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# Bivariate

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### Bivariate

1. 1. Bivariate analysisMx class 2004
2. 2. Univariate ACE model 1 or .5 1 1 1 1 1 1 1 A C E E C A a c e e c a T1 T2
3. 3. Expected Covariance Matrices a2+c2+e2 a2+c2 E MZ = 2x2 a2+c2 a2+c2+e2 a2+c2+e2 .5a2+c2 E DZ = 2x2 .5a2+c2 a2+c2+e2
4. 4. Bivariate Questions In Univariate Analysis: What are the contributions of additive genetic, dominance/shared environmental and unique environmental factors to the variance?n Bivariate Analysis: What are the contributions of genetic and environmental factors to the covariance between two traits?
5. 5. Two Traits AX AC AY X Y EX EC EY
6. 6. Bivariate Questions IIn 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?n With twin data on multiple traits it is possible to partition the covariation into its genetic and environmental componentsn Goal: to understand what factors make sets of variables correlate or co-vary
7. 7. Bivariate Twin Data individual twin within betweentrait within variance twin covariance between trait covariance cross-trait twin covariance
8. 8. Bivariate Twin Covariance Matrix X1 Y1 X2 Y2X1 VX1 CX1Y1 CX1X2 CX1Y2Y1 CY1X1 VY1 CY1X2 CY1Y2X2 CX2X1 CX2Y1 VX2 CX2Y2Y2 CY2X1 CY2Y1 CY2X2 VY2
9. 9. Genetic Correlation 1 or .5 1 or .5 rg rg 1 1 1 1 AX AY AX AY ax ay ax ay X1 Y1 X2 Y2
10. 10. Alternative Representations 1 rg AC1 1 1 1 1 1 AX AY ASX A SY A1 A2 ac ac ax ay asx asy a11 a21 a22 X1 Y1 X1 Y1 X1 Y1
11. 11. Cholesky Decomposition 1 or .5 1 or .5 1 1 1 1 A1 A2 A1 A2 a11 a21 a22 a11 a21 a22 X1 Y1 X2 Y2
12. 12. More Variables 1 1 1 1 1 A1 A2 A3 A4 A5 a11 a21 aa22 a32 a4233 a43 31 a a44 X1 X2 X3 X4 X5
13. 13. Bivariate AE Model 1 or .5 1 or .5 1 1 1 1 A1 A2 A1 A2 a11 a21 a22 a11 a21 a22 X1 Y1 X2 Y2 e11 e21 e22 e11 e21 e22 E1 E2 E1 E2 1 1 1 1
14. 14. MZ Twin Covariance Matrix X1 Y1 X2 Y2X1 a112+e112 a112Y1 a21*a11+ a222+a212+ a21*a11 a222+a212 e21*e11 e222+e212X2Y2
15. 15. DZ Twin Covariance Matrix X1 Y1 X2 Y2X1 a112+e112 .5a112Y1 a21*a11+ a222+a212+ .5a21*a11 .5a222+ e21*e11 e222+e212 .5a212X2Y2
16. 16. Within-Twin Covariances [Mx] 1 1 A1 A2 X Lower 2 2 A1 A2 a11 a21 a22 X1 a11 0 X1 Y1 Y1 a21 a22A=X*X a11 0 a11 a21 a112 a11*a21EA= * = a21 a22 0 a22 a21*a11 a222+a212
17. 17. Within-Twin Covariances a112 a11*a21 EA= a21*a11 a222+a212 e112 e11*e21 EE= e21*e11 e222+e212 a112+ e112 a11*a21 + e11*e21EP= EA+EE = a21*a11 + e21*e11 a222+a212 +e222+e212
18. 18. Cross-Twin Covariances a112 a11*a21MZ EA= a21*a11 a222+a212 .5a112 .5a11*a21DZ .5@EA= .5a21*a11 .5a222+.5a212
19. 19. Cross-Trait Covariancesn Within-twin cross-trait covariances imply common etiological influencesn Cross-twin cross-trait covariances imply familial common etiological influencesn MZ/DZ ratio of cross-twin cross-trait covariances reflects whether common etiological influences are genetic or environmental
20. 20. Univariate Expected Covariances a2+c2+e2 a2+c2 E MZ = 2x2 a2+c2 a2+c2+e2 a2+c2+e2 .5a2+c2 E DZ = 2x2 .5a2+c2 a2+c2+e2
21. 21. Univariate Expected Covariances II EA+EC+EC EA+EC E MZ = 2x2 EA+EC EA+EC+EC EA+EC+EC .5@EA+EC E DZ = 2x2 .5@EA+EC EA+EC+EC
22. 22. Bivariate Expected Covariances EA+EC+EC EA+EC E MZ = 4x4 EA+EC EA+EC+EC EA+EC+EC .5@EA+EC E DZ = 4x4 .5@EA+EC EA+EC+EC
23. 23. Practical Example In Dataset: MCV-CVT Studyn 1983-1993n BMI, skinfolds (bic,tri,calf,sil,ssc)n Longitudinal: 11 yearsn N MZFY: 107, DZF: 60
24. 24. Practical Example IIn Dataset: NL MRI Studyn 1990’sn Working Memory, Gray & White Mattern N MZFY: 68, DZF: 21