Background
Genetic Variance
Simulation
Summary
Implications of Using Identity-by-Descent versus
Identity-by-State Matrices in Genetic Analyses
R.L. Fernando
H. Cheng
X. Sun
Department of Animal Science
Iowa State University
International Plant and Animal Genome XXII
Background
Genetic Variance
Simulation
Summary
IBD Matrices
Pedigree Analyses: IBD probabilities conditional on pedigree
Estimation of genetic variance components: ANOVA, ML,
REML
Prediction of genetic merit: BLP (Selection Index), BLUP
Marker Analyses: IBD probabilities at unobserved QTL,
conditional on markers and pedigree
LE: Chevelat et al. (1984), Fernando and Grossman (1989),
Goddard (1992)
LD: Meuwissen and Goddard (2000)
Background
Genetic Variance
Simulation
Summary
IBD Matrices
Pedigree Analyses: IBD probabilities conditional on pedigree
Estimation of genetic variance components: ANOVA, ML,
REML
Prediction of genetic merit: BLP (Selection Index), BLUP
Marker Analyses: IBD probabilities at unobserved QTL,
conditional on markers and pedigree
LE: Chevelat et al. (1984), Fernando and Grossman (1989),
Goddard (1992)
LD: Meuwissen and Goddard (2000)
Background
Genetic Variance
Simulation
Summary
IBS Matrices
Suppose QTL genotypes are observed:
IBS probabilities for
X\Y AA
AA
1
Aa 0.5
aa
0
individuals X and Y:
Aa aa
0.5 0.0
0.5 0.5
0.5 1
Nejati-Javaremi et al. (1997): Mixed model analysis with A
replaced by T, which is twice the matrix of IBS probabilities
averaged across all QTL.
Background
Genetic Variance
Simulation
Summary
IBS Matrices
Suppose QTL genotypes are observed:
IBS probabilities for
X\Y AA
AA
1
Aa 0.5
aa
0
individuals X and Y:
Aa aa
0.5 0.0
0.5 0.5
0.5 1
Nejati-Javaremi et al. (1997): Mixed model analysis with A
replaced by T, which is twice the matrix of IBS probabilities
averaged across all QTL.
Background
Genetic Variance
Simulation
Summary
Equivalent Models
QTL effects model
y = 1µ + Qα + e
0
T = QQ
2k (Fernando, 1998), for Q with two columns per locus
Assuming α ∼ (0, Iσα2 ) makes this equivalent to model of
Nejati-Javaremi et al.
Breeding value model
y = 1µ + a + e
a =Qα
Var (a|Q) = T2kσα2
Is Tσα2 a genetic covariance matrix?
Background
Genetic Variance
Simulation
Summary
Equivalent Models
QTL effects model
y = 1µ + Qα + e
0
T = QQ
2k (Fernando, 1998), for Q with two columns per locus
Assuming α ∼ (0, Iσα2 ) makes this equivalent to model of
Nejati-Javaremi et al.
Breeding value model
y = 1µ + a + e
a =Qα
Var (a|Q) = T2kσα2
Is Tσα2 a genetic covariance matrix?
Background
Genetic Variance
Simulation
Summary
Two-allele and One-allele Models
Two-allele Model
One-allele Model
Genotype A-column a-column
AA
2
0
Aa
1
1
aa
0
2
2 columns per locus in Q
Genotype A-column
AA
2
Aa
1
aa
0
1 column per locus in Q
The one-allele model is used in most genome analyses:
0
G = QQ
k replaces A
Using T or G in mixed model analysis gives identical
predictions
Background
Genetic Variance
Simulation
Summary
Genetic Covariance Matrices
IBD
Var (a|P) ∝ A
Var (a|A) ∝ A
IBS
Var (a|?) ∝ G
Var (a|Q) ∝ G, given random α, but not a genetic covariance
matrix.
Var (a|G, α) 6∝ G in general
Background
Genetic Variance
Simulation
Summary
Genetic Covariance Matrices
IBD
Var (a|P) ∝ A
Var (a|A) ∝ A
IBS
Var (a|?) ∝ G
Var (a|Q) ∝ G, given random α, but not a genetic covariance
matrix.
Var (a|G, α) 6∝ G in general
Background
Genetic Variance
Simulation
Summary
Compare Var (a|P) to Var (a|Q)
Consider diagonal element i of:
Var (a|P) = Aσa2 : aii σa2
this is the genetic variance among animals with inbreeding
(aii − 1)
expected genetic progress from selection is related to this
variance
Var (a|Q) = Gkσα2 : gii kσα2
this is the variance among animals with genotypes q0i over
different realizations of α
expected genetic progress from selection among such animals
is zero!
so, gii kσα2 is not a genetic variance!
Background
Genetic Variance
Simulation
Summary
Compare Var (a|P) to Var (a|Q)
Consider diagonal element i of:
Var (a|P) = Aσa2 : aii σa2
this is the genetic variance among animals with inbreeding
(aii − 1)
expected genetic progress from selection is related to this
variance
Var (a|Q) = Gkσα2 : gii kσα2
this is the variance among animals with genotypes q0i over
different realizations of α
expected genetic progress from selection among such animals
is zero!
so, gii kσα2 is not a genetic variance!
Background
Genetic Variance
Simulation
Summary
What is Var (a|G, α)?
Recall, G =
QQ0
k
=
∑ qj q0j
k
So, interchanging the columns of Q will give the same G
Also, changing the sign of one or more columns of Q does not
change the value of G
The different Q matrices that result in the same G gives rise
to a distribution for a = Qα
Unfortunately, Var (a|G, α) 6∝ G in general
Background
Genetic Variance
Simulation
Summary
What is Var (a|G, α)?
Recall, G =
QQ0
k
=
∑ qj q0j
k
So, interchanging the columns of Q will give the same G
Also, changing the sign of one or more columns of Q does not
change the value of G
The different Q matrices that result in the same G gives rise
to a distribution for a = Qα
Unfortunately, Var (a|G, α) 6∝ G in general
Background
Genetic Variance
Simulation
Summary
What is Var (a|G, α)?
Recall, G =
QQ0
k
=
∑ qj q0j
k
So, interchanging the columns of Q will give the same G
Also, changing the sign of one or more columns of Q does not
change the value of G
The different Q matrices that result in the same G gives rise
to a distribution for a = Qα
Unfortunately, Var (a|G, α) 6∝ G in general
Background
Genetic Variance
Simulation
Summary
What is Var (a|G, α)?
Recall, G =
QQ0
k
=
∑ qj q0j
k
So, interchanging the columns of Q will give the same G
Also, changing the sign of one or more columns of Q does not
change the value of G
The different Q matrices that result in the same G gives rise
to a distribution for a = Qα
Unfortunately, Var (a|G, α) 6∝ G in general
Background
Genetic Variance
Simulation
Summary
What is Var (a|G, α)?
Recall, G =
QQ0
k
=
∑ qj q0j
k
So, interchanging the columns of Q will give the same G
Also, changing the sign of one or more columns of Q does not
change the value of G
The different Q matrices that result in the same G gives rise
to a distribution for a = Qα
Unfortunately, Var (a|G, α) 6∝ G in general
Background
Genetic Variance
Simulation
Summary
Solution
Following Sorensen et al. (2001), define the genetic variance as
V (a) =
1 n
∑ (ai − ā)2
n i=1
This is the variance of a randomly sampled ai
Estimate V (a) as
V̂ (a) = E [V (a)|y, Q]
Background
Genetic Variance
Simulation
Summary
Alternative Solution
Following Powell et al. (2010), model breeding values as
k
a =
qj
∑ p2pj qj
p
2pj qj αj
j=1
k
=
∑ xj βj
j=1
= Xβ
Then, Var (a|X) becomes
XX0 2
kσβ
k
= Gβ kσβ2
Var (a|X) =
If βj are observed, MLE of σβ2 is
∑ βj2
k
=
∑ 2pj qj αj2
k
Background
Genetic Variance
Simulation
Summary
Alternative Solution (Cont.)
So, given HWE, LE and sufficient data,
MLE(kσβ2 ) = ∑ 2pj qj αj2 = V (a)
Then, Var (a|X) = Gβ V (a)
This is still not a genetic covariance matrix
Background
Genetic Variance
Simulation
Summary
Simulation Results
200 observations from 2 or 40 fullsib families;
V (a) = 1, h2 = 0.5, k = 100
HWE
LE
V̂ (a)
σ̂β2
Families
Y
Y
N
N
Y
N
Y
N
0.99
1.01
0.99
0.99
1.03
1.30∗
1.29∗
1.26∗
40
40
2
2/400
Background
Genetic Variance
Simulation
Summary
Summary
When QTL genotypes are not observed, IBD probabilities are
useful to model genetic covariances.
When QTL genotypes are observed, they can be included in
the model explicitly to estimate substitution effects.
The equivalent breeding-value model may have a
computational advantage.
The G matrix is not a genetic relationship matrix. When two
individuals have identical genotypes, their relationship is
irrelevant
Background
Genetic Variance
Simulation
Summary
Acknowledgements
Colleagues:
Dorian Garrick
Jack Dekkers
Ania Wolc
Jian Zeng
Funding
NIH Grant R01GM099992
USDA/AFRI project EBIGS