introduction to analysis of heritability
DESCRIPTION
Introduction to Analysis of Heritability Journal Club in SYSUTRANSCRIPT
Introduction to Analysis of Heritability
Kou Qiang
Sun Yat-sen University
November 24, 2012
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 1 / 29
Outline
Outline
1 Heritability EstimatesDefinitionThe ACE/ADE modelThe parent-offspring regression
2 The liability-threshold (A∆) modelDefinitionThe genetic relative risk
3 The limiting pathway (LP) model for quantitative traitsDefinitionh2
all and h2pop
4 h2slope(κ0)
DefinitionProperty
5 Detecting epistasis among variantsCochran-Armitage trend test
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 2 / 29
Heritability Estimates Definition
Outline
1 Heritability EstimatesDefinitionThe ACE/ADE modelThe parent-offspring regression
2 The liability-threshold (A∆) modelDefinitionThe genetic relative risk
3 The limiting pathway (LP) model for quantitative traitsDefinitionh2
all and h2pop
4 h2slope(κ0)
DefinitionProperty
5 Detecting epistasis among variantsCochran-Armitage trend test
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 3 / 29
Heritability Estimates Definition
Definition
h2: Narrow-sense heritability
H2: Broad-sense heritability
h2known: Narrow-sense heritability explained by known variants
h2all : True (narrow-sense) heritability
h2pop: Apparent heritability, inferred from population data
πexplained : Proportion of heritability explained
πphantom: Phantom heritability
Ψ: A genetic architecture
βi : Additive effect size of the ith locus
τ : The threshold in A∆ model
IBD: Identity by descent
h2slope(κ0)
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 4 / 29
Heritability Estimates Definition
Definition
h2: Narrow-sense heritability
H2: Broad-sense heritability
h2known: Narrow-sense heritability explained by known variants
h2all : True (narrow-sense) heritability
h2pop: Apparent heritability, inferred from population data
πexplained : Proportion of heritability explained
πphantom: Phantom heritability
Ψ: A genetic architecture
βi : Additive effect size of the ith locus
τ : The threshold in A∆ model
IBD: Identity by descent
h2slope(κ0)
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 4 / 29
Heritability Estimates Definition
Definition
h2: Narrow-sense heritability
H2: Broad-sense heritability
h2known: Narrow-sense heritability explained by known variants
h2all : True (narrow-sense) heritability
h2pop: Apparent heritability, inferred from population data
πexplained : Proportion of heritability explained
πphantom: Phantom heritability
Ψ: A genetic architecture
βi : Additive effect size of the ith locus
τ : The threshold in A∆ model
IBD: Identity by descent
h2slope(κ0)
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 4 / 29
Heritability Estimates Definition
Definition
h2: Narrow-sense heritability
H2: Broad-sense heritability
h2known: Narrow-sense heritability explained by known variants
h2all : True (narrow-sense) heritability
h2pop: Apparent heritability, inferred from population data
πexplained : Proportion of heritability explained
πphantom: Phantom heritability
Ψ: A genetic architecture
βi : Additive effect size of the ith locus
τ : The threshold in A∆ model
IBD: Identity by descent
h2slope(κ0)
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 4 / 29
Heritability Estimates Definition
Definition
h2: Narrow-sense heritability
H2: Broad-sense heritability
h2known: Narrow-sense heritability explained by known variants
h2all : True (narrow-sense) heritability
h2pop: Apparent heritability, inferred from population data
πexplained : Proportion of heritability explained
πphantom: Phantom heritability
Ψ: A genetic architecture
βi : Additive effect size of the ith locus
τ : The threshold in A∆ model
IBD: Identity by descent
h2slope(κ0)
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 4 / 29
Heritability Estimates Definition
Definition
h2: Narrow-sense heritability
H2: Broad-sense heritability
h2known: Narrow-sense heritability explained by known variants
h2all : True (narrow-sense) heritability
h2pop: Apparent heritability, inferred from population data
πexplained : Proportion of heritability explained
πphantom: Phantom heritability
Ψ: A genetic architecture
βi : Additive effect size of the ith locus
τ : The threshold in A∆ model
IBD: Identity by descent
h2slope(κ0)
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 4 / 29
Heritability Estimates Definition
Definition
h2: Narrow-sense heritability
H2: Broad-sense heritability
h2known: Narrow-sense heritability explained by known variants
h2all : True (narrow-sense) heritability
h2pop: Apparent heritability, inferred from population data
πexplained : Proportion of heritability explained
πphantom: Phantom heritability
Ψ: A genetic architecture
βi : Additive effect size of the ith locus
τ : The threshold in A∆ model
IBD: Identity by descent
h2slope(κ0)
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 4 / 29
Heritability Estimates Definition
Definition
h2: Narrow-sense heritability
H2: Broad-sense heritability
h2known: Narrow-sense heritability explained by known variants
h2all : True (narrow-sense) heritability
h2pop: Apparent heritability, inferred from population data
πexplained : Proportion of heritability explained
πphantom: Phantom heritability
Ψ: A genetic architecture
βi : Additive effect size of the ith locus
τ : The threshold in A∆ model
IBD: Identity by descent
h2slope(κ0)
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 4 / 29
Heritability Estimates Definition
Definition
h2: Narrow-sense heritability
H2: Broad-sense heritability
h2known: Narrow-sense heritability explained by known variants
h2all : True (narrow-sense) heritability
h2pop: Apparent heritability, inferred from population data
πexplained : Proportion of heritability explained
πphantom: Phantom heritability
Ψ: A genetic architecture
βi : Additive effect size of the ith locus
τ : The threshold in A∆ model
IBD: Identity by descent
h2slope(κ0)
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 4 / 29
Heritability Estimates Definition
Definition
h2: Narrow-sense heritability
H2: Broad-sense heritability
h2known: Narrow-sense heritability explained by known variants
h2all : True (narrow-sense) heritability
h2pop: Apparent heritability, inferred from population data
πexplained : Proportion of heritability explained
πphantom: Phantom heritability
Ψ: A genetic architecture
βi : Additive effect size of the ith locus
τ : The threshold in A∆ model
IBD: Identity by descent
h2slope(κ0)
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 4 / 29
Heritability Estimates Definition
Definition
h2: Narrow-sense heritability
H2: Broad-sense heritability
h2known: Narrow-sense heritability explained by known variants
h2all : True (narrow-sense) heritability
h2pop: Apparent heritability, inferred from population data
πexplained : Proportion of heritability explained
πphantom: Phantom heritability
Ψ: A genetic architecture
βi : Additive effect size of the ith locus
τ : The threshold in A∆ model
IBD: Identity by descent
h2slope(κ0)
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 4 / 29
Heritability Estimates Definition
Definition
h2: Narrow-sense heritability
H2: Broad-sense heritability
h2known: Narrow-sense heritability explained by known variants
h2all : True (narrow-sense) heritability
h2pop: Apparent heritability, inferred from population data
πexplained : Proportion of heritability explained
πphantom: Phantom heritability
Ψ: A genetic architecture
βi : Additive effect size of the ith locus
τ : The threshold in A∆ model
IBD: Identity by descent
h2slope(κ0)
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 4 / 29
Heritability Estimates Definition
Definition
Z = Ψ(G ,E ) = [α +∑βigi ]
The variance explained by the ith variant Vi = 2fi (1− fi )β2i
Vknown = VS =∑
i∈S Vi
h2all = Vall
h2known =
∑i 2fi (1− fi )β
2i
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 5 / 29
Heritability Estimates Definition
Definition
Z = Ψ(G ,E ) = [α +∑βigi ]
The variance explained by the ith variant Vi = 2fi (1− fi )β2i
Vknown = VS =∑
i∈S Vi
h2all = Vall
h2known =
∑i 2fi (1− fi )β
2i
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 5 / 29
Heritability Estimates Definition
Definition
Z = Ψ(G ,E ) = [α +∑βigi ]
The variance explained by the ith variant Vi = 2fi (1− fi )β2i
Vknown = VS =∑
i∈S Vi
h2all = Vall
h2known =
∑i 2fi (1− fi )β
2i
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 5 / 29
Heritability Estimates Definition
Definition
Z = Ψ(G ,E ) = [α +∑βigi ]
The variance explained by the ith variant Vi = 2fi (1− fi )β2i
Vknown = VS =∑
i∈S Vi
h2all = Vall
h2known =
∑i 2fi (1− fi )β
2i
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 5 / 29
Heritability Estimates Definition
Definition
Z = Ψ(G ,E ) = [α +∑βigi ]
The variance explained by the ith variant Vi = 2fi (1− fi )β2i
Vknown = VS =∑
i∈S Vi
h2all = Vall
h2known =
∑i 2fi (1− fi )β
2i
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 5 / 29
Heritability Estimates Definition
Definition
πexplained = h2known/h
2pop
πmissing = 1− πexplained = 1− h2known/h
2pop
πphantom = 1− h2all/h
2pop
Z = Ψ(G ,E ) = Ψ′(g1, . . . , gn) + ε, where ε ∼ N(0,Ve)
VP = VG + Ve = VA + VD + Ve , if no interaction
VP = VG + Ve =∑n
i ,j=0 VAi D j + Ve , where VAi D j represents theinteraction of additive variance of order i and dominance variance oforder j
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 6 / 29
Heritability Estimates Definition
Definition
πexplained = h2known/h
2pop
πmissing = 1− πexplained = 1− h2known/h
2pop
πphantom = 1− h2all/h
2pop
Z = Ψ(G ,E ) = Ψ′(g1, . . . , gn) + ε, where ε ∼ N(0,Ve)
VP = VG + Ve = VA + VD + Ve , if no interaction
VP = VG + Ve =∑n
i ,j=0 VAi D j + Ve , where VAi D j represents theinteraction of additive variance of order i and dominance variance oforder j
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 6 / 29
Heritability Estimates Definition
Definition
πexplained = h2known/h
2pop
πmissing = 1− πexplained = 1− h2known/h
2pop
πphantom = 1− h2all/h
2pop
Z = Ψ(G ,E ) = Ψ′(g1, . . . , gn) + ε, where ε ∼ N(0,Ve)
VP = VG + Ve = VA + VD + Ve , if no interaction
VP = VG + Ve =∑n
i ,j=0 VAi D j + Ve , where VAi D j represents theinteraction of additive variance of order i and dominance variance oforder j
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 6 / 29
Heritability Estimates Definition
Definition
πexplained = h2known/h
2pop
πmissing = 1− πexplained = 1− h2known/h
2pop
πphantom = 1− h2all/h
2pop
Z = Ψ(G ,E ) = Ψ′(g1, . . . , gn) + ε, where ε ∼ N(0,Ve)
VP = VG + Ve = VA + VD + Ve , if no interaction
VP = VG + Ve =∑n
i ,j=0 VAi D j + Ve , where VAi D j represents theinteraction of additive variance of order i and dominance variance oforder j
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 6 / 29
Heritability Estimates Definition
Definition
πexplained = h2known/h
2pop
πmissing = 1− πexplained = 1− h2known/h
2pop
πphantom = 1− h2all/h
2pop
Z = Ψ(G ,E ) = Ψ′(g1, . . . , gn) + ε, where ε ∼ N(0,Ve)
VP = VG + Ve = VA + VD + Ve , if no interaction
VP = VG + Ve =∑n
i ,j=0 VAi D j + Ve , where VAi D j represents theinteraction of additive variance of order i and dominance variance oforder j
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 6 / 29
Heritability Estimates Definition
Definition
πexplained = h2known/h
2pop
πmissing = 1− πexplained = 1− h2known/h
2pop
πphantom = 1− h2all/h
2pop
Z = Ψ(G ,E ) = Ψ′(g1, . . . , gn) + ε, where ε ∼ N(0,Ve)
VP = VG + Ve = VA + VD + Ve , if no interaction
VP = VG + Ve =∑n
i ,j=0 VAi D j + Ve , where VAi D j represents theinteraction of additive variance of order i and dominance variance oforder j
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 6 / 29
Heritability Estimates The ACE/ADE model
Outline
1 Heritability EstimatesDefinitionThe ACE/ADE modelThe parent-offspring regression
2 The liability-threshold (A∆) modelDefinitionThe genetic relative risk
3 The limiting pathway (LP) model for quantitative traitsDefinitionh2
all and h2pop
4 h2slope(κ0)
DefinitionProperty
5 Detecting epistasis among variantsCochran-Armitage trend test
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 7 / 29
Heritability Estimates The ACE/ADE model
The ACE/ADE model
ACDE: Additive, Common environment, unique Environment andDominance
rMZ : the monozygotic twin correlation
rDZ : the dizygotic twin correlation
The ACE/ADE model
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 8 / 29
Heritability Estimates The ACE/ADE model
The ACE/ADE model
ACDE: Additive, Common environment, unique Environment andDominance
rMZ : the monozygotic twin correlation
rDZ : the dizygotic twin correlation
The ACE/ADE model
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 8 / 29
Heritability Estimates The ACE/ADE model
The ACE/ADE model
ACDE: Additive, Common environment, unique Environment andDominance
rMZ : the monozygotic twin correlation
rDZ : the dizygotic twin correlation
The ACE/ADE model
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 8 / 29
Heritability Estimates The ACE/ADE model
The ACE/ADE model
ACDE: Additive, Common environment, unique Environment andDominance
rMZ : the monozygotic twin correlation
rDZ : the dizygotic twin correlation
The ACE/ADE model
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 8 / 29
Heritability Estimates The ACE/ADE model
The ACE model
rMZ = VA + VC
rDZ = 1/2VA + VC
VA = 2(rMZ − rDZ )
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 9 / 29
Heritability Estimates The ACE/ADE model
The ACE model
rMZ = VC +n∑
i ,j=0
VAi D j
rDZ = VC +n∑
i ,j=0
2−(i+2j)VAi D j
h2pop(ACE ) = 2(rMZ − rDZ )
h2pop(ACE ) =
n∑i ,j=0
(1− 2−(i+2j))VAi D j
= h2all +
n∑(i ,j) 6=(1,0)
(1− 2−(i+2j))VAi D j
= h2all + W (ACE )
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 10 / 29
Heritability Estimates The ACE/ADE model
The ADE model
h2pop(ADE ) = 4rMZ − rDZ
h2pop(ADE ) =
n∑i ,j=0
(4× 2−(i+2j) − 1)VAi D j + 3VC
= h2all +
n∑(i ,j)6=(1,0),(0,1),(2,0)
(4× 2−(i+2j) − 1)VAi D j + 3VC
= h2all + W (ADE )
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 11 / 29
Heritability Estimates The parent-offspring regression
Outline
1 Heritability EstimatesDefinitionThe ACE/ADE modelThe parent-offspring regression
2 The liability-threshold (A∆) modelDefinitionThe genetic relative risk
3 The limiting pathway (LP) model for quantitative traitsDefinitionh2
all and h2pop
4 h2slope(κ0)
DefinitionProperty
5 Detecting epistasis among variantsCochran-Armitage trend test
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 12 / 29
Heritability Estimates The parent-offspring regression
The parent-offspring regression
h2pop(PO) =
√2rPO
rPO = corr(Zoff ,Zf + Zm
2)
h2pop(PO) = VC +
n∑i=0
21−iVAi
= h2all + VC +
n∑i=1
21−iVAi
= h2all + W (PO)
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 13 / 29
The liability-threshold (A∆) model Definition
Outline
1 Heritability EstimatesDefinitionThe ACE/ADE modelThe parent-offspring regression
2 The liability-threshold (A∆) modelDefinitionThe genetic relative risk
3 The limiting pathway (LP) model for quantitative traitsDefinitionh2
all and h2pop
4 h2slope(κ0)
DefinitionProperty
5 Detecting epistasis among variantsCochran-Armitage trend test
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 14 / 29
The liability-threshold (A∆) model Definition
Definition
A∆(h2, τ, cR)
P =∑n
i=1 βig′i + ε
The disease occurs ( Z = 1 ) if and only if P ≥ τ∑ni=1 βig
′i ∼ N(0, h2)
ε ∼ N(0, 1− h2)
ε = εc,R + εu,R
cR = Var(εc,R)/(1− h2)
κR : The kinship coefficient
λR : The relative risk
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 15 / 29
The liability-threshold (A∆) model Definition
Definition
A∆(h2, τ, cR)
P =∑n
i=1 βig′i + ε
The disease occurs ( Z = 1 ) if and only if P ≥ τ∑ni=1 βig
′i ∼ N(0, h2)
ε ∼ N(0, 1− h2)
ε = εc,R + εu,R
cR = Var(εc,R)/(1− h2)
κR : The kinship coefficient
λR : The relative risk
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 15 / 29
The liability-threshold (A∆) model Definition
Definition
A∆(h2, τ, cR)
P =∑n
i=1 βig′i + ε
The disease occurs ( Z = 1 ) if and only if P ≥ τ
∑ni=1 βig
′i ∼ N(0, h2)
ε ∼ N(0, 1− h2)
ε = εc,R + εu,R
cR = Var(εc,R)/(1− h2)
κR : The kinship coefficient
λR : The relative risk
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 15 / 29
The liability-threshold (A∆) model Definition
Definition
A∆(h2, τ, cR)
P =∑n
i=1 βig′i + ε
The disease occurs ( Z = 1 ) if and only if P ≥ τ∑ni=1 βig
′i ∼ N(0, h2)
ε ∼ N(0, 1− h2)
ε = εc,R + εu,R
cR = Var(εc,R)/(1− h2)
κR : The kinship coefficient
λR : The relative risk
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 15 / 29
The liability-threshold (A∆) model Definition
Definition
A∆(h2, τ, cR)
P =∑n
i=1 βig′i + ε
The disease occurs ( Z = 1 ) if and only if P ≥ τ∑ni=1 βig
′i ∼ N(0, h2)
ε ∼ N(0, 1− h2)
ε = εc,R + εu,R
cR = Var(εc,R)/(1− h2)
κR : The kinship coefficient
λR : The relative risk
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 15 / 29
The liability-threshold (A∆) model Definition
Definition
A∆(h2, τ, cR)
P =∑n
i=1 βig′i + ε
The disease occurs ( Z = 1 ) if and only if P ≥ τ∑ni=1 βig
′i ∼ N(0, h2)
ε ∼ N(0, 1− h2)
ε = εc,R + εu,R
cR = Var(εc,R)/(1− h2)
κR : The kinship coefficient
λR : The relative risk
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 15 / 29
The liability-threshold (A∆) model Definition
Definition
A∆(h2, τ, cR)
P =∑n
i=1 βig′i + ε
The disease occurs ( Z = 1 ) if and only if P ≥ τ∑ni=1 βig
′i ∼ N(0, h2)
ε ∼ N(0, 1− h2)
ε = εc,R + εu,R
cR = Var(εc,R)/(1− h2)
κR : The kinship coefficient
λR : The relative risk
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 15 / 29
The liability-threshold (A∆) model Definition
Definition
A∆(h2, τ, cR)
P =∑n
i=1 βig′i + ε
The disease occurs ( Z = 1 ) if and only if P ≥ τ∑ni=1 βig
′i ∼ N(0, h2)
ε ∼ N(0, 1− h2)
ε = εc,R + εu,R
cR = Var(εc,R)/(1− h2)
κR : The kinship coefficient
λR : The relative risk
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 15 / 29
The liability-threshold (A∆) model Definition
Definition
A∆(h2, τ, cR)
P =∑n
i=1 βig′i + ε
The disease occurs ( Z = 1 ) if and only if P ≥ τ∑ni=1 βig
′i ∼ N(0, h2)
ε ∼ N(0, 1− h2)
ε = εc,R + εu,R
cR = Var(εc,R)/(1− h2)
κR : The kinship coefficient
λR : The relative risk
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 15 / 29
The liability-threshold (A∆) model The genetic relative risk
Outline
1 Heritability EstimatesDefinitionThe ACE/ADE modelThe parent-offspring regression
2 The liability-threshold (A∆) modelDefinitionThe genetic relative risk
3 The limiting pathway (LP) model for quantitative traitsDefinitionh2
all and h2pop
4 h2slope(κ0)
DefinitionProperty
5 Detecting epistasis among variantsCochran-Armitage trend test
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 16 / 29
The liability-threshold (A∆) model The genetic relative risk
The genetic relative risk
The prevalence µ = Φ(τ)
(PPR
)∼ N(
(00
),
(1 2κRh
2
2κRh2 1
))
λR = 1µ2Pr(P|PR > τ) = 1
µ2
∫∞x=τ ϕ(x)[1− φ( τ−κR h2x√
1−κ2R h4
)]dx
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 17 / 29
The liability-threshold (A∆) model The genetic relative risk
The genetic relative risk
The prevalence µ = Φ(τ)(PPR
)∼ N(
(00
),
(1 2κRh
2
2κRh2 1
))
λR = 1µ2Pr(P|PR > τ) = 1
µ2
∫∞x=τ ϕ(x)[1− φ( τ−κR h2x√
1−κ2R h4
)]dx
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 17 / 29
The liability-threshold (A∆) model The genetic relative risk
The genetic relative risk
The prevalence µ = Φ(τ)(PPR
)∼ N(
(00
),
(1 2κRh
2
2κRh2 1
))
λR = 1µ2Pr(P|PR > τ) = 1
µ2
∫∞x=τ ϕ(x)[1− φ( τ−κR h2x√
1−κ2R h4
)]dx
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 17 / 29
The liability-threshold (A∆) model The genetic relative risk
The genetic relative risk
The genetic relative risk can be defined as the relative increase inlikelihood of disease given a homozygous risk genotype, compared toa heterozygous state, ηi = Pr(Z=1|gi =2)
Pr(Z=1|gi =1)
It can also define the genetic relative risk in terms of alleles,
ηi =Pr(Z=1|gi,(M)=1)
Pr(Z=1|gi,(M)=0)
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 18 / 29
The liability-threshold (A∆) model The genetic relative risk
The genetic relative risk
The genetic relative risk can be defined as the relative increase inlikelihood of disease given a homozygous risk genotype, compared toa heterozygous state, ηi = Pr(Z=1|gi =2)
Pr(Z=1|gi =1)
It can also define the genetic relative risk in terms of alleles,
ηi =Pr(Z=1|gi,(M)=1)
Pr(Z=1|gi,(M)=0)
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 18 / 29
The limiting pathway (LP) model for quantitative traits Definition
Outline
1 Heritability EstimatesDefinitionThe ACE/ADE modelThe parent-offspring regression
2 The liability-threshold (A∆) modelDefinitionThe genetic relative risk
3 The limiting pathway (LP) model for quantitative traitsDefinitionh2
all and h2pop
4 h2slope(κ0)
DefinitionProperty
5 Detecting epistasis among variantsCochran-Armitage trend test
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 19 / 29
The limiting pathway (LP) model for quantitative traits Definition
Definition
LP(k , h2pathway , cR)
The Limiting Pathways model for quantitative trait is defined as theminimum of k standard Gaussian i.i.d. random variables, Zi , witheach being the sum of genetic, common environmental and uniqueenvironmental components, with respective variances h2
pathway ,
cR(1− h2pathway ) and (1− cR)(1− h2
pathway ).
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 20 / 29
The limiting pathway (LP) model for quantitative traits Definition
Definition
LP(k , h2pathway , cR)
The Limiting Pathways model for quantitative trait is defined as theminimum of k standard Gaussian i.i.d. random variables, Zi , witheach being the sum of genetic, common environmental and uniqueenvironmental components, with respective variances h2
pathway ,
cR(1− h2pathway ) and (1− cR)(1− h2
pathway ).
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 20 / 29
The limiting pathway (LP) model for quantitative traits h2all and h2
pop
Outline
1 Heritability EstimatesDefinitionThe ACE/ADE modelThe parent-offspring regression
2 The liability-threshold (A∆) modelDefinitionThe genetic relative risk
3 The limiting pathway (LP) model for quantitative traitsDefinitionh2
all and h2pop
4 h2slope(κ0)
DefinitionProperty
5 Detecting epistasis among variantsCochran-Armitage trend test
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 21 / 29
The limiting pathway (LP) model for quantitative traits h2all and h2
pop
h2all = kh2
pathway
E [Z1 · Z ]
σ2Z
h2pop = 2(rMZ − rDZ )
rR =E [Z · ZR ]− µ2
Z
σ2Z
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 22 / 29
h2slope(κ0) Definition
Outline
1 Heritability EstimatesDefinitionThe ACE/ADE modelThe parent-offspring regression
2 The liability-threshold (A∆) modelDefinitionThe genetic relative risk
3 The limiting pathway (LP) model for quantitative traitsDefinitionh2
all and h2pop
4 h2slope(κ0)
DefinitionProperty
5 Detecting epistasis among variantsCochran-Armitage trend test
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 23 / 29
h2slope(κ0) Definition
Definition
IBD: Identity by descent
κi ,j = κ(Ii , Ij ): the proportion of their genomes shared in large IBDsegments
ρ(κ): the average phenotypic correlation between pairs of individualswho share proportion κ of their genomes in large IBD blocks
ρ(κ)′: the rate of change of phenotypic correlation around theaverage sharing level of large IBD segments
h2slope = (1− κ0)ρ(κ)′
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 24 / 29
h2slope(κ0) Definition
Definition
IBD: Identity by descent
κi ,j = κ(Ii , Ij ): the proportion of their genomes shared in large IBDsegments
ρ(κ): the average phenotypic correlation between pairs of individualswho share proportion κ of their genomes in large IBD blocks
ρ(κ)′: the rate of change of phenotypic correlation around theaverage sharing level of large IBD segments
h2slope = (1− κ0)ρ(κ)′
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 24 / 29
h2slope(κ0) Definition
Definition
IBD: Identity by descent
κi ,j = κ(Ii , Ij ): the proportion of their genomes shared in large IBDsegments
ρ(κ): the average phenotypic correlation between pairs of individualswho share proportion κ of their genomes in large IBD blocks
ρ(κ)′: the rate of change of phenotypic correlation around theaverage sharing level of large IBD segments
h2slope = (1− κ0)ρ(κ)′
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 24 / 29
h2slope(κ0) Definition
Definition
IBD: Identity by descent
κi ,j = κ(Ii , Ij ): the proportion of their genomes shared in large IBDsegments
ρ(κ): the average phenotypic correlation between pairs of individualswho share proportion κ of their genomes in large IBD blocks
ρ(κ)′: the rate of change of phenotypic correlation around theaverage sharing level of large IBD segments
h2slope = (1− κ0)ρ(κ)′
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 24 / 29
h2slope(κ0) Definition
Definition
IBD: Identity by descent
κi ,j = κ(Ii , Ij ): the proportion of their genomes shared in large IBDsegments
ρ(κ): the average phenotypic correlation between pairs of individualswho share proportion κ of their genomes in large IBD blocks
ρ(κ)′: the rate of change of phenotypic correlation around theaverage sharing level of large IBD segments
h2slope = (1− κ0)ρ(κ)′
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 24 / 29
h2slope(κ0) Property
Outline
1 Heritability EstimatesDefinitionThe ACE/ADE modelThe parent-offspring regression
2 The liability-threshold (A∆) modelDefinitionThe genetic relative risk
3 The limiting pathway (LP) model for quantitative traitsDefinitionh2
all and h2pop
4 h2slope(κ0)
DefinitionProperty
5 Detecting epistasis among variantsCochran-Armitage trend test
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 25 / 29
h2slope(κ0) Property
h2slope(κ0) is a consistent estimator for h2
all
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 26 / 29
Detecting epistasis among variants Cochran-Armitage trend test
Outline
1 Heritability EstimatesDefinitionThe ACE/ADE modelThe parent-offspring regression
2 The liability-threshold (A∆) modelDefinitionThe genetic relative risk
3 The limiting pathway (LP) model for quantitative traitsDefinitionh2
all and h2pop
4 h2slope(κ0)
DefinitionProperty
5 Detecting epistasis among variantsCochran-Armitage trend test
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 27 / 29
Detecting epistasis among variants Cochran-Armitage trend test
Cochran-Armitage trend test
Association of a single locus with disease
Detection of a pairwise interaction between two individual loci
Detection of a pairwise interaction between two pathways
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 28 / 29
Detecting epistasis among variants Cochran-Armitage trend test
Cochran-Armitage trend test
Association of a single locus with disease
Detection of a pairwise interaction between two individual loci
Detection of a pairwise interaction between two pathways
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 28 / 29
Detecting epistasis among variants Cochran-Armitage trend test
Cochran-Armitage trend test
Association of a single locus with disease
Detection of a pairwise interaction between two individual loci
Detection of a pairwise interaction between two pathways
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 28 / 29
Detecting epistasis among variants Thanks
Thanks
Thank you for your time!
Kou Qiang (Sun Yat-sen University) Introduction to Analysis of Heritability November 24, 2012 29 / 29