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

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PE

PA

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EAP

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PP

eerhhrr

eerhhrr

e

h

rrr

r

r

XYXYXY

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XYXYXY

XYXY

XY

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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

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EAP

PYPXPP

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eerhhrr

e

h

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covcovcov

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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

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XYYX

YXXYYX

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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]

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AX

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AYAXY rihCR

AXXX ihR [11.4]

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AX

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AYAXY rihCR

PYAYXY

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rhihCR

hSince

:

AXXX ihR [11.4]

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AX

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PYAYXY

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:

Coheritability

AXXX ihR [11.4]

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AX

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AYAXY rihCR

PYAYXY

PYYAY

rhihCR

hSince

:

Coheritability

PXX ihR 2

Heritability

[11.3]

• Questions?