Introduction to analysis of heritability

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

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

Heritability Estimates Definition

Outline

all and h2pop

4 h2slope(κ0)

DefinitionProperty

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)

Definition

h2slope(κ0)

Definition

h2slope(κ0)

Definition

h2slope(κ0)

Definition

h2slope(κ0)

Definition

h2slope(κ0)

Definition

h2slope(κ0)

Definition

h2slope(κ0)

Definition

h2slope(κ0)

Definition

h2slope(κ0)

Definition

h2slope(κ0)

Definition

h2slope(κ0)

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

Definition

Z = Ψ(G ,E ) = [α +∑βigi ]

Vknown = VS =∑

i∈S Vi

h2all = Vall

h2known =

∑i 2fi (1− fi )β

Definition

Z = Ψ(G ,E ) = [α +∑βigi ]

Vknown = VS =∑

i∈S Vi

h2all = Vall

h2known =

∑i 2fi (1− fi )β

Definition

Z = Ψ(G ,E ) = [α +∑βigi ]

Vknown = VS =∑

i∈S Vi

h2all = Vall

h2known =

∑i 2fi (1− fi )β

Definition

Z = Ψ(G ,E ) = [α +∑βigi ]

Vknown = VS =∑

i∈S Vi

h2all = Vall

h2known =

∑i 2fi (1− fi )β

Definition

πexplained = h2known/h

πmissing = 1− πexplained = 1− h2known/h

πphantom = 1− h2all/h

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

Definition

VP = VG + Ve =∑n

Definition

VP = VG + Ve =∑n

Definition

VP = VG + Ve =∑n

Definition

VP = VG + Ve =∑n

Definition

VP = VG + Ve =∑n

Heritability Estimates The ACE/ADE model

Outline

all and h2pop

4 h2slope(κ0)

DefinitionProperty

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

The ACE model

rMZ = VA + VC

rDZ = 1/2VA + VC

VA = 2(rMZ − rDZ )

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 )

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 )

Heritability Estimates The parent-offspring regression

Outline

all and h2pop

4 h2slope(κ0)

DefinitionProperty

Heritability Estimates The parent-offspring regression

The parent-offspring regression

h2pop(PO) =

√2rPO

rPO = corr(Zoff ,Zf + Zm

h2pop(PO) = VC +

n∑i=0

21−iVAi

= h2all + VC +

n∑i=1

21−iVAi

= h2all + W (PO)

The liability-threshold (A∆) model Definition

Outline

all and h2pop

4 h2slope(κ0)

DefinitionProperty

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

Definition

A∆(h2, τ, cR)

P =∑n

i=1 βig′i + ε

′i ∼ N(0, h2)

ε ∼ N(0, 1− h2)

ε = εc,R + εu,R

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

Definition

A∆(h2, τ, cR)

P =∑n

i=1 βig′i + ε

′i ∼ N(0, h2)

ε ∼ N(0, 1− h2)

ε = εc,R + εu,R

Definition

A∆(h2, τ, cR)

P =∑n

i=1 βig′i + ε

′i ∼ N(0, h2)

ε ∼ N(0, 1− h2)

ε = εc,R + εu,R

Definition

A∆(h2, τ, cR)

P =∑n

i=1 βig′i + ε

′i ∼ N(0, h2)

ε ∼ N(0, 1− h2)

ε = εc,R + εu,R

Definition

A∆(h2, τ, cR)

P =∑n

i=1 βig′i + ε

′i ∼ N(0, h2)

ε ∼ N(0, 1− h2)

ε = εc,R + εu,R

Definition

A∆(h2, τ, cR)

P =∑n

i=1 βig′i + ε

′i ∼ N(0, h2)

ε ∼ N(0, 1− h2)

ε = εc,R + εu,R

Definition

A∆(h2, τ, cR)

P =∑n

i=1 βig′i + ε

′i ∼ N(0, h2)

ε ∼ N(0, 1− h2)

ε = εc,R + εu,R

The liability-threshold (A∆) model The genetic relative risk

Outline

all and h2pop

4 h2slope(κ0)

DefinitionProperty

The genetic relative risk

The prevalence µ = Φ(τ)

)∼ N(

(1 2κRh

2κRh2 1

λR = 1µ2Pr(P|PR > τ) = 1

∫∞x=τ ϕ(x)[1− φ( τ−κR h2x√

1−κ2R h4

The prevalence µ = Φ(τ)(PPR

)∼ N(

(1 2κRh

2κRh2 1

λR = 1µ2Pr(P|PR > τ) = 1

∫∞x=τ ϕ(x)[1− φ( τ−κR h2x√

1−κ2R h4

The prevalence µ = Φ(τ)(PPR

)∼ N(

(1 2κRh

2κRh2 1

λR = 1µ2Pr(P|PR > τ) = 1

∫∞x=τ ϕ(x)[1− φ( τ−κR h2x√

1−κ2R h4

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)

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)

The limiting pathway (LP) model for quantitative traits Definition

Outline

all and h2pop

4 h2slope(κ0)

DefinitionProperty

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 ).

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 ).

The limiting pathway (LP) model for quantitative traits h2all and h2

Outline

all and h2pop

4 h2slope(κ0)

DefinitionProperty

The limiting pathway (LP) model for quantitative traits h2all and h2

h2all = kh2

pathway

E [Z1 · Z ]

h2pop = 2(rMZ − rDZ )

rR =E [Z · ZR ]− µ2

h2slope(κ0) Definition

Outline

all and h2pop

4 h2slope(κ0)

DefinitionProperty

Definition

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)′

Definition

h2slope(κ0) Property

Outline

all and h2pop

4 h2slope(κ0)

DefinitionProperty

h2slope(κ0) Property

h2slope(κ0) is a consistent estimator for h2

Detecting epistasis among variants Cochran-Armitage trend test

Outline

all and h2pop

4 h2slope(κ0)

DefinitionProperty

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

Detecting epistasis among variants Thanks

Thanks

Thank you for your time!

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