resemblance between relatives (part 1)faculty.washington.edu/.../biost551/...relatives.pdf ·...

Resemblance between Relatives (Part 1)

Resemblance Between Relatives (Part 1)


Resemblance of Relatives Intro

I Individuals in a population can vary considerably on physical,cognitive, and behavioral traits. Examples include height,weight, intelligence, personality, etc.

I Individual differences may arise from variation in genes,environmental experiences, or a combination of both

I It is generally accepted that individual differences in mosttraits are due to both genetic and environmental factors.

I To the extent that a trait is influenced by genetic (heritable)effects, phenotypic similarity is correlated with geneticrelatedness.


Resemblance of Relatives Intro

I The degree of resemblance among between relatives allows forthe estimation of genetic variance as well as heritability of atrait.

I We will now focus on how phenotypic correlation of relativescan be used to estimate the relative magnitude of genetic andenvironmental influences on trait variation


Galton 1889 Paper on Resemblance of Relatives

[Galton, 1889] 5


Fisher 1918 Paper on Resemblance of Relatives

RA Fisher (1918). Transactions of the Royal Society of Edinburgh 52: 399-433.

m-a m+d m+a

QQ

Qq

qq

Trait

m-a m+d m+a

QQ

Qq

qq

Trait

7


Partitioning of Phenotypic ValuesI We previously introduced the general model of P = G + E

where P is the phenotypic value, G is the genotypic value,and E is the environmental deviation that is assumed to havea mean of 0 so that E (P) = E (G ).

I If we assume that the genetic and environmental componentsare independent, we have that the variance of P isVar(P) = σ2P = σ2G + σ2E .

I We previously focused on the decomposition of the geneticvariance σ2G into additive and dominance components, i.e.,G = A + D where

Var(G ) = σ2G = σ2A + σ2D

I For multi-locus traits, we showed σ2G can also includeinteraction variance components for the additive anddominance effects. Assume no interaction effects for now.


Partitioning of Phenotypic Values

I So, for any random individual U from the population we havethat Var(PU) = σ2P = σ2G + σ2E = σ2A + σ2D + σ2E .

I We also previously showed that the covariance of thegenotypic values for non-inbred individuals U and V that

Cov(GU ,GV ) = 2θσ2A + k2σ2D

where θ and k2 and the kinship coefficient and probability ofsharing two alleles IBD for individuals U and V

I If we assume that there is no environmental correlation amongrelatives U and V for the trait (potentially a very strongassumption for relatives that lived or were reared in the samehousehold!), we have that

Cov(PU ,PV ) = Cov(GU ,GV ) = 2θσ2A + k2σ2D


Genetic Covariance

I Genetic Covariance for a few outbred relative pairs

Relationship Genetic Covariance

Identical Twins σ2A + σ2DFull Siblings 1

2σ2A + 1

4σ2D

Parent-Offspring 12σ

2A

Half Siblings* 14σ

2A

Uncle-Nephew 14σ

2A

First Cousins 18σ

2A

Double First Cousins 14σ

2A + 1

16σ2D

Second Cousins 132σ

2A

Unrelated 0

* Also grandparent-granchild


Resemblance of Parent and OffspringI Consider the trait values for a parent-offspring pairs. Let Y be

the phenotype value for the offspring and let X be thephenotype value for a parent.

I For a sample of parent-offspring pairs, one can use linearregression to predict offspring phenotype from parentgenotype.

I The linear regression model of Y on X is Y = β0 +β1X where

β1 =Cov(X ,Y )

Var(X )

I From the previous slide we have that the genetic covariancefor parent-offspring pairs is 1

2σ2A. What is β1 in terms of the

additive variance and the variance of the trait forparent-offspring pairs? Assume that there is no correlationamong the genetic and environmental effects.


Resemblance of Parent and Offspring

I We have that

β1 =Cov(X ,Y )

Var(X )=

12σ

2A

σ2P

I From the previous slide we have that the genetic covariancefor parent-offspring pairs is 1

2σ2A.

I Remember that the narrow-sense heritability is

h2 =σ2Aσ2P

I So, from the slope of the regression or parent on offspringtrait value an estimate of the h2 can be obtained from

h2 = 2β1


Resemblance of Mid-Parent and Offspring

I Now consider the trait values for the mid-parent trait value ofthe parents and the offspring pairs. Let Y be the phenotypevalue for the offspring and let X = X1+X2

2 be the mid-parent(or mean of both parents) phenotype value.

I Now obtain an estimate of the narrow-sense heritability of thetrait from the slope. Assume that the different sexes have thesame variance

I Which heritability estimator would be more efficient, the onebased on the linear regression of offspring on a single parentor the estimator based on the regression of offspring onmid-parent?


Resemblance of Mid-Parent and OffspringI We have that the slope of the regression line of offspring on

mid-parent is

β1 =Cov(X ,Y )

Var(X )=

12σ

2A

12σ

2P

=σ2Aσ2P

= h2

I So the slope of the regression of offspring on mid-parentprovides an estimate of h2

I The heritability estimator using the offspring and mid-parentshas a smaller variance than the estimator based on theoffspring and a single parent.

I For some traits, there could be a sex-effect, so using themid-parent value may not be advisable for heritabilityestimation unless some adjustment is made to the phenotypevalues to account for relevant influential confounders such asage, sex, etc.


Resemblance of Half-Siblings

I Now consider the trait values for half-sibling pairs and obtainpairs and obtain

I From the previous slide we have that the genetic covariancefor half-sibling pairs is 1

4σ2A

I Obtain an estimate of the narrow-sense heritability of the traitfrom the slope of the regression.


General FormulaI The correlation of the additive trait values for pairs of

non-inbred individuals X and Y is

ρXY = 2θXY h2

I This relationship between correlation and heritability is whythe slope of a linear regression model for inference onheritability for relationship types that only have an nonzeroadditive variance component and a dominance variancecomponent that is 0, and thus

h2 =ρXY2θXY

=β1

2θXY

where it is assume that Var(X ) = Var(Y ) = σ2P . It iscommon to standardized the phenotype so that σ2P = 1 forheritability estimation.


Resemblance of Full-Siblings

I What is the slope of the regression line for trait values offull-siblings?

I Under what conditions/assumptions can the slope for thisrelative pair type provide an reasonable estimate for theheritability of a trait?


Resemblance of Full-Siblings

I We have that the slope of the regression line for trait valuesof full-sibling pairs is

β1 =12σ

2A + 1

4σ2D

σ2P

I So the narrow-sense heritability can not be estimated from theslope of the regression line (or correlation of phenotypevalues) with full siblings. This relative pair type will onlyprovide an unbiased be estimate if the trait is completelyadditive, i.e., σ2D = 0

resemblance between relatives (part 1)faculty.washington.edu/.../biost551/...relatives.pdf ·...

Documents