqGGOaakcqaIYaGmcqGGPaqkaaGccqGGPaqkaeaacqGHRaWkdaaeqbqaaiabcIcaOiab=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@65AD@
Table 2 Partition of degrees of freedom for two loci with number of alleles n 1 and n 2 (a simplified case without distinguishing the paternal and maternal origins)
For the case of an arbitrary number of loci, the situation will become more complicated. In addition to the additive and dominance effects at each locus and two locus interactions (additive × additive, additive × dominance, dominance × additive, dominance × dominance, with a total number of 22 terms), there are 3 locus interactions (additive × additive × additive, additive × additive × dominance, ..., with a total number of 23 terms), 4 locus interactions (additive × additive × additive × additive, ..., with a total number of 24 terms), and so on. Though the extension is straightforward, the total number of terms will increase dramatically. We will show some models with multiple loci in later examples by ignoring trigenic and higher order epistasis.
Effects and variance components Let p i , p
j
(i , j = 1, 2, ..., n 1 ) be allelic frequencies of paternal and maternal gametes at locus A , respectively. Let also q k , q
l
(k , l = 1, 2, ..., n 2 ) denote allelic frequencies of paternal and maternal gametes at locus B , respectively. In the analysis of variance for the model, it is convenient to use deviations of the indicator variables
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M
i
(
1
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG6bGEdaqhaaWcbaGaemyta00aaSbaaWqaaiabdMgaPbqabaaaleaacqGGOaakcqaIXaqmcqGGPaqkaaaaaa@33AE@
,
z
F
i
(
1
)
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,
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M
j
(
2
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and
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from their expected values. That is
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(
1
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=
z
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(
1
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−
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(
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(
1
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=
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−
p
i
=
{
1
−
p
i
,
for
A
i
allele from paternal gamete
−
p
i
,
otherwise
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Similarly, define
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(
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=
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(
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MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@9DFE@
Taking the constraint conditions on the genetic effects into account, we can show that,
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2
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqabeabbaaaaeaadaaeWbqaaGGaciab=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@355A@
and so on. For example,
∑
i
=
1
n
1
α
i
x
M
i
(
1
)
=
∑
i
=
1
n
1
α
i
(
z
M
i
(
1
)
−
p
i
)
=
∑
i
=
1
n
1
α
i
z
M
i
(
1
)
−
∑
i
=
1
n
1
α
i
p
i
=
∑
i
=
1
n
1
α
i
z
M
i
(
1
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@8DB2@
as
∑
i
=
1
n
1
α
i
p
i
=
0
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaaeWaqaaGGaciab=f7aHnaaCaaaleqabaGaemyAaKgaaOGaemiCaa3aaWbaaSqabeaacqWGPbqAaaaabaGaemyAaKMaeyypa0JaeGymaedabaGaemOBa42aaSbaaWqaaiabigdaXaqabaaaniabggHiLdGccqGH9aqpcqaIWaamaaa@3C9D@
by the constrain condition (2). Therefore, model (5) can be rewritten as
G
=
μ
+
∑
i
=
1
n
1
α
i
x
M
i
(
1
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+
∑
j
=
1
n
1
α
i
x
F
j
(
1
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+
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j
δ
j
i
x
M
i
(
1
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x
F
j
(
1
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+
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k
=
1
n
2
β
k
x
M
k
(
2
)
+
∑
l
=
1
n
2
β
l
x
F
l
(
2
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+
∑
k
,
l
γ
l
k
x
M
k
(
2
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x
F
l
(
2
)
+
[
∑
i
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k
(
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β
k
)
x
M
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(
1
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x
M
k
(
2
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+
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l
(
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1
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x
F
l
(
2
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+
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k
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k
)
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(
1
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2
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α
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(
1
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x
F
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(
2
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]
+
[
∑
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k
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+
[
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l
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2
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(
7
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MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakqaabeqaaiabdEeahjabg2da9GGaciab=X7aTjabgUcaRmaaqahabaGae8xSde2aaWbaaSqabeaacqWGPbqAaaGccqWG4baEdaqhaaWcbaGaemyta00aaSbaaWqaaiabdMgaPbqabaaaleaacqGGOaakcqaIXaqmcqGGPaqkaaaabaGaemyAaKMaeyypa0JaeGymaedabaGaemOBa42aaSbaaWqaaiabigdaXaqabaaaniabggHiLdGccqGHRaWkdaaeWbqaaiab=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j7aInaaBaaaleaacqGFSbaBaeqaaOGaeiykaKIaemiEaG3aa0baaSqaaiabdAeagnaaBaaameaacqWGQbGAaeqaaaWcbaGaeiikaGIaeGymaeJaeiykaKcaaOGaemiEaG3aa0baaSqaaiabdAeagnaaBaaameaacqWGSbaBaeqaaaWcbaGaeiikaGIaeGOmaiJaeiykaKcaaOGaeiyxa0faleaacqWGQbGAcqGGSaalcqWGSbaBaeqaniabggHiLdaakeaacqGHRaWkcqGGBbWwdaaeqbqaaiabcIcaOiab=f7aHnaaCaaaleqabaGaemyAaKgaaOGaeq4SdC2aa0baaSqaaiabdYgaSbqaaiabdUgaRbaakiabcMcaPiabdIha4naaDaaaleaacqWGnbqtdaWgaaadbaGaemyAaKgabeaaaSqaaiabcIcaOiabigdaXiabcMcaPaaakiabdIha4naaDaaaleaacqWGnbqtdaWgaaadbaGaem4AaSgabeaaaSqaaiabcIcaOiabikdaYiabcMcaPaaaaeaacqWGPbqAcqGGSaalcqWGRbWAcqGGSaalcqWGSbaBaeqaniabggHiLdGccqWG4baEdaqhaaWcbaGaemOray0aaSbaaWqaaiabdYgaSbqabaaaleaacqGGOaakcqaIYaGmcqGGPaqkaaGccqGHRaWkdaaeqbqaaiabcIcaOiab=f7aHnaaBaaaleaacqGFQbGAaeqaaOGaeq4SdC2aa0baaSqaaiabdYgaSbqaaiabdUgaRbaakiabcMcaPiabdIha4naaDaaaleaacqWGgbGrdaWgaaadbaGaemOAaOgabeaaaSqaaiabcIcaOiabigdaXiabcMcaPaaakiabdIha4naaDaaaleaacqWGnbqtdaWgaaadbaGaem4AaSgabeaaaSqaaiabcIcaOiabikdaYiabcMcaPaaaaeaacqWGQbGAcqGGSaalcqWGRbWAcqGGSaalcqWGSbaBaeqaniabggHiLdGccqWG4baEdaqhaaWcbaGaemOray0aaSbaaWqaaiabdYgaSbqabaaaleaacqGGOaakcqaIYaGmcqGGPaqkaaGccqGGDbqxaeaacqGHRaWkcqGGBbWwdaaeqbqaaiabcIcaOiab=r7aKnaaDaaaleaacqWGQbGAaeaacqWGPbqAaaGccqWFYoGydaahaaWcbeqaaiabdUgaRbaakiabcMcaPiabdIha4naaDaaaleaacqWGnbqtdaWgaaadbaGaemyAaKgabeaaaSqaaiabcIcaOiabigdaXiabcMcaPaaakiabdIha4naaDaaaleaacqWGgbGrdaWgaaadbaGaemOAaOgabeaaaSqaaiabcIcaOiabigdaXiabcMcaPaaaaeaacqWGPbqAcqGGSaalcqWGQbGAcqGGSaalcqWGRbWAaeqaniabggHiLdGccqWG4baEdaqhaaWcbaGaemyta00aaSbaaWqaaiabdUgaRbqabaaaleaacqGGOaakcqaIYaGmcqGGPaqkaaGccqGHRaWkdaaeqbqaaiabcIcaOiab=r7aKnaaDaaaleaacqWGQbGAaeaacqWGPbqAaaGccqWFYoGydaWgaaWcbaGae4hBaWgabeaakiabcMcaPiabdIha4naaDaaaleaacqWGnbqtdaWgaaadbaGaemyAaKgabeaaaSqaaiabcIcaOiabigdaXiabcMcaPaaakiabdIha4naaDaaaleaacqWGgbGrdaWgaaadbaGaemOAaOgabeaaaSqaaiabcIcaOiabigdaXiabcMcaPaaaaeaacqWGPbqAcqGGSaalcqWGQbGAcqGGSaalcqWGSbaBaeqaniabggHiLdGccqWG4baEdaqhaaWcbaGaemOray0aaSbaaWqaaiabdYgaSbqabaaaleaacqGGOaakcqaIYaGmcqGGPaqkaaGccqGGDbqxaeaacqGHRaWkdaaeqbqaaiabcIcaOiab=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@0729@
When we deal with some reduced models in the next section, we will find that the model of this form is especially helpful as it makes the model parameter constraints built into regression variables which is suited for genetic interpretation. The form of model (7) can also facilitate the demonstration that under Hardy-Weinberg, linkage and genotypic equilibria, the regression coefficients (genetic effects) are Cockerham's least squares effects (3) (Appendix A), and the genotypic variance V
G
has the orthogonal partition (4) (Appendix B).
Now we discuss the properties of model in a disequilibrium situation. As stated in [14 ], there are three types of disequilibria
• Typel: between alleles on the same gametes but at different loci
• Type2: between alleles at the same locus but on different gametes
• Type3: between alleles on different gametes and at different loci.
If we denote
P
j
l
i
k
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGqbaudaqhaaWcbaGaemOAaOMaemiBaWgabaGaemyAaKMaem4AaSgaaaaa@337A@
as the genotypic frequency of A
i
B
k
/A
j
B
l
,
P
j
.
i
.
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGqbaudaqhaaWcbaGaemOAaOMaeiOla4cabaGaemyAaKMaeiOla4caaaaa@3282@
as the genotypic frequency of A
i
/A
j
, and so on, following [14 ], the digenic disequilibria can be written as
D
..
i
k
=
Cov
(
z
M
i
(
1
)
,
z
M
k
(
2
)
)
=
E
(
x
M
i
(
1
)
x
M
k
(
2
)
)
=
P
..
i
k
−
p
i
q
k
D
j
l
..
=
Cov
(
z
F
j
(
1
)
,
z
F
l
(
2
)
)
=
E
(
x
F
j
(
1
)
x
F
l
(
2
)
)
=
P
j
l
..
−
p
j
q
l
D
j
.
i
.
=
E
(
x
M
i
(
1
)
x
F
j
(
1
)
)
=
P
j
.
i
.
−
p
i
p
j
D
.
l
.
k
=
E
(
x
M
k
(
2
)
x
F
l
(
2
)
)
=
P
.
l
.
k
−
q
k
q
l
D
.
j
i
.
=
E
(
x
M
i
(
1
)
x
F
l
(
2
)
)
=
P
.
l
i
.
−
p
i
q
l
D
j
.
.
k
=
E
(
x
F
j
(
1
)
x
M
k
(
2
)
)
=
P
j
.
.
k
−
p
j
q
k
.
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@3973@
And the trigenic disequilibria
D
j
.
i
k
=
E
(
x
M
i
(
1
)
x
F
j
(
1
)
x
M
k
(
2
)
)
=
P
j
.
i
k
−
p
i
P
j
.
.
k
−
q
k
P
j
.
i
.
−
p
j
P
..
i
k
+
2
p
i
p
j
q
k
D
.
l
i
k
=
E
(
x
M
i
(
1
)
x
M
k
(
2
)
x
F
l
(
2
)
)
=
P
.
l
i
k
−
p
i
P
.
l
.
k
−
q
k
P
.
l
i
.
−
q
l
P
..
i
k
+
2
p
i
q
k
q
l
D
j
l
i
.
=
E
(
x
M
i
(
1
)
x
F
j
(
1
)
x
F
l
(
2
)
)
=
P
j
l
i
.
−
p
i
P
j
l
..
−
p
j
P
..
i
.
−
q
l
P
j
.
i
.
+
2
p
i
p
j
q
l
D
j
l
.
k
=
E
(
x
F
j
(
1
)
x
M
k
(
2
)
x
F
l
(
2
)
)
=
P
j
l
.
k
−
p
j
P
.
l
.
k
−
q
k
P
j
.
.
l
−
q
l
P
j
.
.
k
+
2
p
j
q
k
q
l
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@6748@
Similarly for the quadrigenic disequilibrium, we may define
D
j
l
i
k
=
E
(
x
M
i
(
1
)
x
F
j
(
1
)
x
M
k
(
2
)
x
F
l
(
2
)
)
=
P
j
l
i
k
−
p
i
P
j
l
.
k
−
p
j
P
.
l
i
k
−
q
k
P
j
l
i
.
−
q
l
P
j
.
i
k
+
p
i
p
j
P
.
l
.
k
−
p
i
q
k
P
j
l
..
+
p
i
q
l
P
j
.
.
k
+
p
j
q
k
P
.
l
i
.
+
p
j
q
l
P
..
i
k
+
q
k
q
l
P
j
.
i
.
−
3
p
i
p
j
q
k
q
l
.
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@DFBE@
If we express
D
j
l
i
k
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGebardaqhaaWcbaGaemOAaOMaemiBaWgabaGaemyAaKMaem4AaSgaaaaa@3362@
as a function of lower-order linkage disequilibria, we have
D
j
l
i
k
=
P
j
l
i
k
−
p
i
D
j
l
.
k
−
p
j
D
.
l
i
k
−
q
k
D
j
l
i
.
−
q
l
D
j
.
i
k
−
p
i
p
j
D
.
l
.
k
−
p
i
q
k
D
j
l
..
−
p
i
q
l
D
j
.
.
k
−
p
j
q
k
D
.
l
i
.
−
p
j
q
l
D
..
i
k
−
q
k
q
l
D
j
.
i
.
−
p
i
p
j
q
k
q
l
.
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@BE30@
This definition is the same as that given by [15 , 16 ]. Note that
∑
i
z
M
i
(
1
)
=
1
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaaeqaqaaiabdQha6naaDaaaleaacqWGnbqtdaWgaaadbaGaemyAaKgabeaaaSqaaiabcIcaOiabigdaXiabcMcaPaaaaeaacqWGPbqAaeqaniabggHiLdGccqGH9aqpcqaIXaqmaaa@38E1@
. Then, we have
∑
i
D
j
l
i
k
=
E
[
(
∑
i
x
M
i
(
1
)
)
x
F
j
(
1
)
x
M
k
(
2
)
x
F
l
(
2
)
]
=
E
[
∑
i
(
z
M
i
(
1
)
−
p
i
)
x
F
j
(
1
)
x
M
k
(
2
)
x
F
l
(
2
)
=
0
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakqaabeqaamaaqafabaGaemiraq0aa0baaSqaaiabdQgaQjabdYgaSbqaaiabdMgaPjabdUgaRbaaaeaacqWGPbqAaeqaniabggHiLdGccqGH9aqpcqWGfbqrcqGGBbWwcqGGOaakdaaeqbqaaiabdIha4naaDaaaleaacqWGnbqtdaWgaaadbaGaemyAaKgabeaaaSqaaiabcIcaOiabigdaXiabcMcaPaaakiabcMcaPiabdIha4naaDaaaleaacqWGgbGrdaWgaaadbaGaemOAaOgabeaaaSqaaiabcIcaOiabigdaXiabcMcaPaaaaeaacqWGPbqAaeqaniabggHiLdGccqWG4baEdaqhaaWcbaGaemyta00aaSbaaWqaaiabdUgaRbqabaaaleaacqGGOaakcqaIYaGmcqGGPaqkaaGccqWG4baEdaqhaaWcbaGaemOray0aaSbaaWqaaiabdYgaSbqabaaaleaacqGGOaakcqaIYaGmcqGGPaqkaaGccqGGDbqxaeaacqGH9aqpcqWGfbqrcqGGBbWwdaaeqbqaaiabcIcaOiabdQha6naaDaaaleaacqWGnbqtdaWgaaadbaGaemyAaKgabeaaaSqaaiabcIcaOiabigdaXiabcMcaPaaakiabgkHiTiabdchaWnaaCaaaleqabaGaemyAaKgaaOGaeiykaKIaemiEaG3aa0baaSqaaiabdAeagnaaBaaameaacqWGQbGAaeqaaaWcbaGaeiikaGIaeGymaeJaeiykaKcaaaqaaiabdMgaPbqab0GaeyyeIuoakiabdIha4naaDaaaleaacqWGnbqtdaWgaaadbaGaem4AaSgabeaaaSqaaiabcIcaOiabikdaYiabcMcaPaaakiabdIha4naaDaaaleaacqWGgbGrdaWgaaadbaGaemiBaWgabeaaaSqaaiabcIcaOiabikdaYiabcMcaPaaakiabg2da9iabicdaWaaaaa@8728@
In general,
D
j
l
i
k
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGebardaqhaaWcbaGaemOAaOMaemiBaWgabaGaemyAaKMaem4AaSgaaaaa@3362@
is summed to zero over any allele involved, so are
D
j
.
i
k
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGebardaqhaaWcbaGaemOAaOMaeiOla4cabaGaemyAaKMaem4AaSgaaaaa@32E5@
and other disequilibrium measurements.
With Hardy-Weinberg and genotypic equilibria but linkage disequilibrium, model (7) leads to the following expression for the overall mean
E
(
G
)
=
μ
+
∑
i
,
k
(
α
i
β
k
)
D
..
i
k
+
∑
j
,
l
(
α
j
β
l
)
D
j
l
..
+
∑
i
,
j
,
k
,
l
(
δ
j
i
γ
l
k
)
D
i
k
D
j
l
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqadeGabaaabaGaemyrauKaeiikaGIaem4raCKaeiykaKIaeyypa0dcciGae8hVd0Maey4kaSYaaabuaeaacqGGOaakcqWFXoqydaahaaWcbeqaaiabdMgaPbaakiab=j7aInaaCaaaleqabaGaem4AaSgaaOGaeiykaKIaemiraq0aa0baaSqaaiabc6caUiabc6caUaqaaiabdMgaPjabdUgaRbaaaeaacqWGPbqAcqGGSaalcqWGRbWAaeqaniabggHiLdGccqGHRaWkdaaeqbqaaiabcIcaOiab=f7aHnaaBaaaleaacqWGQbGAaeqaaOGae8NSdi2aaSbaaSqaaiabdYgaSbqabaGccqGGPaqkcqWGebardaqhaaWcbaGaemOAaOMaemiBaWgabaGaeiOla4IaeiOla4caaaqaaiabdQgaQjabcYcaSiabdYgaSbqab0GaeyyeIuoaaOqaaiabgUcaRmaaqafabaGaeiikaGIae8hTdq2aa0baaSqaaiabdQgaQbqaaiabdMgaPbaakiabeo7aNnaaDaaaleaacqWGSbaBaeaacqWGRbWAaaGccqGGPaqkcqWGebardaahaaWcbeqaaiabdMgaPjabdUgaRbaakiabdseaenaaBaaaleaacqWGQbGAcqWGSbaBaeqaaaqaaiabdMgaPjabcYcaSiabdQgaQjabcYcaSiabdUgaRjabcYcaSiabdYgaSbqab0GaeyyeIuoaaaaaaa@7A2E@
where μ is the mean genotypic value under linkage equilibrium, and
Δ
μ
=
∑
i
,
k
(
α
i
β
k
)
D
..
i
k
+
∑
j
,
l
(
α
j
β
l
)
D
j
l
..
+
∑
i
,
j
,
k
,
l
(
δ
j
i
γ
l
k
)
D
..
i
k
D
j
l
..
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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n7aNnaaDaaaleaacqWGSbaBaeaacqWGRbWAaaGccqGGPaqkaSqaaiabdMgaPjabcYcaSiabdQgaQjabcYcaSiabdUgaRjabcYcaSiabdYgaSbqab0GaeyyeIuoakiabdseaenaaDaaaleaacqGGUaGlcqGGUaGlaeaacqWGPbqAcqWGRbWAaaGccqWGebardaqhaaWcbaGaemOAaOMaemiBaWgabaGaeiOla4IaeiOla4caaaaa@7A04@
represents the departure from μ due to linkage disequilibrium and epistasis. If there is no epistasis, linkage disequilibrium does not affect the mean genotypic value. Similar results were given by [17 ]. Note that for marginal means of the genotypic values, we have
G
..
..
=
∑
i
,
j
,
k
,
l
P
..
i
k
P
j
l
..
G
j
l
i
k
,
G
..
i
.
=
1
p
i
∑
j
,
k
,
l
P
..
i
k
P
j
l
..
G
j
l
i
k
,
G
..
i
k
=
∑
j
,
l
P
j
l
..
G
j
l
i
k
,
G
j
.
i
.
=
1
p
i
p
j
∑
k
,
l
P
j
l
..
P
..
i
k
G
j
l
i
k
,
G
j
.
i
k
=
1
p
j
∑
l
P
j
l
..
G
j
l
i
k
,
...
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbfv3ySLgzGueE0jxyaibaieYdOi=BH8vipeYdI8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0lXdbba9pGe9qqFf0dXdHuk9fr=xfr=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@DA34@
and so on.
Then the question is what the genetic effects are in a disequilibrium population. Do Hardy-Weinberg and linkage disequilibria change the definition and values of genetic effects? The short answer to this question is "no" in a fully characterized model, but "yes" in a model that ignores some QTL or genetic effects. This is proved and discussed in [10 ]. With Hardy-Weinberg and linkage disequilibria, the genetic effects no longer correspond to the deviations from marginal means of genotypic values in a disequilibrium population. In a multiple regression model (7), the genetic effects are partial regression coefficients. These partial regression coefficients correspond to the simple regression coefficients, or deviations from marginal means of genotypic values, only in an equilibrium population. In a disequilibrium population, a direct analysis on the partial regression coefficients can be very complex (see the appendix of [10 ] for a relatively simple example). However, in a full model which includes all relevant loci and genetic effects, the model parameters depend only on how the regressors, i.e. x variables in (7), are defined and are independent of correlations between x variables, i.e. Hardy-Weinberg and linkage disequilibria. So, the genetic effects are still the same as those defined in the equilibrium population, although the population mean and marginal means of genotypic values are changed in a disequilibrium population.
Hardy-Weinberg and linkage disequilibria introduce correlation between x variables, thus covariances between different genetic effect components. Define
A
1
=
∑
i
=
1
n
1
α
i
x
M
i
(
1
)
+
∑
j
=
1
n
1
α
j
x
F
j
(
1
)
D
1
=
∑
i
,
j
δ
j
i
x
M
i
(
1
)
x
F
j
(
1
)
A
2
=
∑
k
=
1
n
2
β
k
x
M
k
(
2
)
+
∑
l
=
1
n
2
β
l
x
F
l
(
2
)
D
2
=
∑
k
,
l
γ
l
k
x
M
k
(
2
)
x
F
l
(
2
)
A
1
A
2
=
∑
i
,
k
(
α
i
β
k
)
x
M
i
(
1
)
x
M
k
(
2
)
+
∑
i
,
l
(
α
i
β
l
)
x
M
i
(
1
)
x
F
l
(
2
)
+
∑
j
,
k
(
α
j
β
k
)
x
F
j
(
1
)
x
M
k
(
2
)
+
∑
j
,
l
(
α
j
β
l
)
x
F
j
(
1
)
x
F
l
(
2
)
A
1
D
2
=
∑
i
,
k
,
l
(
α
i
γ
l
k
)
x
M
i
(
1
)
x
M
k
(
2
)
x
F
l
(
2
)
+
∑
j
,
k
,
l
(
α
j
γ
l
k
)
x
F
j
(
1
)
x
M
k
(
2
)
x
F
l
(
2
)
A
2
D
1
=
∑
i
,
j
,
k
(
δ
j
i
β
k
)
x
M
i
(
1
)
x
F
j
(
1
)
x
M
k
(
2
)
+
∑
i
,
j
,
l
(
δ
j
i
β
l
)
x
M
i
(
1
)
x
F
j
(
1
)
x
F
l
(
2
)
D
1
D
2
=
∑
i
,
j
,
k
,
l
(
δ
j
i
γ
l
k
)
x
M
i
(
1
)
x
F
j
(
1
)
x
M
k
(
2
)
x
F
l
(
2
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbfv3ySLgzGueE0jxyaibaieYdOi=BH8vipeYdI8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0lXdbba9pGe9qqFf0dXdHuk9fr=xfr=xfrpiWZqaaeaabiGaaiaacaqabeaabeqacmaaaOabaeqabaGaemyqae0aaSbaaSqaaiabigdaXaqabaGccqGH9aqpdaaeWbqaaGGaciab=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f7aHnaaCaaaleqabaGaemyAaKgaaOGae83SdC2aa0baaSqaaiabdYgaSbqaaiabdUgaRbaaaeaacqWGPbqAcqGGSaalcqWGRbWAcqGGSaalcqWGSbaBaeqaniabggHiLdGccqGGPaqkcqWG4baEdaqhaaWcbaGaemyta00aaSbaaWqaaiabdMgaPbqabaaaleaacqGGOaakcqaIXaqmcqGGPaqkaaGccqWG4baEdaqhaaWcbaGaemyta00aaSbaaWqaaiabdUgaRbqabaaaleaacqGGOaakcqaIYaGmcqGGPaqkaaGccqWG4baEdaqhaaWcbaGaemOray0aaSbaaWqaaiabdYgaSbqabaaaleaacqGGOaakcqaIYaGmcqGGPaqkaaaakeaacaWLjaGaey4kaSYaaabuaeaacqGGOaakcqWFXoqydaWgaaWcbaGaemOAaOgabeaakiab=n7aNnaaDaaaleaacqWGSbaBaeaacqWGRbWAaaaabaGaemOAaOMaeiilaWIaem4AaSMaeiilaWIaemiBaWgabeqdcqGHris5aOGaeiykaKIaemiEaG3aa0baaSqaaiabdAeagnaaBaaameaacqWGQbGAaeqaaaWcbaGaeiikaGIaeGymaeJaeiykaKcaaOGaemiEaG3aa0baaSqaaiabd2eannaaBaaameaacqWGRbWAaeqaaaWcbaGaeiikaGIaeGOmaiJaeiykaKcaaOGaemiEaG3aa0baaSqaaiabdAeagnaaBaaameaacqWGSbaBaeqaaaWcbaGaeiikaGIaeGOmaiJaeiykaKcaaaGcbaGaemyqae0aaSbaaSqaaiabikdaYaqabaGccqWGebardaWgaaWcbaGaeGymaedabeaakiabg2da9maaqafabaGaeiikaGIae8hTdq2aa0baaSqaaiabdQgaQbqaaiabdMgaPbaakiab=j7aInaaCaaaleqabaGaem4AaSgaaaqaaiabdMgaPjabcYcaSiabdQgaQjabcYcaSiabdUgaRbqab0GaeyyeIuoakiabcMcaPiabdIha4naaDaaaleaacqWGnbqtdaWgaaadbaGaemyAaKgabeaaaSqaaiabcIcaOiabigdaXiabcMcaPaaakiabdIha4naaDaaaleaacqWGgbGrdaWgaaadbaGaemOAaOgabeaaaSqaaiabcIcaOiabigdaXiabcMcaPaaakiabdIha4naaDaaaleaacqWGnbqtdaWgaaadbaGaem4AaSgabeaaaSqaaiabcIcaOiabikdaYiabcMcaPaaaaOqaaiaaxMaacqGHRaWkdaaeqbqaaiabcIcaOiab=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@0D44@
Then we can write
G = μ + A 1 + A 2 + D 1 + D 2 + A 1 A 2 + A 1 D 2 + A 2 D 1 + D 1 D 2
In a disequilibrium population, the partition of the genotypic variance becomes
V
G
=
∑
i
=
1
8
∑
j
=
1
8
V
i
j
=
1
T
V
1
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbfv3ySLgzGueE0jxyaibaieYdOi=BH8vipeYdI8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0lXdbba9pGe9qqFf0dXdHuk9fr=xfr=xfrpiWZqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabdAfawnaaBaaaleaacqWGhbWraeqaaOGaeyypa0ZaaabCaeaadaaeWbqaaiabdAfawnaaBaaaleaacqWGPbqAcqWGQbGAaeqaaaqaaiabdQgaQjabg2da9iabigdaXaqaaiabiIda4aqdcqGHris5aaWcbaGaemyAaKMaeyypa0JaeGymaedabaGaeGioaGdaniabggHiLdGccqGH9aqpieqacqWFXaqmdaahaaWcbeqaaiab=rfaubaakiab=zfawjab=fdaXaaa@4898@
where
V = (V
ij
)8 × 8
It is a symmetric matrix. In Appendix C, we give the detailed result for each component of the matrix with linkage disequilibrium, but assuming Hardy-Weinberg equilibrium.
For the rest of paper, when we discuss disequilibrium, we mainly discuss linkage disequilibrium and assume Hardy-Weinberg and genotypic equilibria which can be achieved by random mating in one generation. Hardy-Weinberg disequilibrium can be taken into account which will make results more complex and is thus omitted.
Reduced models In many genetic applications, experimental population has some regular genetic structure by design. In these cases, the genetic model can be further simplified to reflect the experimental design structure. Also sometimes we may want to simplify the genetic model by imposing certain constrains or assumptions, such as the number of alleles, to increase the feasibility of analysis. In this section, we give a few reduced genetic models that are relevant to many genetic applications.
1. Backcross population or recombinant inbred population (haploid model) Backcross population or recombinant inbred population is a common experimental design for QTL mapping study. By crossing two inbred lines, we can create a F 1 population. If we randomly backcross F 1 to one of the inbred lines, we have a backcross population. Let us assume that the cross is AA (paternal) × Aa (maternal). In a random-mating backcross population, there are only two possible genotypes at each segregating locus A
r
A
r
or A
r
a
r
, for r = 1, 2, ..., m , where m is the number of QTL. Since for the paternal gametes,
z
M
1
(
r
)
=
{
1
,
for
A
r
allele from paternal gamete
0
,
otherwise
=
1
,
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbfv3ySLgzGueE0jxyaibaieYdOi=BH8vipeYdI8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0lXdbba9pGe9qqFf0dXdHuk9fr=xfr=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@7692@
and
z
M
2
(
r
)
=
{
1
,
for
a
r
allele from paternal gamete
0
,
otherwise
=
0
,
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbfv3ySLgzGueE0jxyaibaieYdOi=BH8vipeYdI8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0lXdbba9pGe9qqFf0dXdHuk9fr=xfr=xfrpiWZqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabdQha6naaDaaaleaacqWGnbqtdaWgaaadbaGaeGOmaidabeaaaSqaaiabcIcaOiabdkhaYjabcMcaPaaakiabg2da9maaceaabaqbaeaabiGaaaqaaiabigdaXiabcYcaSaqaaiabbAgaMjabb+gaVjabbkhaYjabbccaGGqaciab=fgaHnaaBaaaleaacqWGYbGCaeqaaOGaeeiiaaIaeeyyaeMaeeiBaWMaeeiBaWMaeeyzauMaeeiBaWMaeeyzauMaeeiiaaIaeeOzayMaeeOCaiNaee4Ba8MaeeyBa0MaeeiiaaIaeeiCaaNaeeyyaeMaeeiDaqNaeeyzauMaeeOCaiNaeeOBa4MaeeyyaeMaeeiBaWMaeeiiaaIaee4zaCMaeeyyaeMaeeyBa0MaeeyzauMaeeiDaqNaeeyzaugabaGaeGimaaJaeiilaWcabaGaee4Ba8MaeeiDaqNaeeiAaGMaeeyzauMaeeOCaiNaee4DaCNaeeyAaKMaee4CamNaeeyzauMaeeiiaacaaaGaay5EaaGaeyypa0JaeGimaaJaeiilaWcaaa@76D9@
thus
x
M
1
(
r
)
=
z
M
1
(
r
)
−
1
=
0
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbfv3ySLgzGueE0jxyaibaieYdOi=BH8vipeYdI8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0lXdbba9pGe9qqFf0dXdHuk9fr=xfr=xfrpiWZqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabdIha4naaDaaaleaacqWGnbqtdaWgaaadbaGaeGymaedabeaaaSqaaiabcIcaOiabdkhaYjabcMcaPaaakiabg2da9iabdQha6naaDaaaleaacqWGnbqtdaWgaaadbaGaeGymaedabeaaaSqaaiabcIcaOiabdkhaYjabcMcaPaaakiabgkHiTiabigdaXiabg2da9iabicdaWaaa@4170@
and
x
M
2
(
r
)
=
z
M
2
(
r
)
=
0
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbfv3ySLgzGueE0jxyaibaieYdOi=BH8vipeYdI8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0lXdbba9pGe9qqFf0dXdHuk9fr=xfr=xfrpiWZqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabdIha4naaDaaaleaacqWGnbqtdaWgaaadbaGaeGOmaidabeaaaSqaaiabcIcaOiabdkhaYjabcMcaPaaakiabg2da9iabdQha6naaDaaaleaacqWGnbqtdaWgaaadbaGaeGOmaidabeaaaSqaaiabcIcaOiabdkhaYjabcMcaPaaakiabg2da9iabicdaWaaa@3F97@
for r = 1, 2, ..., m . For maternal gametes however,
x
F
1
(
r
)
=
{
1
/
2
,
for
A
r
from maternal gamete
−
1
/
2
,
otherwise
=
−
x
F
2
(
r
)
,
for
r
=
1
,
2
,
⋅
⋅
⋅
,
m
.
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@8FA6@
Thus the model becomes
G
=
μ
+
∑
r
=
1
m
a
r
x
F
1
(
r
)
+
∑
r
<
s
b
r
s
(
x
F
1
(
r
)
x
F
1
(
s
)
)
+
∑
r
<
s
<
t
c
r
s
t
(
x
F
1
(
r
)
x
F
1
(
s
)
x
F
1
(
t
)
)
+
⋯
(
8
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@8A3E@
where
a
r
=
α
1
(
r
)
−
α
2
(
r
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbfv3ySLgzGueE0jxyaibaieYdOi=BH8vipeYdI8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0lXdbba9pGe9qqFf0dXdHuk9fr=xfr=xfrpiWZqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabdggaHnaaBaaaleaacqWGYbGCaeqaaOGaeyypa0dcciGae8xSde2aa0baaSqaaiabigdaXaqaaiabcIcaOiabdkhaYjabcMcaPaaakiabgkHiTiab=f7aHnaaDaaaleaacqaIYaGmaeaacqGGOaakcqWGYbGCcqGGPaqkaaaaaa@3F06@
is the substitution effect between homozygote genotype A
r
A
r
and heterozygote genotype
A
r
a
r
,
b
r
s
=
(
α
1
(
r
)
α
1
(
s
)
)
−
(
α
1
(
r
)
α
2
(
s
)
)
−
(
α
2
(
r
)
α
1
(
s
)
)
+
(
α
2
(
r
)
α
2
(
s
)
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbfv3ySLgzGueE0jxyaibaieYdOi=BH8vipeYdI8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0lXdbba9pGe9qqFf0dXdHuk9fr=xfr=xfrpiWZqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabdgeabnaaBaaaleaacqWGYbGCaeqaaOGaemyyae2aaSbaaSqaaiabdkhaYbqabaGccqGGSaalcqWGIbGydaWgaaWcbaGaemOCaiNaem4Camhabeaakiabg2da9iabcIcaOGGaciab=f7aHnaaDaaaleaacqaIXaqmaeaacqGGOaakcqWGYbGCcqGGPaqkaaGccqWFXoqydaqhaaWcbaGaeGymaedabaGaeiikaGIaem4CamNaeiykaKcaaOGaeiykaKIaeyOeI0IaeiikaGIae8xSde2aa0baaSqaaiabigdaXaqaaiabcIcaOiabdkhaYjabcMcaPaaakiab=f7aHnaaDaaaleaacqaIYaGmaeaacqGGOaakcqWGZbWCcqGGPaqkaaGccqGGPaqkcqGHsislcqGGOaakcqWFXoqydaqhaaWcbaGaeGOmaidabaGaeiikaGIaemOCaiNaeiykaKcaaOGae8xSde2aa0baaSqaaiabigdaXaqaaiabcIcaOiabdohaZjabcMcaPaaakiabcMcaPiabgUcaRiabcIcaOiab=f7aHnaaDaaaleaacqaIYaGmaeaacqGGOaakcqWGYbGCcqGGPaqkaaGccqWFXoqydaqhaaWcbaGaeGOmaidabaGaeiikaGIaem4CamNaeiykaKcaaOGaeiykaKcaaa@72E2@
is the interaction effect between loci r and
s
,
c
r
s
t
=
(
α
1
(
r
)
α
1
(
s
)
α
1
(
t
)
)
−
(
α
1
(
r
)
α
1
(
s
)
α
2
(
t
)
)
−
(
α
1
(
r
)
α
2
(
s
)
α
1
(
t
)
)
+
(
α
1
(
r
)
α
2
(
s
)
α
2
(
t
)
)
−
(
α
2
(
r
)
α
1
(
s
)
α
1
(
t
)
)
+
(
α
2
(
r
)
α
1
(
s
)
α
2
(
t
)
)
+
(
α
2
(
r
)
α
2
(
s
)
α
1
(
t
)
)
−
(
α
2
(
r
)
α
2
(
s
)
α
2
(
t
)
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbfv3ySLgzGueE0jxyaibaieYdOi=BH8vipeYdI8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0lXdbba9pGe9qqFf0dXdHuk9fr=xfr=xfrpiWZqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabdohaZjabcYcaSiabdogaJnaaBaaaleaacqWGYbGCcqWGZbWCcqWG0baDaeqaaOGaeyypa0JaeiikaGccciGae8xSde2aa0baaSqaaiabigdaXaqaaiabcIcaOiabdkhaYjabcMcaPaaakiab=f7aHnaaDaaaleaacqaIXaqmaeaacqGGOaakcqWGZbWCcqGGPaqkaaGccqWFXoqydaqhaaWcbaGaeGymaedabaGaeiikaGIaemiDaqNaeiykaKcaaOGaeiykaKIaeyOeI0IaeiikaGIae8xSde2aa0baaSqaaiabigdaXaqaaiabcIcaOiabdkhaYjabcMcaPaaakiab=f7aHnaaDaaaleaacqaIXaqmaeaacqGGOaakcqWGZbWCcqGGPaqkaaGccqWFXoqydaqhaaWcbaGaeGOmaidabaGaeiikaGIaemiDaqNaeiykaKcaaOGaeiykaKIaeyOeI0IaeiikaGIae8xSde2aa0baaSqaaiabigdaXaqaaiabcIcaOiabdkhaYjabcMcaPaaakiab=f7aHnaaDaaaleaacqaIYaGmaeaacqGGOaakcqWGZbWCcqGGPaqkaaGccqWFXoqydaqhaaWcbaGaeGymaedabaGaeiikaGIaemiDaqNaeiykaKcaaOGaeiykaKIaey4kaSIaeiikaGIae8xSde2aa0baaSqaaiabigdaXaqaaiabcIcaOiabdkhaYjabcMcaPaaakiab=f7aHnaaDaaaleaacqaIYaGmaeaacqGGOaakcqWGZbWCcqGGPaqkaaGccqWFXoqydaqhaaWcbaGaeGOmaidabaGaeiikaGIaemiDaqNaeiykaKcaaOGaeiykaKIaeyOeI0IaeiikaGIae8xSde2aa0baaSqaaiabikdaYaqaaiabcIcaOiabdkhaYjabcMcaPaaakiab=f7aHnaaDaaaleaacqaIXaqmaeaacqGGOaakcqWGZbWCcqGGPaqkaaGccqWFXoqydaqhaaWcbaGaeGymaedabaGaeiikaGIaemiDaqNaeiykaKcaaOGaeiykaKIaey4kaSIaeiikaGIae8xSde2aa0baaSqaaiabikdaYaqaaiabcIcaOiabdkhaYjabcMcaPaaakiab=f7aHnaaDaaaleaacqaIXaqmaeaacqGGOaakcqWGZbWCcqGGPaqkaaGccqWFXoqydaqhaaWcbaGaeGOmaidabaGaeiikaGIaemiDaqNaeiykaKcaaOGaeiykaKIaey4kaSIaeiikaGIae8xSde2aa0baaSqaaiabikdaYaqaaiabcIcaOiabdkhaYjabcMcaPaaakiab=f7aHnaaDaaaleaacqaIYaGmaeaacqGGOaakcqWGZbWCcqGGPaqkaaGccqWFXoqydaqhaaWcbaGaeGymaedabaGaeiikaGIaemiDaqNaeiykaKcaaOGaeiykaKIaeyOeI0IaeiikaGIae8xSde2aa0baaSqaaiabikdaYaqaaiabcIcaOiabdkhaYjabcMcaPaaakiab=f7aHnaaDaaaleaacqaIYaGmaeaacqGGOaakcqWGZbWCcqGGPaqkaaGccqWFXoqydaqhaaWcbaGaeGOmaidabaGaeiikaGIaemiDaqNaeiykaKcaaOGaeiykaKcaaa@D8C6@
, ..., and so on. Taking constraint conditions into account, we have α 1 = -α 2 , β 1 = -β 2 , and so on. Then, a
r
=
2
α
1
(
r
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbfv3ySLgzGueE0jxyaibaieYdOi=BH8vipeYdI8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0lXdbba9pGe9qqFf0dXdHuk9fr=xfr=xfrpiWZqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabikdaYGGaciab=f7aHnaaDaaaleaacqaIXaqmaeaacqGGOaakcqWGYbGCcqGGPaqkaaaaaa@3535@
, b
rs
= 4(
α
1
(
r
)
α
1
(
s
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbfv3ySLgzGueE0jxyaibaieYdOi=BH8vipeYdI8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0lXdbba9pGe9qqFf0dXdHuk9fr=xfr=xfrpiWZqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaGGaciab=f7aHnaaDaaaleaacqaIXaqmaeaacqGGOaakcqWGYbGCcqGGPaqkaaGccqWFXoqydaqhaaWcbaGaeGymaedabaGaeiikaGIaem4CamNaeiykaKcaaaaa@3A25@
), and c
rst
= 8(
α
1
(
r
)
α
1
(
s
)
α
1
(
t
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbfv3ySLgzGueE0jxyaibaieYdOi=BH8vipeYdI8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0lXdbba9pGe9qqFf0dXdHuk9fr=xfr=xfrpiWZqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaGGaciab=f7aHnaaDaaaleaacqaIXaqmaeaacqGGOaakcqWGYbGCcqGGPaqkaaGccqWFXoqydaqhaaWcbaGaeGymaedabaGaeiikaGIaem4CamNaeiykaKcaaOGae8xSde2aa0baaSqaaiabigdaXaqaaiabcIcaOiabdsha0jabcMcaPaaaaaa@4009@
), and so on. With linkage equilibrium, the genetic effects as the partial regression coefficients of the model correspond to the simple regression coefficients. For example, for the substitution effect of locus r , a
r
, it is the covariance between genotypic value G and substitution effect design variable
x
F
1
(
r
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG4baEdaqhaaWcbaGaemOray0aaSbaaWqaaiabigdaXaqabaaaleaacqGGOaakcqWGYbGCcqGGPaqkaaaaaa@33AE@
divided by the variance of
x
F
1
(
r
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG4baEdaqhaaWcbaGaemOray0aaSbaaWqaaiabigdaXaqabaaaleaacqGGOaakcqWGYbGCcqGGPaqkaaaaaa@33AE@
. So in general, we have
a
r
=
E
[
(
G
−
μ
)
x
F
1
(
r
)
]
/
E
(
x
F
1
(
r
)
2
)
=
E
[
G
(
z
F
1
(
r
)
−
1
/
2
)
]
/
(
1
/
4
)
=
2
[
E
(
G
|
z
F
1
(
r
)
=
1
)
−
E
(
G
)
]
b
r
s
=
E
[
(
G
−
μ
)
x
F
1
(
r
)
x
F
1
(
s
)
]
/
[
E
(
x
F
1
(
r
)
2
)
E
(
x
F
1
(
s
)
2
)
]
=
4
[
E
(
G
|
z
F
1
(
r
)
=
z
F
1
(
s
)
=
1
)
−
E
(
G
|
z
F
1
(
r
)
=
1
)
−
E
(
G
|
z
F
1
(
s
)
=
1
)
+
E
(
G
)
]
c
r
s
t
=
E
[
(
G
−
μ
)
x
F
1
(
r
)
x
F
1
(
s
)
x
F
1
(
t
)
]
/
[
E
(
x
F
1
(
r
)
2
)
E
(
x
F
1
(
s
)
2
)
E
(
x
F
1
(
t
)
2
)
]
=
8
[
E
(
G
|
z
F
1
(
r
)
=
z
F
1
(
s
)
=
z
F
1
(
t
)
=
1
)
−
E
(
G
|
z
F
1
(
r
)
=
z
F
1
(
s
)
=
1
)
−
E
(
G
|
z
F
1
(
r
)
=
z
F
1
(
t
)
=
1
)
−
E
(
G
|
z
F
1
(
s
)
=
z
F
1
(
t
)
=
1
)
+
E
(
G
|
z
F
1
(
r
)
=
1
)
+
E
(
G
|
z
F
1
(
s
)
=
1
)
+
E
(
G
|
z
F
1
(
t
)
=
1
)
−
E
(
G
)
]
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@D9C6@
The orthogonal partition of the genotypic variance in an equilibrium population is
V
G
=
1
4
∑
r
=
1
l
a
r
2
+
1
4
2
∑
r
<
s
b
r
s
2
+
1
4
3
∑
r
<
s
<
t
c
r
s
t
2
+
⋯
(
9
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@6623@
As noted above, linkage disequilibrium does not change the values of genetic effects in a full model. The model parameters are still the same as those defined in the equilibrium population. However, in this case there is a simple relationship between the substitution effects at multiple loci and marginal means of genotypic values in a disequilibrium population [18 ]. This is noted here. Let
P
r
s
=
P
{
z
F
1
(
r
)
=
z
F
1
(
s
)
=
1
}
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGqbaudaWgaaWcbaGaemOCaiNaem4Camhabeaakiabg2da9iabdcfaqjabcUha7jabdQha6naaDaaaleaacqWGgbGrdaWgaaadbaGaeGymaedabeaaaSqaaiabcIcaOiabdkhaYjabcMcaPaaakiabg2da9iabdQha6naaDaaaleaacqWGgbGrdaWgaaadbaGaeGymaedabeaaaSqaaiabcIcaOiabdohaZjabcMcaPaaakiabg2da9iabigdaXiabc2ha9baa@4734@
, and the digenic linkage disequilibrium be defined as
D
r
s
=
Cov(
z
F
1
(
r
)
,
z
F
1
(
s
)
)
=
E
(
x
F
1
(
r
)
x
F
1
(
s
)
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@5852@
Ignoring trigenic and higher order linkage disequilibria, we have
E
[
(
G
−
μ
)
x
F
1
(
r
)
]
=
1
4
a
r
+
∑
r
'
≠
r
D
r
r
'
a
r
'
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGfbqrdaWadaqaaiabcIcaOiabdEeahjabgkHiTGGaciab=X7aTjabcMcaPiabdIha4naaDaaaleaacqWGgbGrdaWgaaadbaGaeGymaedabeaaaSqaaiabcIcaOiabdkhaYjabcMcaPaaaaOGaay5waiaaw2faaiabg2da9maalaaabaGaeGymaedabaGaeGinaqdaaiabdggaHnaaBaaaleaacqWGYbGCaeqaaOGaey4kaSYaaabuaeaacqWGebardaWgaaWcbaGaemOCaiNaemOCaiNaei4jaCcabeaakiabdggaHnaaBaaaleaacqWGYbGCcqGGNaWjaeqaaaqaaiabdkhaYjabcEcaNiabgcMi5kabdkhaYbqab0GaeyyeIuoaaaa@533C@
E
[
(
G
−
μ
)
x
F
1
(
r
)
x
F
1
(
s
)
]
=
1
4
2
b
r
s
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGfbqrcqGGBbWwcqGGOaakcqWGhbWrcqGHsisliiGacqWF8oqBcqGGPaqkcqWG4baEdaqhaaWcbaGaemOray0aaSbaaWqaaiabigdaXaqabaaaleaacqGGOaakcqWGYbGCcqGGPaqkaaGccqWG4baEdaqhaaWcbaGaemOray0aaSbaaWqaaiabigdaXaqabaaaleaacqGGOaakcqWGZbWCcqGGPaqkaaGccqGGDbqxcqGH9aqpdaWcaaqaaiabigdaXaqaaiabisda0maaCaaaleqabaGaeGOmaidaaaaakiabdkgaInaaBaaaleaacqWGYbGCcqWGZbWCaeqaaaaa@4C46@
Therefore, the digenic interaction effects can be expressed as
b
r
s
=
4
2
[
P
r
s
E
(
G
|
z
F
1
(
r
)
=
z
F
1
(
s
)
=
1
)
−
D
r
s
μ
]
−
4
[
E
(
G
|
z
F
1
(
r
)
=
1
)
+
E
(
G
|
z
F
1
(
s
)
=
1
)
−
E
(
G
)
]
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@7FB5@
Then the substitution effects can be expressed as a function of marginal means in the disequilibrium population as
(
a
1
a
2
⋮
a
l
)
=
(
I
+
4
D
)
−
1
(
2
q
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaqadaqaauaabeqaeeaaaaqaaiabdggaHnaaBaaaleaacqaIXaqmaeqaaaGcbaGaemyyae2aaSbaaSqaaiabikdaYaqabaaakeaacqWIUlstaeaacqWGHbqydaWgaaWcbaGaemiBaWgabeaaaaaakiaawIcacaGLPaaacqGH9aqpcqGGOaakcqWGjbqscqGHRaWkcqaI0aancqWGebarcqGGPaqkdaahaaWcbeqaaiabgkHiTiabigdaXaaakiabcIcaOiabikdaYmXvP5wqSXMqHnxAJn0BKvguHDwzZbqegyvzYrwyUfgaiqqacaWFXbaceaGaa4xkaaaa@4EC7@
where I is a m × m identity matrix, D = (D
ij
)m × m with all diagonal elements being zeros;
q = (q 1 , q 2 , ..., q
m
,)T , with q
i
= E (G |
z
F
1
(
i
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG6bGEdaqhaaWcbaGaemOray0aaSbaaWqaaiabigdaXaqabaaaleaacqGGOaakcqWGPbqAcqGGPaqkaaaaaa@33A0@
= 1) - E (G ), for i = 1, 2, ..., m .
The partition of genetic variance with linkage disequilibrium is complex. Here we give details of the partition of genotypic variance for the following model
G
=
μ
+
∑
r
=
1
m
a
r
x
F
1
(
r
)
+
∑
r
<
s
b
r
s
(
x
F
1
(
r
)
x
F
1
(
s
)
)
(
10
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@620F@
Let x
r
=
x
F
1
(
r
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG4baEdaqhaaWcbaGaemOray0aaSbaaWqaaiabigdaXaqabaaaleaacqGGOaakcqWGYbGCcqGGPaqkaaaaaa@33AE@
and x
s
=
x
F
1
(
s
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG4baEdaqhaaWcbaGaemOray0aaSbaaWqaaiabigdaXaqabaaaleaacqGGOaakcqWGZbWCcqGGPaqkaaaaaa@33B0@
to simplify the notation here. The genotypic variance is
V
G
=
V
(
∑
r
a
r
x
r
)
+
2
Cov(
∑
r
a
r
x
r
,
∑
r
<
s
b
r
s
x
r
x
s
)
+
V
(
∑
r
<
s
b
r
s
x
r
x
s
)
=
∑
r
a
r
2
p
r
(
1
−
p
r
)
+
2
∑
r
<
s
a
r
a
s
D
r
s
+
2
∑
r
<
s
[
a
r
b
r
s
(
1
−
2
p
r
)
D
r
s
+
a
s
b
r
s
(
1
−
2
p
s
)
D
r
s
]
+
2
∑
r
<
s
<
t
(
a
r
b
s
t
+
a
s
b
r
t
+
a
t
b
r
s
)
D
r
s
t
+
∑
r
<
s
b
r
s
2
[
(
1
−
2
p
r
)
(
1
−
2
p
s
)
D
r
s
+
p
r
(
1
−
p
r
)
p
s
(
1
−
p
s
)
−
D
r
s
2
]
+
2
∑
r
<
s
<
t
{
b
r
s
b
r
t
[
(
1
−
2
p
r
)
D
r
s
t
−
p
r
(
1
−
p
r
)
D
s
t
−
D
r
s
D
r
t
]
+
b
r
s
b
s
t
[
(
1
−
2
p
s
)
D
r
s
t
+
p
s
(
1
−
p
s
)
D
r
t
−
D
r
s
D
s
t
]
+
b
r
t
b
s
t
[
(
1
−
2
p
t
)
D
r
s
t
+
p
t
(
1
−
p
t
)
D
r
s
−
D
r
t
D
s
t
]
}
+
2
∑
r
<
s
<
t
<
u
[
b
r
t
b
s
u
(
D
r
s
t
u
−
D
r
t
D
s
u
)
+
b
r
u
b
s
t
(
D
r
s
t
u
−
D
r
u
D
s
t
)
+
b
r
s
b
t
u
(
D
r
s
t
u
−
D
r
s
D
t
u
)
]
(
11
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakqaabeqaaiabdAfawnaaBaaaleaacqWGhbWraeqaaOGaeyypa0JaemOvayLaeiikaGYaaabuaeaacqWGHbqydaWgaaWcbaGaemOCaihabeaaaeaacqWGYbGCaeqaniabggHiLdGccqWG4baEdaWgaaWcbaGaemOCaihabeaakiabcMcaPiabgUcaRiabikdaYiabboeadjabb+gaVjabbAha2jabbIcaOmaaqafabaGaemyyae2aaSbaaSqaaiabdkhaYbqabaGccqWG4baEdaWgaaWcbaGaemOCaihabeaakiabcYcaSmaaqafabaGaemOyai2aaSbaaSqaaiabdkhaYjabdohaZbqabaGccqWG4baEdaWgaaWcbaGaemOCaihabeaakiabdIha4naaBaaaleaacqWGZbWCaeqaaOGaeiykaKcaleaacqWGYbGCcqGH8aapcqWGZbWCaeqaniabggHiLdaaleaacqWGYbGCaeqaniabggHiLdaakeaacqGHRaWkcqqGwbGvcqGGOaakdaaeqbqaaiabdkgaInaaBaaaleaacqWGYbGCcqWGZbWCaeqaaOGaemiEaG3aaSbaaSqaaiabdkhaYbqabaGccqWG4baEdaWgaaWcbaGaem4CamhabeaakiabcMcaPaWcbaGaemOCaiNaeyipaWJaem4CamhabeqdcqGHris5aaGcbaGaeyypa0ZaaabuaeaacqWGHbqydaqhaaWcbaGaemOCaihabaGaeGOmaidaaOGaemiCaa3aaSbaaSqaaiabdkhaYbqabaGccqGGOaakcqaIXaqmcqGHsislcqWGWbaCdaWgaaWcbaGaemOCaihabeaakiabcMcaPaWcbaGaemOCaihabeqdcqGHris5aOGaey4kaSIaeGOmaiZaaabuaeaacqWGHbqydaWgaaWcbaGaemOCaihabeaakiabdggaHnaaBaaaleaacqWGZbWCaeqaaOGaemiraq0aaSbaaSqaaiabdkhaYjabdohaZbqabaaabaGaemOCaiNaeyipaWJaem4CamhabeqdcqGHris5aaGcbaGaey4kaSIaeGOmaiZaaabuaeaacqGGBbWwcqWGHbqydaWgaaWcbaGaemOCaihabeaakiabdkgaInaaBaaaleaacqWGYbGCcqWGZbWCaeqaaOGaeiikaGIaeGymaeJaeyOeI0IaeGOmaiJaemiCaa3aaSbaaSqaaiabdkhaYbqabaGccqGGPaqkcqWGebardaWgaaWcbaGaemOCaiNaem4CamhabeaakiabgUcaRiabdggaHnaaBaaaleaacqWGZbWCaeqaaOGaemOyai2aaSbaaSqaaiabdkhaYjabdohaZbqabaGccqGGOaakcqaIXaqmcqGHsislcqaIYaGmcqWGWbaCdaWgaaWcbaGaem4CamhabeaakiabcMcaPiabdseaenaaBaaaleaacqWGYbGCcqWGZbWCaeqaaOGaeiyxa0faleaacqWGYbGCcqGH8aapcqWGZbWCaeqaniabggHiLdaakeaacqGHRaWkcqaIYaGmdaaeqbqaaiabcIcaOiabdggaHnaaBaaaleaacqWGYbGCaeqaaOGaemOyai2aaSbaaSqaaiabdohaZjabdsha0bqabaGccqGHRaWkcqWGHbqydaWgaaWcbaGaem4CamhabeaakiabdkgaInaaBaaaleaacqWGYbGCcqWG0baDaeqaaOGaey4kaSIaemyyae2aaSbaaSqaaiabdsha0bqabaGccqWGIbGydaWgaaWcbaGaemOCaiNaem4CamhabeaakiabcMcaPiabdseaenaaBaaaleaacqWGYbGCcqWGZbWCcqWG0baDaeqaaaqaaiabdkhaYjabgYda8iabdohaZjabgYda8iabdsha0bqab0GaeyyeIuoaaOqaaiabgUcaRmaaqafabaGaemOyai2aa0baaSqaaiabdkhaYjabdohaZbqaaiabikdaYaaaaeaacqWGYbGCcqGH8aapcqWGZbWCaeqaniabggHiLdGcdaWabaqaaiabcIcaOiabigdaXiabgkHiTiabikdaYiabdchaWnaaBaaaleaacqWGYbGCaeqaaOGaeiykaKIaeiikaGIaeGymaeJaeyOeI0IaeGOmaiJaemiCaa3aaSbaaSqaaiabdohaZbqabaGccqGGPaqkcqWGebardaWgaaWcbaGaemOCaiNaem4CamhabeaaaOGaay5waaaabaWaamGaaeaacqGHRaWkcqWGWbaCdaWgaaWcbaGaemOCaihabeaakiabcIcaOiabigdaXiabgkHiTiabdchaWnaaBaaaleaacqWGYbGCaeqaaOGaeiykaKIaemiCaa3aaSbaaSqaaiabdohaZbqabaGccqGGOaakcqaIXaqmcqGHsislcqWGWbaCdaWgaaWcbaGaem4CamhabeaakiabcMcaPiabgkHiTiabdseaenaaDaaaleaacqWGYbGCcqWGZbWCaeaacqaIYaGmaaaakiaaw2faaaqaaiabgUcaRiabikdaYmaaqafabaWaaiqaaeaacqWGIbGydaWgaaWcbaGaemOCaiNaem4CamhabeaakiabdkgaInaaBaaaleaacqWGYbGCcqWG0baDaeqaaOGaei4waSLaeiikaGIaeGymaeJaeyOeI0IaeGOmaiJaemiCaa3aaSbaaSqaaiabdkhaYbqabaGccqGGPaqkcqWGebardaWgaaWcbaGaemOCaiNaem4CamNaemiDaqhabeaaaOGaay5EaaaaleaacqWGYbGCcqGH8aapcqWGZbWCcqGH8aapcqWG0baDaeqaniabggHiLdaakeaacqGHsislcqWGWbaCdaWgaaWcbaGaemOCaihabeaakiabcIcaOiabigdaXiabgkHiTiabdchaWnaaBaaaleaacqWGYbGCaeqaaOGaeiykaKIaemiraq0aaSbaaSqaaiabdohaZjabdsha0bqabaGccqGHsislcqWGebardaWgaaWcbaGaemOCaiNaem4CamhabeaakiabdseaenaaBaaaleaacqWGYbGCcqWG0baDaeqaaOGaeiyxa0fabaGaey4kaSIaemOyai2aaSbaaSqaaiabdkhaYjabdohaZbqabaGccqWGIbGydaWgaaWcbaGaem4CamNaemiDaqhabeaakiabcUfaBjabcIcaOiabigdaXiabgkHiTiabikdaYiabdchaWnaaBaaaleaacqWGZbWCaeqaaOGaeiykaKIaemiraq0aaSbaaSqaaiabdkhaYjabdohaZjabdsha0bqabaGccqGHRaWkcqWGWbaCdaWgaaWcbaGaem4CamhabeaakiabcIcaOiabigdaXiabgkHiTiabdchaWnaaBaaaleaacqWGZbWCaeqaaOGaeiykaKIaemiraq0aaSbaaSqaaiabdkhaYjabdsha0bqabaGccqGHsislcqWGebardaWgaaWcbaGaemOCaiNaem4CamhabeaakiabdseaenaaBaaaleaacqWGZbWCcqWG0baDaeqaaOGaeiyxa0fabaGaey4kaSIaemOyai2aaSbaaSqaaiabdkhaYjabdsha0bqabaGccqWGIbGydaWgaaWcbaGaem4CamNaemiDaqhabeaakiabcUfaBjabcIcaOiabigdaXiabgkHiTiabikdaYiabdchaWnaaBaaaleaacqWG0baDaeqaaOGaeiykaKIaemiraq0aaSbaaSqaaiabdkhaYjabdohaZjabdsha0bqabaaakeaadaGacaqaaiabgUcaRiabdchaWnaaBaaaleaacqWG0baDaeqaaOGaeiikaGIaeGymaeJaeyOeI0IaemiCaa3aaSbaaSqaaiabdsha0bqabaGccqGGPaqkcqWGebardaWgaaWcbaGaemOCaiNaem4CamhabeaakiabgkHiTiabdseaenaaBaaaleaacqWGYbGCcqWG0baDaeqaaOGaemiraq0aaSbaaSqaaiabdohaZjabdsha0bqabaGccqGGDbqxaiaaw2haaaqaaiabgUcaRiabikdaYmaaqafabaGaei4waSLaemOyai2aaSbaaSqaaiabdkhaYjabdsha0bqabaGccqWGIbGydaWgaaWcbaGaem4CamNaemyDauhabeaakiabcIcaOiabdseaenaaBaaaleaacqWGYbGCcqWGZbWCcqWG0baDcqWG1bqDaeqaaOGaeyOeI0Iaemiraq0aaSbaaSqaaiabdkhaYjabdsha0bqabaaabaGaemOCaiNaeyipaWJaem4CamNaeyipaWJaemiDaqNaeyipaWJaemyDauhabeqdcqGHris5aOGaemiraq0aaSbaaSqaaiabdohaZjabdwha1bqabaGccqGGPaqkaeaacqGHRaWkcqWGIbGydaWgaaWcbaGaemOCaiNaemyDauhabeaakiabdkgaInaaBaaaleaacqWGZbWCcqWG0baDaeqaaOGaeiikaGIaemiraq0aaSbaaSqaaiabdkhaYjabdohaZjabdsha0jabdwha1bqabaGccqGHsislcqWGebardaWgaaWcbaGaemOCaiNaemyDauhabeaakiabdseaenaaBaaaleaacqWGZbWCcqWG0baDaeqaaOGaeiykaKcabaGaey4kaSIaemOyai2aaSbaaSqaaiabdkhaYjabdohaZbqabaGccqWGIbGydaWgaaWcbaGaemiDaqNaemyDauhabeaakiabcIcaOiabdseaenaaBaaaleaacqWGYbGCcqWGZbWCcqWG0baDcqWG1bqDaeqaaOGaeyOeI0Iaemiraq0aaSbaaSqaaiabdkhaYjabdohaZbqabaGccqWGebardaWgaaWcbaGaemiDaqNaemyDauhabeaakiabcMcaPiabc2faDjaaxMaacaWLjaWaaeWaaeaacqaIXaqmcqaIXaqmaiaawIcacaGLPaaaaaaa@32E1@
where
D
rst
= E (x
r
x
s
x
t
) and D
rstu
= E (x
r
x
s
x
t
x
u
)
are three locus and four locus linkage disequilibria. This is a general partition of genetic variance for a haploid model.
For the backcross population, it can be shown that D
rst
= 0 (see Appendix D for both backcross and F 2 populations) and D
rstu
= D
rs
D
tu
for loci r , s , t and u in this order under the assumption of no crossing-over interference. Also with this assumption, D
rt
= 4D
rs
D
st
and D
rs
= (1 - 2λ
rs
)/4, where λ
rs
is the recombination frequency between loci r and s . Since, p
r
= p
s
= 1/2, the variance becomes
V
G
=
1
4
∑
r
a
r
2
+
2
∑
r
<
s
a
r
a
s
D
r
s
+
1
16
∑
r
<
s
b
r
s
2
(
1
−
16
D
r
s
2
)
+
1
2
∑
r
<
s
<
t
[
b
r
s
b
r
t
(
1
−
16
D
r
s
2
)
D
s
t
+
b
r
t
b
s
t
(
1
−
16
D
s
t
2
)
D
r
s
]
+
2
∑
r
<
s
<
t
<
u
(
b
r
t
b
s
u
+
b
r
u
b
s
t
)
(
1
−
16
D
s
t
2
)
D
r
s
D
t
u
(
12
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@DAF7@
In this partition of variance, the first summation term is the genetic variance due to the substitution effect of each QTL, the second summation term is the covariance between substitution effects of QTL pairs due to linkage disequilibrium, the third summation term is the genetic variance due to epistatic effects of QTL, and the fourth and fifth summation terms are the covariance between different epistatic effects of QTL due to linkage disequilibrium. There is no covariance between the main substitution effects and epistatic effects (see also [19 ]).
For a backcross population, the genetic interpretation of the substitution effect a
r
depends on which parental line is backcrossed. In one backcross AA × Aa , the substitution effect is traditionally defined as the difference between the additive effect and dominance effects, and in the other backcross Aa × aa , it is the sum of the additive and dominance effects. Only with both backcrosses, can one estimate both additive and dominance effects separately (for example [20 ]).
The same model also applies to a recombinant inbred population which is another very popular experimental design for QTL mapping study. For a recombinant inbred population, the substitution effects of QTL are the additive effects and the epistatic effects are the additive × additive interaction effects. Statistical methods to map QTL and to estimate various components of the genetic variance due to QTL including epistasis has been developed through the maximum likelihood approach [19 , 21 ]. In a few cases where the method was applied, we estimated, for the first time, how the quantitative genetic variance was partitioned into various components in designed experimental populations. For example, Weber et al. [22 ] reported the result of QTL mapping for wing shape on the third chromosome of Drosophila melanogaster from a cross of divergent selection lines. From 519 recombinant inbred lines, 11 QTL were mapped on the third chromosome. Nine QTL pairs showed significant epistatic effects. The total genetic variance amounts to 95.5% of the phenotypic variance in the recombinant inbred lines with phenotypes measured and averaged over 50 male flies for each recombinant inbred line. The partition of the genetic variance is as follows (see Table 6 and 7 of [22 ]): 27.4% due to the variances of additive effects (equivalent to the first summation term of (12)); 67.3% due to the covariances between additive effects (the second summation term); 7.2% due to the variances of epistatic effects (the third summation term); and -6.0% due to the covariances between epistatic effects (the fourth and fifth summation terms). The covariances between additive and epistatic effects, expected to be 0, account for -0.4% due to sampling. Similar kind of partition of the genetic variance is also observed in a group of 701 second chromosome recombinant inbred lines from a cross of the same divergent selection lines (see Table 4 and 5 of [23 ]). See also [20 ] for another example.
2. F 2 population F 2 is created from a cross between pairs of F 1 individuals. It is also a very popular experimental design for QTL mapping study. The advantage of this design is that both additive and dominance effects of a QTL can be estimated as well as various epistatic effects. The design also has more statistical power for QTL detection as compared to a backcross population. In a random-mating F 2 population, there are only two alleles at each segregating locus and allelic frequencies are expected to be one half if there is no segregation distortion.
Let us consider only two loci first. Let A and a denote the two alleles at locus 1, and B and b at locus 2. In this case,
x
M
1
(
1
)
=
−
x
M
2
(
1
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG4baEdaqhaaWcbaGaemyta00aaSbaaWqaaiabigdaXaqabaaaleaacqGGOaakcqaIXaqmcqGGPaqkaaGccqGH9aqpcqGHsislcqWG4baEdaqhaaWcbaGaemyta00aaSbaaWqaaiabikdaYaqabaaaleaacqGGOaakcqaIXaqmcqGGPaqkaaaaaa@3BD1@
and
x
F
1
(
1
)
=
−
x
F
2
(
1
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG4baEdaqhaaWcbaGaemOray0aaSbaaWqaaiabigdaXaqabaaaleaacqGGOaakcqaIXaqmcqGGPaqkaaGccqGH9aqpcqGHsislcqWG4baEdaqhaaWcbaGaemOray0aaSbaaWqaaiabikdaYaqabaaaleaacqGGOaakcqaIXaqmcqGGPaqkaaaaaa@3BB5@
. Assuming
G
j
l
i
j
=
G
i
k
j
l
=
G
j
k
i
l
=
G
i
l
j
k
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGhbWrdaqhaaWcbaGaemOAaOMaemiBaWgabaGaemyAaKMaemOAaOgaaOGaeyypa0Jaem4raC0aa0baaSqaaiabdMgaPjabdUgaRbqaaiabdQgaQjabdYgaSbaakiabg2da9iabdEeahnaaDaaaleaacqWGQbGAcqWGRbWAaeaacqWGPbqAcqWGSbaBaaGccqGH9aqpcqWGhbWrdaqhaaWcbaGaemyAaKMaemiBaWgabaGaemOAaOMaem4AaSgaaaaa@4ACA@
, it also holds that α 1 = α 1 = -α 2 = -α 2 ,
δ
1
1
=
δ
2
2
=
−
δ
1
2
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWF0oazdaqhaaWcbaGaeGymaedabaGaeGymaedaaOGaeyypa0Jae8hTdq2aa0baaSqaaiabikdaYaqaaiabikdaYaaakiabg2da9iabgkHiTiab=r7aKnaaDaaaleaacqaIXaqmaeaacqaIYaGmaaaaaa@3AD2@
, and so on. The additive term for locus 1 then becomes
A
1
=
α
1
(
x
M
1
(
1
)
−
x
M
2
(
1
)
)
+
α
1
(
x
F
1
(
1
)
−
x
F
2
(
1
)
)
=
2
α
1
w
1
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGbbqqdaWgaaWcbaGaeGymaedabeaakiabg2da9GGaciab=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@5DFE@
with
w
1
=
x
M
1
(
1
)
+
x
F
1
(
1
)
=
{
1
,
for
A
A
at locus 1
0, for
A
a
at locus 1
−
1
,
for
a
a
at locus 1
(
13
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@8A53@
and the dominance term is
D
1
=
2
(
δ
1
1
)
(
x
M
1
(
1
)
x
F
1
(
1
)
+
x
M
1
(
1
)
x
F
1
(
1
)
)
=
(
−
2
)
(
δ
1
1
)
v
1
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGebardaWgaaWcbaGaeGymaedabeaakiabg2da9iabikdaYiabcIcaOGGaciab=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@5FCD@
with
v
1
=
(
−
2
)
x
M
1
(
1
)
x
F
1
(
1
)
=
{
−
1
/
2
,
for
A
A
at locus 1
1
/
2
,
for
A
a
at locus 1
−
1
/
2
,
for
a
a
at locus 1
(
14
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@9382@
Note that the v variable in this section for F 2 differ, by a factor -2, from the v variable in the next section for a general two-allele model to conform to the usual definition for the F 2 model. Similarly, for locus 2
A 2 = 2β 1 w 2 and D 2 = (-2)(
γ
1
2
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFZoWzdaqhaaWcbaGaeeymaedabaGaeeOmaidaaaaa@305B@
)v 2
with
w
2
=
{
1
,
for
B
B
at locus 2
0
,
for
B
b
at locus 2
−
1
,
for
b
b
at locus 2
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@76AB@
and
v
2
=
{
−
1
/
2
,
for
B
B
at locus 2
1
/
2
,
for
B
b
at locus 2
−
1
/
2
,
for
b
b
at locus 2
(
15
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG2bGDdaWgaaWcbaGaeGOmaidabeaakiabg2da9maaceaabaqbaeqabmqaaaqaaiabgkHiTiabigdaXiabc+caViabikdaYiabcYcaSiabbccaGiabbAgaMjabb+gaVjabbkhaYjabbccaGiabdkeacjabdkeacjabbccaGiabbggaHjabbsha0jabbccaGiabbYgaSjabb+gaVjabbogaJjabbwha1jabbohaZjabbccaGiabbkdaYaqaaiabigdaXiabc+caViabikdaYiabcYcaSiabbccaGiabbAgaMjabb+gaVjabbkhaYjabbccaGiabdkeacjabdkgaIjabbccaGiabbggaHjabbsha0jabbccaGiabbYgaSjabb+gaVjabbogaJjabbwha1jabbohaZjabbccaGiabbkdaYaqaaiabgkHiTiabigdaXiabc+caViabikdaYiabcYcaSiabbccaGiabbAgaMjabb+gaVjabbkhaYjabbccaGiabdkgaIjabdkgaIjabbccaGiabbggaHjabbsha0jabbccaGiabbYgaSjabb+gaVjabbogaJjabbwha1jabbohaZjabbccaGiabbkdaYaaaaiaawUhaaiaaxMaacaWLjaWaaeWaaeaacqaIXaqmcqaI1aqnaiaawIcacaGLPaaaaaa@81D5@
The model can then be written as
G
=
μ
+
a
1
w
1
+
d
1
v
1
+
a
2
w
2
+
d
2
v
2
+
(
a
a
)
12
(
w
1
w
2
)
+
(
a
d
)
12
(
w
1
v
2
)
+
(
d
a
)
12
(
v
1
w
2
)
+
(
d
d
)
12
(
v
1
v
2
)
(
16
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@85EC@
where the parameters are related as a 1 = 2α 1 , a 2 = 2β 2 , d 1 = -
−
2
δ
1
1
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqGHsislcqaIYaGmiiGacqWF0oazdaqhaaWcbaGaeGymaedabaGaeGymaedaaaaa@3244@
, d 2 = -
−
2
γ
1
1
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqGHsislcqaIYaGmiiGacqWFZoWzdaqhaaWcbaGaeGymaedabaGaeGymaedaaaaa@3246@
, (aa )12 = 4(α 1 β 1 ), (ad )12 = 8(
α
1
γ
1
1
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqee0evGueE0jxyaibaiKI8=vI8tuQ8FMI8Gi=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciGacaGaaeqabaqadeqadaaakeaaiiGacqWFXoqydaWgaaWcbaGaaGymaaqabaGccqWFZoWzdaqhaaWcbaGaaGymaaqaaiaaigdaaaaaaa@3909@
), (da )12 = 8(
δ
1
1
β
1
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWF0oazdaqhaaWcbaGaeGymaedabaGaeGymaedaaOGae8NSdi2aaSbaaSqaaiabigdaXaqabaaaaa@3327@
), (dd )12 = 16(
δ
1
1
γ
1
1
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWF0oazdaqhaaWcbaGaeGymaedabaGaeGymaedaaOGae83SdC2aa0baaSqaaiabigdaXaqaaiabigdaXaaaaaa@341E@
). With random mating and linkage equilibrium, we have
a
1
=
E
[
G
−
μ
)
w
1
]
/
E
(
w
1
2
)
=
2
(
G
1.
..
−
G
..
..
)
d
1
=
E
[
(
G
−
μ
)
v
1
]
/
E
(
v
2
1
)
=
(
−
2
)
(
G
1.
1.
−
2
G
1.
..
+
G
..
..
)
a
2
=
2
(
G
.1
..
−
G
..
..
)
d
2
=
(
−
2
)
(
G
.1
.1
−
2
G
.1
..
+
G
..
..
)
(
a
a
)
12
=
E
[
(
G
−
μ
)
w
1
w
2
]
/
E
(
w
1
2
w
2
2
)
=
4
(
G
..
11
−
G
1.
..
−
G
.1
..
+
G
..
..
(
a
d
)
12
=
E
[
(
G
−
μ
)
w
1
v
2
]
/
E
(
w
1
2
v
2
2
)
=
(
−
4
)
(
G
11
.1
−
2
G
11
..
−
G
.1
.1
+
G
1.
..
+
2
G
.1
..
−
G
..
..
)
(
d
a
)
12
=
E
[
(
G
−
μ
)
v
1
w
2
]
/
E
(
v
1
2
w
2
2
)
=
(
−
4
)
(
G
11
1.
−
2
G
11
..
−
G
1.
1.
+
G
.1
..
+
2
G
1.
..
−
G
..
..
)
(
d
d
)
12
=
E
[
(
G
−
μ
)
v
1
v
2
]
/
E
(
v
1
2
v
2
2
)
=
4
(
G
11
11
−
2
G
11
1.
−
2
G
11
.1
+
G
1.
1.
+
G
.1
.1
+
4
G
11
..
−
2
G
1.
..
−
2
G
.1
..
+
G
..
..
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@D683@
The orthogonal partition of the genotypic variance is
V
G
=
1
2
a
1
2
+
1
4
d
1
2
+
1
2
a
2
2
+
1
4
d
2
2
+
1
4
(
a
a
)
12
2
+
1
8
(
a
d
)
12
2
+
1
8
(
d
a
)
12
2
+
1
16
(
d
d
)
12
2
.
(
17
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@77BF@
Recently, Kao and Zeng [18 ] have examined many genetic and statistical issues of the above F 2 model and the effects of linkage disequilibrium. As we have shown here, the F 2 model is a special case of Cockerham model with two alleles at each locus and all allelic frequencies being 1/2.
Now we give the partition of genetic variance for m loci with epistasis and linkage disequilibrium in the F 2 population. Generalizing model (16) to m loci and ignoring the trigenic and higher order epistasis, we have the following model
G
=
μ
+
∑
r
=
1
m
a
r
w
r
+
∑
r
=
1
m
d
r
v
r
+
∑
r
<
s
(
a
a
)
r
s
(
w
r
w
s
)
+
∑
r
≠
s
(
a
d
)
r
s
(
w
r
v
s
)
+
∑
r
<
s
(
d
d
)
r
s
(
v
r
v
s
)
=
μ
+
A
+
D
+
A
A
+
A
d
+
D
D
(
18
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@A465@
The partition of genetic variance for this model under the assumption of Hardy-Weinberg equilibrium is
V
G
=
V
A
+
V
D
+
V
A
A
+
V
A
D
+
V
D
D
+
2
Cov
(
A
,
D
)
+
2
Cov
(
A
,
A
A
)
+
2
Cov
(
A
,
A
D
)
+
2
Cov(A,DD)+2Cov
(
D
,
A
A
)
+
2
Cov
(
D
,
A
D
)
+
2
Cov(
D
,
D
D
)
+
2
C
o
v
(
A
A
,
A
D
)
+
2
Cov(
A
A
,
D
D
)
+
2
Cov
(
A
D
,
D
D
)
.
(
19
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@BD98@
The detail of each component is presented in Appendix D.
The F 2 model is a special case of the general two-allele model with p
r
= 1/2. Note the difference on the v variable used for the F 2 model and for the general two-allele model below. This partition of genetic variance can provide a basis for the interpretation of genetic variance estimation by multiple interval mapping in a F 2 population [19 , 21 ].
3. A general two-allele model Here, we provide details of a general two-allele model for multiple loci. This model is probably useful for studying genetic architecture of a quantitative trait in natural populations. Let the two alleles at locus r be A
r
and a
r
for r = 1, 2, ..., m with m the number of QTL. Assume that the frequencies and genetic effects of alleles are the same for both paternal and maternal gametes. Let p
r
denote the frequency of allele A
r
at locus r . Note that in this case
z
M
1
(
r
)
=
1
−
z
M
2
(
r
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG6bGEdaqhaaWcbaGaemyta00aaSbaaWqaaiabigdaXaqabaaaleaacqGGOaakcqWGYbGCcqGGPaqkaaGccqGH9aqpcqaIXaqmcqGHsislcqWG6bGEdaqhaaWcbaGaemyta00aaSbaaWqaaiabikdaYaqabaaaleaacqGGOaakcqWGYbGCcqGGPaqkaaaaaa@3DC3@
, r = 1, 2, ..., m . Also
x
M
1
(
r
)
=
z
M
1
(
r
)
−
E
[
z
M
1
(
r
)
]
=
(
1
−
z
M
2
(
r
)
)
−
E
(
1
−
z
M
2
(
r
)
)
=
−
x
M
2
(
r
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@68F1@
. Similarly,
x
F
2
(
r
)
=
−
x
F
1
(
r
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG4baEdaqhaaWcbaGaemOray0aaSbaaWqaaiabikdaYaqabaaaleaacqGGOaakcqWGYbGCcqGGPaqkaaGccqGH9aqpcqGHsislcqWG4baEdaqhaaWcbaGaemOray0aaSbaaWqaaiabigdaXaqabaaaleaacqGGOaakcqWGYbGCcqGGPaqkaaaaaa@3CAF@
Ignoring higher order epistasis involving at least three loci, we can define a two-allele model as
G
=
μ
+
∑
r
=
1
m
a
r
w
r
+
∑
r
=
1
m
d
r
v
r
+
∑
r
<
s
(
a
a
)
r
s
(
w
r
w
s
)
+
∑
r
≠
s
(
a
d
)
r
s
(
w
r
v
s
)
+
∑
r
<
s
(
d
d
)
r
s
(
v
r
v
s
)
(
20
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@9483@
where
w
r
=
x
M
1
(
r
)
+
x
F
1
(
r
)
=
{
(
1
−
p
r
)
for
A
r
A
r
at locus
r
1
−
2
p
r
for
A
r
a
r
at locus
r
−
2
p
r
for
a
r
a
r
at locus
r
(
21
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@9FEE@
v
r
=
x
M
1
(
x
)
x
F
1
(
x
)
=
{
(
1
−
p
r
)
2
for
A
r
A
r
at locus
r
−
p
r
(
1
−
p
r
)
for
A
r
a
r
at locus
r
p
r
2
for
a
r
a
r
at locus
r
(
22
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@A419@
for r = 1, 2, ..., m . The coefficients are associated with the original parameters in Cockerham model as follows.
a
r
=
α
1
(
r
)
−
α
2
(
r
)
d
r
=
δ
1
1
(
r
)
−
δ
2
1
(
r
)
−
δ
1
2
(
r
)
+
δ
2
2
(
r
)
(
a
a
)
r
s
=
(
α
1
(
r
)
α
1
(
s
)
)
−
(
α
1
(
r
)
α
2
(
s
)
)
−
(
α
2
(
r
)
α
1
(
s
)
)
+
(
α
2
α
2
(
s
)
)
(
a
d
)
r
s
=
(
α
1
(
r
)
δ
1
1
(
s
)
)
−
(
α
1
(
r
)
δ
2
1
(
s
)
)
−
(
α
1
(
r
)
δ
1
2
(
s
)
)
+
(
α
1
(
r
)
δ
2
2
(
s
)
)
−
(
α
2
(
r
)
δ
1
1
(
s
)
)
+
(
α
2
(
r
)
δ
2
1
(
s
)
)
+
(
α
2
(
r
)
δ
1
2
(
s
)
)
−
(
α
2
(
r
)
δ
2
2
(
s
)
)
(
d
d
)
r
s
=
(
δ
1
1
(
r
)
δ
1
1
(
s
)
)
−
(
δ
1
1
(
r
)
δ
2
1
(
s
)
)
−
(
δ
1
1
(
r
)
δ
1
2
(
s
)
)
+
(
δ
1
1
(
r
)
δ
2
2
(
s
)
)
−
(
δ
2
1
(
r
)
δ
1
1
(
s
)
)
+
(
δ
2
1
(
r
)
δ
2
1
(
s
)
)
+
(
δ
2
1
(
r
)
δ
1
2
(
s
)
)
−
(
δ
2
1
(
r
)
δ
2
2
(
s
)
)
−
(
δ
1
2
(
r
)
δ
1
1
(
s
)
)
+
(
δ
1
2
(
r
)
δ
2
1
(
s
)
)
+
(
δ
1
2
(
r
)
δ
1
2
(
s
)
)
−
(
δ
1
2
(
r
)
δ
2
2
(
s
)
)
+
(
δ
2
2
(
r
)
δ
1
1
(
s
)
)
−
(
δ
2
2
(
r
)
δ
2
1
(
s
)
)
−
(
δ
2
2
(
r
)
δ
1
2
(
s
)
)
+
(
δ
2
2
(
r
)
δ
2
2
(
s
)
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqadeadcaaaaaaabaGaemyyae2aaSbaaSqaaiabdkhaYbqabaGccqGH9aqpaeaaiiGacqWFXoqydaqhaaWcbaGaeGymaedabaGaeiikaGIaemOCaiNaeiykaKcaaOGaeyOeI0Iae8xSde2aa0baaSqaaiabikdaYaqaaiabcIcaOiabdkhaYjabcMcaPaaaaOqaaiabdsgaKnaaBaaaleaacqWGYbGCaeqaaOGaeyypa0dabaGae8hTdq2aa0baaSqaaiabigdaXaqaaiabigdaXiabcIcaOiabdkhaYjabcMcaPaaakiabgkHiTiab=r7aKnaaDaaaleaacqaIYaGmaeaacqaIXaqmcqGGOaakcqWGYbGCcqGGPaqkaaGccqGHsislcqWF0oazdaqhaaWcbaGaeGymaedabaGaeGOmaiJaeiikaGIaemOCaiNaeiykaKcaaOGaey4kaSIae8hTdq2aa0baaSqaaiabikdaYaqaaiabikdaYiabcIcaOiabdkhaYjabcMcaPaaaaOqaaiabcIcaOiabdggaHjabdggaHjabcMcaPmaaBaaaleaacqWGYbGCcqWGZbWCaeqaaOGaeyypa0dabaGaeiikaGIae8xSde2aa0baaSqaaiabigdaXaqaaiabcIcaOiabdkhaYjabcMcaPaaakiab=f7aHnaaDaaaleaacqaIXaqmaeaacqGGOaakcqWGZbWCcqGGPaqkaaGccqGGPaqkcqGHsislcqGGOaakcqWFXoqydaqhaaWcbaGaeGymaedabaGaeiikaGIaemOCaiNaeiykaKcaaOGae8xSde2aa0baaSqaaiabikdaYaqaaiabcIcaOiabdohaZjabcMcaPaaakiabcMcaPiabgkHiTiabcIcaOiab=f7aHnaaDaaaleaacqaIYaGmaeaacqGGOaakcqWGYbGCcqGGPaqkaaGccqWFXoqydaqhaaWcbaGaeGymaedabaGaeiikaGIaem4CamNaeiykaKcaaOGaeiykaKcabaaabaGaey4kaSIaeiikaGIae8xSde2aaSbaaSqaaiabikdaYaqabaGccqWFXoqydaqhaaWcbaGaeGOmaidabaGaeiikaGIaem4CamNaeiykaKcaaOGaeiykaKcabaGaeiikaGIaemyyaeMaemizaqMaeiykaKYaaSbaaSqaaiabdkhaYjabdohaZbqabaGccqGH9aqpaeaacqGGOaakcqWFXoqydaqhaaWcbaGaeGymaedabaGaeiikaGIaemOCaiNaeiykaKcaaOGae8hTdq2aa0baaSqaaiabigdaXaqaaiabigdaXiabcIcaOiabdohaZjabcMcaPaaakiabcMcaPiabgkHiTiabcIcaOiab=f7aHnaaDaaaleaacqaIXaqmaeaacqGGOaakcqWGYbGCcqGGPaqkaaGccqWF0oazdaqhaaWcbaGaeGOmaidabaGaeGymaeJaeiikaGIaem4CamNaeiykaKcaaOGaeiykaKIaeyOeI0IaeiikaGIae8xSde2aa0baaSqaaiabigdaXaqaaiabcIcaOiabdkhaYjabcMcaPaaakiab=r7aKnaaDaaaleaacqaIXaqmaeaacqaIYaGmcqGGOaakcqWGZbWCcqGGPaqkaaGccqGGPaqkaeaaaeaacqGHRaWkcqGGOaakcqWFXoqydaqhaaWcbaGaeGymaedabaGaeiikaGIaemOCaiNaeiykaKcaaOGae8hTdq2aa0baaSqaaiabikdaYaqaaiabikdaYiabcIcaOiabdohaZjabcMcaPaaakiabcMcaPiabgkHiTiabcIcaOiab=f7aHnaaDaaaleaacqaIYaGmaeaacqGGOaakcqWGYbGCcqGGPaqkaaGccqWF0oazdaqhaaWcbaGaeGymaedabaGaeGymaeJaeiikaGIaem4CamNaeiykaKcaaOGaeiykaKIaey4kaSIaeiikaGIae8xSde2aa0baaSqaaiabikdaYaqaaiabcIcaOiabdkhaYjabcMcaPaaakiab=r7aKnaaDaaaleaacqaIYaGmaeaacqaIXaqmcqGGOaakcqWGZbWCcqGGPaqkaaGccqGGPaqkaeaaaeaacqGHRaWkcqGGOaakcqWFXoqydaqhaaWcbaGaeGOmaidabaGaeiikaGIaemOCaiNaeiykaKcaaOGae8hTdq2aa0baaSqaaiabigdaXaqaaiabikdaYiabcIcaOiabdohaZjabcMcaPaaakiabcMcaPiabgkHiTiabcIcaOiab=f7aHnaaDaaaleaacqaIYaGmaeaacqGGOaakcqWGYbGCcqGGPaqkaaGccqWF0oazdaqhaaWcbaGaeGOmaidabaGaeGOmaiJaeiikaGIaem4CamNaeiykaKcaaOGaeiykaKcabaGaeiikaGIaemizaqMaemizaqMaeiykaKYaaSbaaSqaaiabdkhaYjabdohaZbqabaGccqGH9aqpaeaacqGGOaakcqWF0oazdaqhaaWcbaGaeGymaedabaGaeGymaeJaeiikaGIaemOCaiNaeiykaKcaaOGae8hTdq2aa0baaSqaaiabigdaXaqaaiabigdaXiabcIcaOiabdohaZjabcMcaPaaakiabcMcaPiabgkHiTiabcIcaOiab=r7aKnaaDaaaleaacqaIXaqmaeaacqaIXaqmcqGGOaakcqWGYbGCcqGGPaqkaaGccqWF0oazdaqhaaWcbaGaeGOmaidabaGaeGymaeJaeiikaGIaem4CamNaeiykaKcaaOGaeiykaKIaeyOeI0IaeiikaGIae8hTdq2aa0baaSqaaiabigdaXaqaaiabigdaXiabcIcaOiabdkhaYjabcMcaPaaakiab=r7aKnaaDaaaleaacqaIXaqmaeaacqaIYaGmcqGGOaakcqWGZbWCcqGGPaqkaaGccqGGPaqkaeaaaeaacqGHRaWkcqGGOaakcqWF0oazdaqhaaWcbaGaeGymaedabaGaeGymaeJaeiikaGIaemOCaiNaeiykaKcaaOGae8hTdq2aa0baaSqaaiabikdaYaqaaiabikdaYiabcIcaOiabdohaZjabcMcaPaaakiabcMcaPiabgkHiTiabcIcaOiab=r7aKnaaDaaaleaacqaIYaGmaeaacqaIXaqmcqGGOaakcqWGYbGCcqGGPaqkaaGccqWF0oazdaqhaaWcbaGaeGymaedabaGaeGymaeJaeiikaGIaem4CamNaeiykaKcaaOGaeiykaKIaey4kaSIaeiikaGIae8hTdq2aa0baaSqaaiabikdaYaqaaiabigdaXiabcIcaOiabdkhaYjabcMcaPaaakiab=r7aKnaaDaaaleaacqaIYaGmaeaacqaIXaqmcqGGOaakcqWGZbWCcqGGPaqkaaGccqGGPaqkaeaaaeaacqGHRaWkcqGGOaakcqWF0oazdaqhaaWcbaGaeGOmaidabaGaeGymaeJaeiikaGIaemOCaiNaeiykaKcaaOGae8hTdq2aa0baaSqaaiabigdaXaqaaiabikdaYiabcIcaOiabdohaZjabcMcaPaaakiabcMcaPiabgkHiTiabcIcaOiab=r7aKnaaDaaaleaacqaIYaGmaeaacqaIXaqmcqGGOaakcqWGYbGCcqGGPaqkaaGccqWF0oazdaqhaaWcbaGaeGOmaidabaGaeGOmaiJaeiikaGIaem4CamNaeiykaKcaaOGaeiykaKIaeyOeI0IaeiikaGIae8hTdq2aa0baaSqaaiabigdaXaqaaiabikdaYiabcIcaOiabdkhaYjabcMcaPaaakiab=r7aKnaaDaaaleaacqaIXaqmaeaacqaIXaqmcqGGOaakcqWGZbWCcqGGPaqkaaGccqGGPaqkaeaaaeaacqGHRaWkcqGGOaakcqWF0oazdaqhaaWcbaGaeGymaedabaGaeGOmaiJaeiikaGIaemOCaiNaeiykaKcaaOGae8hTdq2aa0baaSqaaiabikdaYaqaaiabigdaXiabcIcaOiabdohaZjabcMcaPaaakiabcMcaPiabgUcaRiabcIcaOiab=r7aKnaaDaaaleaacqaIXaqmaeaacqaIYaGmcqGGOaakcqWGYbGCcqGGPaqkaaGccqWF0oazdaqhaaWcbaGaeGymaedabaGaeGOmaiJaeiikaGIaem4CamNaeiykaKcaaOGaeiykaKIaeyOeI0IaeiikaGIae8hTdq2aa0baaSqaaiabigdaXaqaaiabikdaYiabcIcaOiabdkhaYjabcMcaPaaakiab=r7aKnaaDaaaleaacqaIYaGmaeaacqaIYaGmcqGGOaakcqWGZbWCcqGGPaqkaaGccqGGPaqkaeaaaqaabeqaaiabgUcaRiabcIcaOiab=r7aKnaaDaaaleaacqaIYaGmaeaacqaIYaGmcqGGOaakcqWGYbGCcqGGPaqkaaGccqWF0oazdaqhaaWcbaGaeGymaedabaGaeGymaeJaeiikaGIaem4CamNaeiykaKcaaOGaeiykaKIaeyOeI0IaeiikaGIae8hTdq2aa0baaSqaaiabikdaYaqaaiabikdaYiabcIcaOiabdkhaYjabcMcaPaaakiab=r7aKnaaDaaaleaacqaIYaGmaeaacqaIXaqmcqGGOaakcqWGZbWCcqGGPaqkaaGccqGGPaqkcqGHsislcqGGOaakcqWF0oazdaqhaaWcbaGaeGOmaidabaGaeGOmaiJaeiikaGIaemOCaiNaeiykaKcaaOGae8hTdq2aa0baaSqaaiabigdaXaqaaiabikdaYiabcIcaOiabdohaZjabcMcaPaaakiabcMcaPaqaaiabgUcaRiabcIcaOiab=r7aKnaaDaaaleaacqaIYaGmaeaacqaIYaGmcqGGOaakcqWGYbGCcqGGPaqkaaGccqWF0oazdaqhaaWcbaGaeGOmaidabaGaeGOmaiJaeiikaGIaem4CamNaeiykaKcaaOGaeiykaKcaaaaaaa@2BE6@
The constraint conditions further lead to
a
r
=
α
1
(
r
)
−
(
−
p
r
)
(
1
−
p
r
)
α
1
(
r
)
=
1
(
1
−
p
r
)
α
1
(
r
)
d
r
=
1
(
1
−
p
r
)
2
δ
1
1
(
r
)
(
a
a
)
r
s
=
1
(
1
−
p
r
)
(
1
−
p
s
)
(
α
1
(
r
)
α
1
(
s
)
)
(
a
d
)
r
s
=
1
(
1
−
p
r
)
(
1
−
p
s
)
2
(
α
1
(
r
)
δ
1
1
(
s
)
)
(
d
d
)
r
s
=
1
(
1
−
p
r
)
2
(
1
−
p
s
)
2
(
δ
1
1
(
r
)
δ
1
1
(
s
)
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@DE64@
With Hardy-Weinberg, linkage and genotypic equilibria, the partial regression coefficients in the above model correspond to the simple regression coefficients
a
r
=
1
(
1
−
p
r
)
[
E
(
G
|
z
M
1
(
r
)
=
1
)
−
E
(
G
)
]
d
r
=
1
(
1
−
p
r
)
2
[
E
(
G
|
z
M
1
(
r
)
=
z
F
1
(
r
)
=
1
)
−
2
E
(
G
|
z
M
1
(
r
)
=
1
)
+
E
(
G
)
]
(
a
a
)
r
s
=
1
(
1
−
p
r
)
(
1
−
p
s
)
[
E
(
G
|
z
M
1
(
r
)
=
z
M
1
(
s
)
=
1
)
−
E
(
G
|
z
M
1
(
r
)
=
1
)
−
E
(
G
|
z
M
1
(
s
)
=
1
)
+
E
(
G
)
]
(
a
d
)
r
s
=
1
(
1
−
p
r
)
(
1
−
p
s
)
2
[
E
(
G
|
z
M
1
(
r
)
=
z
M
1
(
s
)
=
z
F
1
(
s
)
=
1
)
−
2
E
(
G
|
z
M
1
(
r
)
=
z
M
1
(
s
)
=
1
)
−
E
(
G
|
z
M
1
(
s
)
=
z
F
1
(
s
)
=
1
)
+
E
(
G
|
z
M
1
(
r
)
=
1
)
+
2
E
(
G
|
z
M
1
(
s
)
=
1
)
−
E
(
G
)
]
(
d
d
)
r
s
=
1
(
1
−
p
r
)
2
(
1
−
p
s
)
2
[
E
(
G
|
z
M
1
(
r
)
=
z
F
1
(
r
)
=
z
M
1
(
s
)
=
z
F
1
(
s
)
=
1
)
−
2
E
(
G
|
z
M
1
(
r
)
=
z
F
1
(
r
)
=
z
M
1
(
s
)
=
1
)
−
2
E
(
G
|
z
M
1
(
r
)
=
z
M
1
(
s
)
=
z
F
1
(
s
)
=
1
)
−
E
(
G
|
z
M
1
(
r
)
=
z
F
1
(
r
)
=
1
)
+
E
(
G
|
z
M
1
(
s
)
=
z
F
1
(
r
)
=
1
)
+
4
E
(
G
|
z
M
1
(
r
)
=
z
F
1
(
s
)
=
1
)
−
2
E
(
G
|
z
M
1
(
r
)
=
1
)
−
2
E
(
G
|
z
M
1
(
s
)
=
1
)
+
E
(
G
)
]
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@596E@
Note that in this case the genetic effects in the original model are
α
1
(
r
)
=
E
(
G
|
z
M
1
(
r
)
=
1
)
−
E
(
G
)
δ
1
1
(
r
)
=
E
(
G
|
z
M
1
(
r
)
=
z
F
1
(
r
)
=
1
)
−
2
E
(
G
|
z
M
1
(
r
)
=
1
)
+
E
(
G
)
(
α
1
(
r
)
α
1
(
s
)
)
=
E
(
G
|
z
M
1
(
r
)
=
z
M
1
(
s
)
=
1
)
−
E
(
G
|
z
M
1
(
r
)
=
1
)
−
E
(
G
|
z
M
1
(
s
)
=
1
)
+
E
(
G
)
(
α
1
(
r
)
δ
1
1
(
s
)
)
=
E
(
G
|
z
M
1
(
r
)
=
z
M
1
(
s
)
=
z
F
1
(
s
)
=
1
)
−
2
E
(
G
|
z
M
1
(
r
)
=
z
M
1
(
s
)
=
1
)
−
E
(
G
|
z
M
1
(
s
)
=
z
F
1
(
s
)
=
1
)
+
E
(
G
|
z
M
1
(
r
)
=
1
)
+
2
E
(
G
|
z
M
1
(
s
)
=
1
)
−
E
(
G
)
(
δ
1
1
(
r
)
δ
1
1
(
s
)
)
=
E
(
G
|
z
M
1
(
r
)
=
z
F
1
(
r
)
=
z
M
1
(
s
)
=
z
F
1
(
s
)
=
1
)
−
2
E
(
G
|
z
M
1
(
r
)
=
z
F
1
(
r
)
=
z
M
1
(
s
)
=
1
)
−
2
E
(
G
|
z
M
1
(
r
)
=
z
M
1
(
s
)
=
z
F
1
(
s
)
=
1
)
+
E
(
G
|
z
M
1
(
r
)
=
z
F
1
(
r
)
=
1
]
+
E
(
G
|
z
M
1
(
s
)
=
z
F
1
(
s
)
=
1
)
+
4
E
(
G
|
z
M
1
(
r
)
=
z
F
1
(
s
)
=
1
)
−
2
E
(
G
|
z
M
1
(
r
)
=
1
)
−
2
E
(
G
|
z
M
1
(
s
)
=
1
)
+
E
(
G
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@2ACD@
They are the same as the least squares definition.
Yet another form of this result is shown in Table 1 of [10 ]. Zeng et al. [10 ] also show that linkage disequilibrium does not change the values of genetic effects in a full model. This means that the partial regression coefficients in a disequilibrium population equal to the simple regression coefficients in a corresponding equilibrium population with the same allelic frequency configuration.
The partition of genotypic variance in an equilibrium population is
V
G
=
2
∑
r
=
1
m
p
r
(
1
−
p
r
)
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r
2
+
∑
r
=
1
m
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r
2
(
1
−
p
r
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2
d
r
2
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4
∑
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s
p
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s
(
1
−
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r
)
(
1
−
p
s
)
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a
a
)
r
s
2
+
2
∑
r
≠
s
p
r
(
1
−
p
r
)
p
s
2
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1
−
p
s
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2
(
a
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)
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s
2
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s
p
r
2
p
s
2
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1
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2
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1
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(
d
d
)
r
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2
(
23
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@D284@
or
V
G
=
2
p
r
(
1
−
p
r
)
∑
r
=
1
m
(
α
1
(
r
)
)
2
+
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r
2
(
1
−
p
r
)
2
∑
r
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1
m
(
δ
1
1
(
r
)
)
2
+
4
p
r
p
s
(
1
−
p
r
)
(
1
−
p
s
)
∑
r
<
s
(
α
1
(
r
)
α
1
(
s
)
)
2
+
2
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r
p
s
2
(
1
−
p
r
)
(
1
−
p
s
)
2
∑
r
≠
s
(
α
1
(
r
)
δ
1
1
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2
+
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r
2
p
s
2
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r
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24
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MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqadeabbaaaaeaacqWGwbGvdaWgaaWcbaGaem4raCeabeaakiabg2da9maalaaabaGaeGOmaiJaemiCaa3aaSbaaSqaaiabdkhaYbqabaaakeaacqGGOaakcqaIXaqmcqGHsislcqWGWbaCdaWgaaWcbaGaemOCaihabeaakiabcMcaPaaadaaeWbqaaiabcIcaOGGaciab=f7aHnaaDaaaleaacqaIXaqmaeaacqGGOaakcqWGYbGCcqGGPaqkaaGccqGGPaqkdaahaaWcbeqaaiabikdaYaaakiabgUcaRmaalaaabaGaemiCaa3aa0baaSqaaiabdkhaYbqaaiabikdaYaaaaOqaaiabcIcaOiabigdaXiabgkHiTiabdchaWnaaBaaaleaacqWGYbGCaeqaaOGaeiykaKYaaWbaaSqabeaacqaIYaGmaaaaaaqaaiabdkhaYjabg2da9iabigdaXaqaaiabd2gaTbqdcqGHris5aOWaaabCaeaacqGGOaakcqWF0oazdaqhaaWcbaGaeGymaedabaGaeGymaeJaeiikaGIaemOCaiNaeiykaKcaaOGaeiykaKYaaWbaaSqabeaacqaIYaGmaaaabaGaemOCaiNaeyypa0JaeGymaedabaGaemyBa0ganiabggHiLdaakeaacqGHRaWkdaWcaaqaaiabisda0iabdchaWnaaBaaaleaacqWGYbGCaeqaaOGaemiCaa3aaSbaaSqaaiabdohaZbqabaaakeaacqGGOaakcqaIXaqmcqGHsislcqWGWbaCdaWgaaWcbaGaemOCaihabeaakiabcMcaPiabcIcaOiabigdaXiabgkHiTiabdchaWnaaBaaaleaacqWGZbWCaeqaaOGaeiykaKcaamaaqafabaGaeiikaGIae8xSde2aa0baaSqaaiabigdaXaqaaiabcIcaOiabdkhaYjabcMcaPaaakiab=f7aHnaaDaaaleaacqaIXaqmaeaacqGGOaakcqWGZbWCcqGGPaqkaaGccqGGPaqkdaahaaWcbeqaaiabikdaYaaaaeaacqWGYbGCcqGH8aapcqWGZbWCaeqaniabggHiLdaakeaacqGHRaWkdaWcaaqaaiabikdaYiabdchaWnaaBaaaleaacqWGYbGCaeqaaOGaemiCaa3aa0baaSqaaiabdohaZbqaaiabikdaYaaaaOqaaiabcIcaOiabigdaXiabgkHiTiabdchaWnaaBaaaleaacqWGYbGCaeqaaOGaeiykaKIaeiikaGIaeGymaeJaeyOeI0IaemiCaa3aaSbaaSqaaiabdohaZbqabaGccqGGPaqkdaahaaWcbeqaaiabikdaYaaaaaGcdaaeqbqaaiabcIcaOiab=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r7aKnaaDaaaleaacqaIXaqmaeaacqaIXaqmcqGGOaakcqWGZbWCcqGGPaqkaaGccqGGPaqkdaahaaWcbeqaaiabikdaYaaaaeaacqWGYbGCcqGH8aapcqWGZbWCaeqaniabggHiLdaaaOGaaCzcaiaaxMaadaqadaqaaiabikdaYiabisda0aGaayjkaiaawMcaaaaa@F338@
The partition of the genetic variance with epistasis and linkage disequilibrium is complex. We give the result with trigenic and quadrigenic linkage disequilibria included as well. The partition of variance has a similar form as (19). The detail of each component is presented in Appendix E.