1. Correlations Between Characters In “Genetics and Analysis of Quantitative traits” by Lynch, M. and Walsh, B. Presented Sansak Nakavisut

2. Topics Covariance and Correlation Genetic Covariance Estimation of the Genetic Correlation Pairwise Comparison of Relatives Nested Analysis of Variance and Covariance Regression of Family Means

3. Covariance Covariance measures how much 2 variables vary together wt & age, age & grey hair; ADG & NBA If 2 variables vary in opposite direction, Cov can be –ve eg. ADG & FCR Cov of a variable with itself = Variance

4. Covariance & Variance

5. Example

6. Correlation A measure of the strength of a bivariate linear relationship –1 < r < +1

7. Correlation & Regression

8. Properties of covariance "The expected value of the cross product" Cov(a,Y) = 0 Cov(aX,Y) = aCov(X,Y) Cov(X+W,Y) = Cov(X,Y) + Cov(W,Y) Cov(X,X) = Var(x) Cov(a+X,Y) = Cov(a,Y) + Cov(X,Y) = Cov(X,Y) Cov(X,Y) = Cov(Y,X)

9. Correlations between Characters Phenotypic correlations ie height & feet size Environmental correlations Genetic correlations  pleiotropy  gametic phase disequilibrium

10. Genetic covariance G1 G2

11. Estimation of the genetic correlation Three methods Pairwise Comparison of Relatives Nested Analysis of Variance and Covariance Regression of Family Means Extra method not in the book

12. Pairwise Comparison of Relatives Data from pairs of relatives ie mid-parent (x) values for Trait1 and Trait2 And progeny means (y) for Trait1 and Trait2 Four phenotypic Cov. can be computed Cov(x1,y1); Cov(x2,y2) >>> heritabilities T1&T2 Cov(x1,y2); Cov(x2,y1) >>> r g(1,2)

13. Pairwise Comparison of Relatives

14. Genetic correlation

15. Example from my real data

16. Estimate of additive genetic correlation between ADG & FCR

17. Nested Analysis of Var and Cov Nested full-sib and half-sib designs (Ch 18) Provide nested analysis of genetic variance Mean squared deviations of individual traits A parallel analysis > add. genetic covariance Mean cross-products of the deviations of traits 1 and 2 rather than MS

18. Full-sib design T1 T2 1 2 11 12 T1 T2 21

19. Half-sib design T1 T2 1 2 11 12 T1 T2 21

20. Analysis of Variance (half-sib) Factordf SS MS E(MS) SireN-1 SSs/(N-1) Within sireT-N SSs/(T-N) TotalT-1 SSt(T-1)

21. Analysis of Covariance (half-sib) Factordf Sum cross-prod. MCP E(MCP) SireN-1 SCPs/(N-1) Within sireT-N SCPe/(T-N) TotalT-1 SCPt(T-1)

22. ANOVA (half-sib) ADG & FCR Factordf SS MS E(MS) Sire650 42101986477 Within sire1094 23879332182 Total1744 6598132 Factordf SS MS E(MS) Sire650 1160.180 Within sire1094 940.087 Total1744 211 ADG FCR

23. Analysis of Cov(ADG,FCR) (half-sib) FactordfS cross-prod. MCP E(MCP) Sire650-10715.1 -16.48 Within sire1094-6605.7 -6.04 Total1744-17320.8 -9.93

24. Regression of Family means Correlation between family mean phenotypes The Family size , the sampling errors  Family mean phenotype  Family mean genotype value

25. Regression of family means in practice

26. This is how we do it now (REML) correlation between ADG & FCR Anim !P Sire !P Dam !P ADG FCR chapter21.ped !ALPHA data.dat !MAXIT 30 ADG FCR ~ Trait !r Tr.Anim 1 2 1 0 Tr 0 US 1 0.1 1 Tr.Anim 2 Tr 0 US 1 0.1 1 Anim h1 = 0.5998  0.0198 h2 = 0.5109  0.0229 rp = -0.4391  0.0111 rg = -0.4001  0.0302

27. THE END