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Correlated characters Sanja Franic VU 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? rP XY cov PXY PX PY cov PXY rP XY PX PY cov PXY cov AXY cov E XY rP XY PX PY rAXY AX AY rE XY EX EY A h P (phenotypic correlation) (phenotypic covariance) (phenotypic covariance expressed in terms of A and E) (substitution gives) (because σ2P= σ2A+ σ2E σP= σA+ σE σP=hσP+eσP) E e P rP XY PX PY rAXY hX PX hY PY rE XY e X PX eY PY rP XY rAXY hX hY rE XY e X eY (substitution gives) (phenotypic correlation expressed in terms of A and E) • • We can only observe the phenotypic correlation How to decompose it into genetic and environmental causal components? rP XY cov PXY PX PY cov PXY rP XY PX PY cov PXY cov AXY cov E XY rP XY PX PY rAXY AX AY rE XY EX EY A h P (phenotypic correlation) (phenotypic covariance) (phenotypic covariance expressed in terms of A and E) (substitution gives) (because σ2P= σ2A+ σ2E σP= σA+ σE σP=hσP+eσP) E e P rP XY PX PY rAXY hX PX hY PY rE XY e X PX eY PY rP XY rAXY hX hY rE XY e X eY (substitution gives) (phenotypic correlation expressed 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 sires d=number of dames per sire k=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 sires d=number of dames per sire k=number of offspring per dam 2 P 2 S 2 D 2W between-sire between-dam within-sire within-progeny observational components 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 sires d=number of dames per sire k=number of offspring per dam 2 P 2 S 2 D 2W between-sire between-dam within-sire within-progeny A D E observational components causal components 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 = σ2T-σ2S -σ2W = 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 = σ2T-σ2S -σ2W = 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 rA • • • cov XY XY cov XY var X varY covS = ¼ covA varSX = ¼ σ2AX varSY = ¼ σ2AY rA 1 cov A 4 1 cov 1 cov AX AY 4 4 rA • • • cov XY XY cov XY var X varY covS = ¼ covA varSX = ¼ σ2AX varSY = ¼ σ2AY rA 1 cov A 4 1 cov 1 cov AX AY 4 4 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 rA • • • cov XY XY cov XY var X varY covS = ¼ covA varSX = ¼ σ2AX varSY = ¼ σ2AY rA 1 cov A 4 1 cov 1 cov AX AY 4 4 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 rA cov XY cov XX covYY 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 b( A)YX cov A 2 AX r AY AX 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 b( A)YX cov A 2 AX r AY AX because: r cov XY XY bYX , cov XY r X Y cov XY 2 , cov XY bYX 2 X X r X Y bYX 2 X b r X Y 2X r Y X RX ihX AX [11.4] RX ihX AX CRY b( A)YX RX [11.4] [11.4] RX ihX AX CRY b( A)YX RX CRY ih X AX rA AY AX [11.4] RX ihX AX CRY b( A)YX RX CRY ih X AX rA CRY ihX rA AY AY AX [11.4] RX ihX AX CRY b( A)YX RX CRY ih X AX rA AY AX CRY ihX rA AY Since AY hY PY : CRY ih X hY rA PY [11.4] RX ihX AX CRY b( A)YX RX CRY ih X AX rA AY AX CRY ihX rA AY Since AY hY PY : CRY ih X hY rA PY Coheritability [11.4] RX ihX AX CRY b( A)YX RX CRY ih X AX rA AY AX CRY ihX rA AY Since AY hY PY : CRY ih X hY rA PY Coheritability RX ih 2 PX Heritability [11.3] • Questions?