lecture 4: basic designs for estimation of genetic parameters

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Lecture 4: Basic Designs for Estimation of Genetic Parameters

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Lecture 4:Basic Designs for

Estimation of Genetic Parameters

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 values

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.

r(A;P)=æ(A;P)æAæP=æ2AæAæP=æAæP=h

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

Decreasing VP increases h2.

Heritability values measured in 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 is

The residual variance is also a function of h2:

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

A=æ(P;A)æ2P(P°πp)+e=h2(P°πp)+eæ2e=(1°h2)æ2A

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) groups

The variance among family effects equals the covariance between full sibs

The within-family variance 2w = 2

P - 2f,

Cov(FullSibs)=æ(zij;zik)=æ[(π+fi+wij);(π+fi+wik)]=æ(fi;fi)+æ(fi;wik)+æ(wij;fi)+æ(wij;wik)=æ2fæ2f=æ2A=2+æ2D=4+æ2Ecæ2w(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) 2w

NX

i=1

nX

j =1

(zi j ° z i)2

nN

i=1

(z i ° z)2X

Estimating the variance components:

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

Var(f)=MSf°MSwnVar(w)=MSwVar(z)=Var(f)+Var(w)Sinceæ2f=æ2A=2+æ2D=4+æ2Ec

Assigning standard errors ( = square root of Var)

Nice fact: Under normality, the (large-sample) variance for a mean-square is given by

æ2(MSx) '

2(MSx)2

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

))

Var[Var(f)]=Var∑MSf°MSwn∏'2n2µ(MSf)2N+1+(MSw)2T°N+2∂

Estimating heritability

Hence, h2 < 2 tFS

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

tFS=Var(f)Var(z)=12h2+æ2D=4+æ2Ecæ2z---SE(h2)'2(1°tFS)[1+(n°1)tFS]p2=[Nn(n°1)]

Worked example

Factor Df SS MS EMS

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

f

Within-families 40 SSw = 800 20 2w

10 full-sib families, each with 5 offspring are measured

VA < 10

h2 < 2 (5/25) = 0.4

Var(f)=MSf°MSwn=45°205=5Var(w) = MSw = 20

Var(z) = Var(f) + Var(w) = 25SE(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)

MnNX

i=1

M iX

j=1

(z i ° z)2

nNX

i =1

MX

j=1

(z i j ° z i)2

NX

i=1

MX

j =1

nX

k=1

(zi jk ° zi j )2

æ2w+næ2d+Mnæ2sæ2w+næ2dæ2w

Estimation 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)

Var(s)=MSs°MSdMnVar(d)=MSd°MSwnVar(e)=MSwæ2d=æ2z°æ2s°æ2w=æ(FS)°æ(PHS)

Summarizing,

Expressing these in terms of the genetic and environmental variances,

æ2s=æ(PHS)æ2w=æ2z°æ(FS)æ2d=æ2z°æ2s°æ2w=æ(FS)°æ(PHS)æ2s'æ2A4æ2d'æ2A4+æ2D4+æ2Ecæ2w'æ2A2+3æ2D4+æ2Es

4tPHS = h2

h2 < 2tFS

Intraclass correlations and estimating heritability

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

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

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+10æ2d+30æ2sæ2w+10æ2dæ2wæ2w=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 regression

Shared environmental values

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

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

Residual error variance (spread around expected values)

æ2e=µ1°h22∂æ2z

The expected slope of this regression is h2

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

Midparent - offspring regressionzoi=π+bojMPµzmi+zfi2°π∂+ei

Residual error variance (spread around expected values)æ2e=µ1°h22∂æ2zbokMP=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)=2bojpKey: 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

Squared regression slopeTotal number of offspring

Midparent-offspring regression, N sets of parents, each with n offspring

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

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

Sib correlation t=8<tHS=h2=4forhalf-sibstFS=h2=2+æ2D+æ2Ecæ2zforfullsibs:Var(h2)=Var(2bojp)=4Var(bojp)Var(h2)=Var(bojMP)'2[n(tFS°b2ojMP=2)+(1°tFS)])Nn

Estimating Heritability in Natural Populations

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

Why is this a lower bound?

Covariance between breeding value in nature and BV in lab

A lower bound can be placed of heritability usingparents from nature and their lab-reared offspring,h2min=(b0ojMP)2Varn(z)Varl(A)

(b0ojMP)2Varn(z)Varl(A)=∑Covl;n(A)Varn(z)∏2Varn(z)Varl(A)=∞2h2nwhere

is the additive genetic covariance between

environments and hence 2 < 1

∞=Covl;n(A)Varn(A)Varl(A)p

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.

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 becomes

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

zijk̀=Gi+Ej+GEij+pijk+eijk̀æ2(zi)=æ2G+æ2E+æ2GEe+æ2per+æ2eern