lecture 4: basic designs for estimation of genetic parameters
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Lecture 4: Basic Designs for Estimation of Genetic Parameters. Heritability. Narrow vs. board sense. Narrow sense: h 2 = V A /V P. Slope of midparent-offspring regression (sexual reproduction). Board sense: H 2 = V G /V P. Slope of a parent - cloned offspring regression - PowerPoint PPT PresentationTRANSCRIPT
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