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Twin Analysis 104

Twin Analysis 104. October 2010. Bivariate Questions I. Univariate Analysis: What are the contributions of additive genetic, dominance/shared environmental and unique environmental factors to the variance?

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Twin Analysis 104

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  1. Twin Analysis 104 • October 2010

  2. Bivariate Questions I • Univariate Analysis: What are the contributions of additive genetic, dominance/shared environmental and unique environmental factors to the variance? • Bivariate Analysis: What are the contributions of genetic and environmental factors to the covariance between two traits?

  3. Two Traits

  4. Bivariate Questions II • Two or more traits can be correlated because they share common genes or common environmental influences • e.g. Are the same genetic/environmental factors influencing the traits? • With twin data on multiple traits it is possible to partition the covariation into its genetic and environmental components • Goal: to understand what factors make sets of variables correlate or co-vary

  5. Bivariate Twin Data

  6. Bivariate Twin Covariance Matrix

  7. Genetic Correlation

  8. Alternative Representations

  9. Cholesky Decomposition

  10. Bivariate AE Model

  11. MZ Twin Covariance Matrix a112+e112 a112 a21*a11+ e21*e11 a222+a212+ e222+e212 a21*a11 a222+a212

  12. DZ Twin Covariance Matrix a112+e112 .5a112 a21*a11+ e21*e11 a222+a212+ e222+e212 .5a21*a11 .5a222+ .5a212

  13. Within-Twin Covariances [Mx]

  14. Within-Twin Covariances

  15. Cross-Twin Covariances

  16. Cross-Trait Covariances • Within-twin cross-trait covariances imply common etiological influences • Cross-twin cross-trait covariances imply familial common etiological influences • MZ/DZ ratio of cross-twin cross-trait covariances reflects whether common etiological influences are genetic or environmental

  17. Practical Example I • Dataset: MCV-CVT Study • 1983-1993 • BMI, skinfolds (bic,tri,calf,sil,ssc) • Longitudinal: 11 years • N MZF: 107, DZF: 60

  18. Bivariate • Saturated Model • equality of means/variances • Genetic Models (ACE) • bivariate -> Cholesky Decomposition

  19. > parameterSpecifications(bivACEFit)model:ACE, matrix:a [,1] [,2] [1,] [a11] 0 [2,] [a21] [a22]model:ACE, matrix:c [,1] [,2] [1,] [c11] 0 [2,] [c21] [c22]model:ACE, matrix:e [,1] [,2] [1,] [e11] 0 [2,] [e21] [e22]model:ACE, matrix:Mean [,1] [,2][1,] [NA] [NA]

  20. Multivariate • Saturated Model • equality of means/variances • Genetic Models (ACE) • multivariate -> Cholesky Decomposition • Independent Pathway • Common Pathway

  21. Scientific Questions • Are these measures influenced by the same genes (single common factor)? • Is there more than one factor (overall fat- fat distribution)? • What is the structure of C and E? • Contribution of A, C, E factors to covariance between traits

  22. Cholesky F1 F2 F3 F4 P1 f11 P2 f21 P3 f31 P4 f41 Text Text

  23. F1 F2 F3 F4 P1 f11 0 P2 f21 f22 P3 f31 f32 P4 f41 f42

  24. F1 F2 F3 F4 P1 f11 0 0 P2 f21 f22 0 P3 f31 f32 f33 P4 f41 f42 f43

  25. F1 F2 F3 F4 P1 f11 0 0 0 P2 f21 f22 00 P3 f31 f32 f33 0 P4 f41 f42 f43 f44

  26. Phenotypic

  27. Cholesky Decomposition Estimate covariance matrix, fully saturated F1 F2 F3 F4 P1 f11 0 0 0 P2 f21 f22 00 P3 f31 f32 f33 0 P4 f41 f42 f43 f44 F1 F2 F3 F4 P1 f11 f21 f31 f41 P2 0 f22 f32 f42 P3 00 f33 f43 P4 0 00 f44 %*% F %*% t(F)

  28. Genetic

  29. & Environmental

  30. model:ACE, matrix:a [,1] [,2] [,3] [,4] [,5] [1,] [a11] 0 0 0 0 [2,] [a21] [a22] 0 0 0 [3,] [a31] [a32] [a33] 0 0 [4,] [a41] [a42] [a43] [a44] 0 [5,] [a51] [a52] [a53] [a54] [a55]model:ACE, matrix:c [,1] [,2] [,3] [,4] [,5] [1,] [c11] 0 0 0 0 [2,] [c21] [c22] 0 0 0 [3,] [c31] [c32] [c33] 0 0 [4,] [c41] [c42] [c43] [c44] 0 [5,] [c51] [c52] [c53] [c54] [c55]model:ACE, matrix:e [,1] [,2] [,3] [,4] [,5] [1,] [e11] 0 0 0 0 [2,] [e21] [e22] 0 0 0 [3,] [e31] [e32] [e33] 0 0 [4,] [e41] [e42] [e43] [e44] 0 [5,] [e51] [e52] [e53] [e54] [e55]model:ACE, matrix:Mean [,1] [,2] [,3] [,4] [,5][1,] [NA] [NA] [NA] [NA] [NA]

  31. IP

  32. Common Factor

  33. F1 P1 f11 P2 f21 P3 f31 P4 f41 F1 F2 F3 F4 P1 f11 f21 f31 f41 %*% F %*% t(F)

  34. Residuals

  35. E1 E2 E3 E4 P1 f11 0 0 0 P2 0 f22 00 P3 0 0 f33 0 P4 0 0 0 f44 E1 E2 E3 E4 P1 f11 0 0 0 P2 0 f22 00 P3 0 0 f33 0 P4 0 0 0 f44 %*% E %*% t(E)

  36. F %*% t(F) + E %*% t(E)

  37. with Twin Data

  38. twin 1

  39. GeneticCommon Factor

  40. Shared environmentalCommon Factor

  41. Unique environmental Common Factor

  42. Genetic Residuals& C & E Residuals

  43. A1 P1 a11 P2 a21 P3 a31 P4 a41 P1 P2 P3 P4 A1 a11 a21 a31 a41 %*% ac %*% t(ac)

  44. A1 A2 A3 A4 P1 a11 0 0 0 P2 0 a22 00 P3 0 0 a33 0 P4 0 0 0 a44 P1 P2 P3 P4 A1 a11 0 0 0 A2 0 a22 00 A3 0 0 a33 0 A4 0 0 0 a44 %*% as %*% t(as)

  45. model:ACE, matrix:as [,1] [,2] [,3] [,4] [,5] [1,] [as11] 0 0 0 0 [2,] 0 [as21] 0 0 0 [3,] 0 0 [as31] 0 0 [4,] 0 0 0 [as41] 0 [5,] 0 0 0 0 [as51]model:ACE, matrix:cs [,1] [,2] [,3] [,4] [,5] [1,] [cs11] 0 0 0 0 [2,] 0 [cs21] 0 0 0 [3,] 0 0 [cs31] 0 0 [4,] 0 0 0 [cs41] 0 [5,] 0 0 0 0 [cs51]model:ACE, matrix:es [,1] [,2] [,3] [,4] [,5] [1,] [es11] 0 0 0 0 [2,] 0 [es21] 0 0 0 [3,] 0 0 [es31] 0 0 [4,] 0 0 0 [es41] 0 [5,] 0 0 0 0 [es51]model:ACE, matrix:M [,1] [,2] [,3] [,4] [,5][1,] [NA] [NA] [NA] [NA] [NA] model:ACE, matrix:ac [,1] [1,] [ac11][2,] [ac21][3,] [ac31][4,] [ac41][5,] [ac51]model:ACE, matrix:cc [,1] [1,] [cc11][2,] [cc21][3,] [cc31][4,] [cc41][5,] [cc51]model:ACE, matrix:ec [,1] [1,] [ec11][2,] [ec21][3,] [ec31][4,] [ec41][5,] [ec51]

  46. Independent Pathway • Biometric model • Different covariance structure for A, C and E

  47. IP Model

  48. CP

  49. Latent Phenotype

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