Correlated characters Sanja Franic VU University Amsterdam 2008.
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Transcript of Correlated characters Sanja Franic VU University Amsterdam 2008.
Correlated characters
Sanja FranicVU University Amsterdam 2008
• Relationship between 2 metric characters whose values are correlated in the individuals of a population
• Relationship between 2 metric characters whose values are correlated in the individuals of a population
• Why are correlated characters important?
• Effects of pleiotropy in quantitative genetics
– Pleiotropy – gene affects 2 or more characters – (e.g. genes that increase growth rate increase both height and weight)
• Selection – how will the improvement in one character cause simultaneous changes in other characters?
• Relationship between 2 metric characters whose values are correlated in the individuals of a population
• Why are correlated characters important?
• Effects of pleiotropy in quantitative genetics
– Pleiotropy – gene affects 2 or more characters – (e.g. genes that increase growth rate increase both height and weight)
• Selection – how will the improvement in one character cause simultaneous changes in other characters?
• Causes of correlation:
• Genetic– mainly pleiotropy– but some genes may cause +r, while some cause –r, so overall effect not
always detectable
• Environmental– two characters influenced by the same differences in the environment
• We can only observe the phenotypic correlation • How to decompose it into genetic and environmental causal components?
• We can only observe the phenotypic correlation • How to decompose it into genetic and environmental causal components?
YXEYXAP
PYYPXXEPYYPXXAPYPXP
PE
PA
EYEXEAYAXAPYPXP
EAP
PYPXPP
PYPX
PP
eerhhrr
eerhhrr
e
h
rrr
r
r
XYXYXY
XYXYXY
XYXYXY
XYXYXY
XYXY
XY
XY
covcovcov
cov
cov (phenotypic correlation)
(phenotypic covariance)
(phenotypic covarianceexpressed in terms of A and E)
(substitution gives)
(because σ2P= σ2
A+ σ2E
σP= σA+ σE
σP=hσP+eσP)
(substitution gives)
(phenotypic correlationexpressed in terms of A and E)
• We can only observe the phenotypic correlation • How to decompose it into genetic and environmental causal components?
YXEYXAP
PYYPXXEPYYPXXAPYPXP
PE
PA
EYEXEAYAXAPYPXP
EAP
PYPXPP
PYPX
PP
eerhhrr
eerhhrr
e
h
rrr
r
r
XYXYXY
XYXYXY
XYXYXY
XYXYXY
XYXY
XY
XY
covcovcov
cov
cov (phenotypic correlation)
(phenotypic covariance)
(phenotypic covarianceexpressed in terms of A and E)
(substitution gives)
(because σ2P= σ2
A+ σ2E
σP= σA+ σE
σP=hσP+eσP)
(substitution gives)
(phenotypic correlationexpressed in terms of A and E)
Estimation of the genetic correlation
• Analogous to estimation of heritabilities, but instead of ANOVA we use an ANCOVA
Estimation of the genetic correlation
• Analogous to estimation of heritabilities, but instead of ANOVA we use an ANCOVA
Half-sib families• Design: a number of sires each mated to several dames (random mating) • A number of offspring from each dam are measured
Estimation of the genetic correlation
• Analogous to estimation of heritabilities, but instead of ANOVA we use an ANCOVA
Half-sib families• Design: a number of sires each mated to several dames (random mating) • A number of offspring from each dam are measured
s=number of siresd=number of dames per sirek=number of offspring per dam
Estimation of the genetic correlation
• Analogous to estimation of heritabilities, but instead of ANOVA we use an ANCOVA
Half-sib families• Design: a number of sires each mated to several dames (random mating) • A number of offspring from each dam are measured
s=number of siresd=number of dames per sirek=number of offspring per dam
observational components
WDSP2222
between-sire
between-dam
within-sire
within-progeny
Estimation of the genetic correlation
• Analogous to estimation of heritabilities, but instead of ANOVA we use an ANCOVA
Half-sib families• Design: a number of sires each mated to several dames (random mating) • A number of offspring from each dam are measured
s=number of siresd=number of dames per sirek=number of offspring per dam
observational components
causal components
WDSP2222
between-sire
between-dam
within-sire
within-progeny
A D E
σ2S = variance between means of half-sib families (phenotypic covariance of half-sibs) = ¼ VA
σ2S = variance between means of half-sib families (phenotypic covariance of half-sibs) = ¼ VA
σ2W VT = VBG + VWG
VWG = VT – VBG
VBG = covFS
covFS = ½ VA + ¼ VD
σ2W = VWG = VT - ½ VA - ¼ VD
= VA + VD +VE - ½ VA - ¼ VD
= ½ VA + ¾ VD + VEW
σ2S = variance between means of half-sib families (phenotypic covariance of half-sibs) = ¼ VA
σ2W VT = VBG + VWG
VWG = VT – VBG
VBG = covFS
covFS = ½ VA + ¼ VD
σ2W = VWG = VT - ½ VA - ¼ VD
= VA + VD +VE - ½ VA - ¼ VD
= ½ VA + ¾ VD + VEW
σ2D = σ2
T-σ2S -σ2
W
= VA + VD +VE - ¼ VA - ½ VA – ¾ VD - VEW
= ¼ VA + ¼ VD + VEC
(VE = VEC +VEW)
σ2S = variance between means of half-sib families (phenotypic covariance of half-sibs) = ¼ VA
σ2W VT = VBG + VWG
VWG = VT – VBG
VBG = covFS
covFS = ½ VA + ¼ VD
σ2W = VWG = VT - ½ VA - ¼ VD
= VA + VD +VE - ½ VA - ¼ VD
= ½ VA + ¾ VD + VEW
σ2D = σ2
T-σ2S -σ2
W
= VA + VD +VE - ¼ VA - ½ VA – ¾ VD - VEW
= ¼ VA + ¼ VD + VEC
(VE = VEC +VEW)
• In partitioning the covariance, instead of starting from individual values we start from the product of the values of the 2 characters
covS = ¼ covA
• covS = ¼ covA
• varSX = ¼ σ2AX
• varSY = ¼ σ2AY
YX
XY
YX
XYAr
varvar
covcov
AYAX
A
Arcov4
1cov41
cov41
• covS = ¼ covA
• varSX = ¼ σ2AX
• varSY = ¼ σ2AY
Offspring-parent relationship
• To estimate the heritability of one character, we compute the covariance of offspring and parent
• To estimate the genetic correlation between 2 characters we compute the “cross-variance”: product of value of X in offspring and value of Y in parents
• Cross-variance = ½ covA
YX
XY
YX
XYAr
varvar
covcov
AYAX
A
Arcov4
1cov41
cov41
• covS = ¼ covA
• varSX = ¼ σ2AX
• varSY = ¼ σ2AY
Offspring-parent relationship
• To estimate the heritability of one character, we compute the covariance of offspring and parent
• To estimate the genetic correlation between 2 characters we compute the “cross-variance”: product of value of X in offspring and value of Y in parents
• Cross-variance = ½ covA
YX
XY
YX
XYAr
varvar
covcov
AYAX
A
Arcov4
1cov41
cov41
YYXX
XYAr
covcov
cov
Correlated response to selection
• If we select for X, what will be the change in Y?
Correlated response to selection
• If we select for X, what will be the change in Y?
• The response in X – the mean breeding value of the selected individuals• The consequent change in Y – regression of breeding value of Y on breeding
value of X
Correlated response to selection
• If we select for X, what will be the change in Y?
• The response in X – the mean breeding value of the selected individuals• The consequent change in Y – regression of breeding value of Y on breeding
value of X
AX
AY
AX
AA rbYX
2)(
cov
Correlated response to selection
• If we select for X, what will be the change in Y?
• The response in X – the mean breeding value of the selected individuals• The consequent change in Y – regression of breeding value of Y on breeding
value of X
because:
AX
AY
AX
AA rbYX
2)(
cov
X
Y
X
YX
XYXYX
XYXXYX
XYYX
YXXYYX
XY
rr
b
br
bb
rr
2
2
22
cov,cov
cov,cov
AXXX ihR
[11.4]
AXXX ihR [11.4]
XYXAY RbCR )(
AXXX ihR [11.4]
XYXAY RbCR )(
AX
AYAAXXY rihCR
AXXX ihR [11.4]
XYXAY RbCR )(
AX
AYAAXXY rihCR
AYAXY rihCR
AXXX ihR [11.4]
XYXAY RbCR )(
AX
AYAAXXY rihCR
AYAXY rihCR
PYAYXY
PYYAY
rhihCR
hSince
:
AXXX ihR [11.4]
XYXAY RbCR )(
AX
AYAAXXY rihCR
AYAXY rihCR
PYAYXY
PYYAY
rhihCR
hSince
:
Coheritability
AXXX ihR [11.4]
XYXAY RbCR )(
AX
AYAAXXY rihCR
AYAXY rihCR
PYAYXY
PYYAY
rhihCR
hSince
:
Coheritability
PXX ihR 2
Heritability
[11.3]
• Questions?