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!

introduction to analysis of heritability

Education

broadsense heritability

apparent heritability

proportion of heritability

variants h2

phantom heritability

h2 slope0 denition property

model denition

variants kou qiang