Lecture 4:Heritability

HeritabilityNarrow vs. board sense

Narrow sense: h2 = VA/VP

Board sense: H2 = VG/VP

Slope of midparent-offspring regression (sexual reproduction)

Slope of a parent - cloned offspring regression (asexual reproduction)

When one refers to heritability, the default is narrow-sense, h2

h2 is the measure of (easily) usable genetic variation under sexual reproduction

Why h2 instead of h?

Blame Sewall Wright, who used h to denote the correlation between phenotype and breeding value. Hence, h2 is the total fraction of phenotypic variance due to breeding valuesr(A;P)=æ(A;P)æAæP=æ2AæAæP=æAæP=h

Heritabilities are functions of populations

Heritability measures the standing genetic variation of a population,A zero heritability DOES NOT imply that the trait is not geneticallydetermined

Heritability values only make sense in the content of the populationfor which it was measured.

Heritabilities are functions of the distribution ofenvironmental values (i.e., the universe of E values)

Decreasing VP increases h2.

Heritability values measured one one environment(or distribution of environments) may not be valid under another

Measures of heritability for lab-reared individualsmay be very different from heritability in nature

Heritability and the prediction of breeding values

If P denotes an individual's phenotype, then best linear predictor of their breeding value A isA=æ(P;A)æ2P(P°πp)+e=h2(P°πp)+eThe residual variance is also a function of h2:æ2e=(1°h2)æ2A

The larger the heritability, the tighter the distribution of true breeding values around the value h2(P - P) predicted by an individual’s phenotype.

Heritability and population divergence

Heritability is a completely unreliable predictor of long-term response

Measuring heritability values in two populations that show a difference in their means provides no informationon whether the underlying difference is genetic

Sample heritabilities

People hs

Height 0.65

Serum IG 0.45

Pigs

Back-fat 0.70

Weight gain 0.30

Litter size 0.05

Fruit Flies

Abdominal Bristles

0.50

Body size 0.40

Ovary size 0.3

Egg production 0.20

Traits more closelyassociated with fitnesstend to have lower heritabilities

Estimation: One-way ANOVA

Simple (balanced) full-sib design: N full-sib families, each with n offspring

One-way ANOVA model:

zij = + fi + wij

Trait value in sib j from family iCommon MeanEffect for family i =deviation of mean of i from The common mean

Deviation of sib j from the family mean

2f = between-family variance = variance among family means

2w = within-family variance

2P = Total phenotypic variance = 2

f + 2w

Covariance between members of the same group equals the variance among (between) groupsCov(FullSibs)=æ(zij;zik)=æ[(π+fi+wij);(π+fi+wik)]=æ(fi;fi)+æ(fi;wik)+æ(wij;fi)+æ(wij;wik)=æ2fThe variance among family effects equals the

covariance between full sibsæ2f=æ2A=2+æ2D=4+æ2Ec The within-family variance 2

w = 2P - 2

f,æ2w(FS)=æ2P°(æ2A=2+æ2D=4+æ2Ec)=æ2A+æ2D+æ2E°(æ2A=2+æ2D=4+æ2Ec)=(1=2)æ2A+(3=4)æ2D+æ2E°æ2Ec

One-way Anova: N families with n sibs, T = Nn

Factor Degrees of freedom, df

Sums ofSquares (SS)

Mean sum of squares (MS)

E[ MS ]

Among-family N-1 SSf = SSf/(N-1) 2w + n 2

f

Within-family T-N SSW = SSw/(T-N) 2wNXi=1nXj=1(zij°zi)2nNi=1(zi°z)2X

Estimating the variance components:æ2f=æ2A=2+æ2D=4+æ2EcSince

2Var(f) is an upper bound for the additive variance

Var(f)=MSf°MSwnVar(w)=MSwVar(z)=Var(f)+Var(w)

Assigning standard errors ( = square root of Var)æ2(MSx)'2(MSx)2dfx+2Nice fact: Under normality, the (large-sample) variance for a mean-square is given byVar[Var(f)]=Var∑MSf°MSwn∏'2n2µ(MSf)2N+1+(MSw)2T°N+2∂(

()

)Var[Var(w(FS))]=Var(MSw)'2(MSw)2TN+2

SE(h2)'2(1°tFS)[1+(n°1)tFS]p2=[Nn(n°1)]- - -

tFS=Var(f)Var(z)=12h2+æ2D=4+æ2Ecæ2zEstimating heritability

Hence, h2 < 2 tFS

An approximate large-sample standard error for h2 is given by

Worked example

Factor Df SS MS EMS

Among-familes 9 SSf = 405 45 2f + 5 2

f

Within-families 40 SSw = 800 20 2w

10 full-sib families, each with 5 offspring are measuredVar(f)=MSf°MSwn=45°205=5Var(w)=MSw=20Var(z)=Var(f)+Var(w)=25VA < 10

h2 < 2 (5/25) = 0.4SE(h2)'2(1°0:4)[1+(5°1)0:4]p2=[50(5°1)]=0:312

Full sib-half sib design: Nested ANOVA

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Full-sibs

Half-sibs

Estimation: Nested ANOVA

Balanced full-sib / half-sib design: N males (sires)are crossed to M dams each of which has n offspring

Nested ANOVA model:

zijk = + si + dij+ wijk

Value of the kth offspringfrom the kth dam for sire i

Overall meanEffect of sire i = deviationof mean of i’s family fromoverall mean

Effect of dam j of sire i = deviationof mean of dam j from sire and overallmean

Within-family deviation of kthoffspring from the mean of theij-th family

2s = between-sire variance = variance in sire family means

2d = variance among dams within sires =

variance of dam means for the same sire

2w = within-family variance

2T = 2

s + 2d + 2

w

Nested Anova: N sires crossed to M dams, each with n sibs, T = NMn

Factor Df SS MS EMS

Sires N-1 SSs = SSs/(N-1)

Dams(Sires) N(M-1) SSs = SSd /(N[M-1])

Sibs(Dams) T-NM SSw = SSw/(T-NM)

MnNXi=1MiXj=1(zi°z)2æ2w+næ2d+Mnæ2snNXi=1MXj=1(zij°zi)2æ2w+næ2dæ2wNXi=1MXj=1nXk=1(zijk°zij)2

Var(s)=MSs°MSdMnVar(d)=MSd°MSwnVar(e)=MSwEstimation of sire, dam, and family variances:

Translating these into the desired variance components

• Var(Total) = Var(between FS families) + Var(Within FS)

• Var(Sires) = Cov(Paternal half-sibs)

2w = 2

z - Cov(FS)æ2d=æ2z°æ2s°æ2w=æ(FS)°æ(PHS)

æ2w=æ2z°æ(FS)æ2s=æ(PHS)æ2d=æ2z°æ2s°æ2w=æ(FS)°æ(PHS)æ2s'æ2A4æ2d'æ2A4+æ2D4+æ2Ecæ2w'æ2A2+3æ2D4+æ2EsSummarizing,

Expressing these in terms of the genetic and environmental variances,

tPHS=Cov(PHS)Var(z)=Var(s)Var(z)tFS=Cov(FS)Var(z)=Var(s)+Var(d)Var(z)4tPHS = h2

h2 < 2tFS

Intraclass correlations and estimating heritability

Note that 4tPHS = 2tFS implies no dominance or shared family environmental effects

Worked Example: N=10 sires, M = 3 dams, n = 10 sibs/dam

Factor Df SS MS EMS

Sires 9 4,230 470

Dams(Sires) 20 3,400 170

Within Dams 270 5,400 20

æ2w+10æ2dæ2w+10æ2d+30æ2sæ2wæ2w=MSw=20æ2d=MSd°MSwn=170°2010=15æ2s=MSs°MSdNn=470°17030=10æ2P=æ2s+æ2d+æ2w=45æ2A=4æ2s=40h2=æ2Aæ2P=4045=0:89æ2d=15=(1=4)æ2A+(1=4)æ2D+æ2Ec=10+(1=4)æ2D+æ2Ecæ2D+4æ2Ec=20

Parent-offspring regressionzoi=π+bojp(zpi°π)+eiSingle parent - offspring regressionE(bojp)=æ(zo;zp)æ2(zp)'(æ2A=2)+æ(Eo;Ep)æ2z=h22+æ(Eo;Ep)æ2zThe expected slope of this regression is:

Shared environmental values

To avoid this term, typically regressions are male-offspring, as female-offspring more likely to share environmental values

æ2e=µ1°h22∂æ2z

Residual error variance (spread around expected values)

zoi=π+bojMPµzmi+zfi2°π∂+ei

Midparent - offspring regression

The expected slope of this regression is h2, as bokMP=Cov[zo;(zm+zf)=2]Var[(zm+zf)=2]=[Cov(zo;zm)+Cov(zo;zf)]=2[Var(z)+Var(z)]=4=2Cov(zo;zp)Var(z)=2bojpæ2e=µ1°h22∂æ2z

Residual error variance (spread around expected values)

Hence, even when heritability is one, there is considerable spread of the true offspring values about their midparent values, with Var = VA/2, the segregation variance

Key: Var(MP) = (1/2)Var(P)bokMP=Cov[zo;(zm+zf)=2]Var[(zm+zf)=2]=[Cov(zo;zm)+Cov(zo;zf)]=2[Var(z)+Var(z)]=4=2Cov(zo;zp)Var(z)=2bojp

Standard errors Single parent-offspring regression, N parents, each with n offspringVar(bojp)'n(t°b2ojp)+(1°t)Nn

Squared regression slopeTotal number of offspringt=8<tHS=h2=4forhalf-sibstFS=h2=2+æ2D+æ2Ecæ2zforfullsibs:Sib correlation

Var(h2)=Var(2bojp)=4Var(bojp)Midparent-offspring regression, N sets of parents, each with n offspringVar(h2)=Var(bojMP)'2[n(tFS°b2ojMP=2)+(1°tFS)])Nn

• Midparent-offspring variance half that of single parent-offspring variance

Estimating Heritability in Natural Populations

Often, sibs are reared in a laboratory environment, making parent-offspring regressions and sib ANOVAproblematic for estimating heritability

A lower bound can be placed of heritability usingparents from nature and their lab-reared offspring,h2min=(b0ojMP)2Varn(z)Varl(A)Why is this a lower bound?(b0ojMP)2Varn(z)Varl(A)=∑Covl;n(A)Varn(z)∏2Varn(z)Varl(A)

Covariance between breeding value in nature and BV in lab

=∞2h2nwhere∞=Covl;n(A)Varn(A)Varl(A)p

is the additive genetic covariance between

environments and hence 2 < 1

Defining H2 for Plant PopulationsPlant breeders often do not measure individual plants (especially with pure lines), but instead measure a plot or a block of individuals.

This can result in inconsistent measures of H2 even for otherwise identical populations.zijk̀=Gi+Ej+GEij+pijk+eijk̀

Genotype iEnvironment jInteraction between Genotype Iand environment j

Effect of plot k for Genotype Iin environment j

deviations of individualPlants within this plot

If we set our unit of measurement as the average over all plots,the phenotypic variance becomesæ2(zi)=æ2G+æ2E+æ2GEe+æ2per+æ2eern

Number of EnvironmentsNumber of plots/environmentNumber of individuals/plotHence, VP, and hence H2, depends on our choice of e, r, and n