1Bivariate analysisHGEN619 class 2007Univariate ACE modelT2AEA Ca ac eT11 111Ee1Cc11 or .512Expected Covariance Matricesa2+c2+e2 .5a2+c2.5a2+c2 a2+c2+e2E DZ =a2+c2+e2 a2+c2a2+c2 a2+c2+e2E MZ = 2 x 22 x 2Bivariate Questions I Univariate Analysis: What are the contributionsof additive genetic, dominance/sharedenvironmental and unique environmental factorsto the variance? Bivariate Analysis: What are the contributions ofgenetic and environmental factors to thecovariance between two traits?3Two TraitsEYEXX YAXAYACECBivariate Questions II Two or more traits can be correlated becausethey share common genes or commonenvironmental influences e.g. Are the same genetic/environmental factorsinfluencing the traits? With twin data on multiple traits it is possible topartition the covariation into its genetic andenvironmental components Goal: to understand what factors make sets ofvariables correlate or co-vary4Bivariate Twin Databetweenwithintraitbetweenwithinindividual twin(within-twin within-trait co)variance(cross-twin within-trait) covariance(cross-twin within-trait) covariancecross-twin cross-traitcovarianceBivariate Twin Covariance MatrixY2X2Y1X1Y2X2Y1X1VX1 CX1X2CX2X1 VX2VY1 CY1Y2CY2Y1 VY2CX1Y1 CX2Y2CY1X1 CY2X2 CX1Y2CX2Y1 CY1X2CY2X15Genetic CorrelationY2AYAXAYaxayayX11 11AXax11 or .5 1 or .5X2Y1rgrgAlternative RepresentationsAXAYaxayX11 1Y1rgASXASYasxasyX11 1Y1AC1acacA1A2a11a22X11 1Y1a216Cholesky DecompositionA1A2a11a22X11 1Y1a21A1A2a11a22X21 1Y2a211 or .5 1 or .5More VariablesA1A2a11a22X11 1X2a21A3a331X3A4a441X4a32a43a31a42A51X57Bivariate AE ModelA1A2a11a22X11 1Y1a21A1A2a11a22X21 1Y2a211 or .5 1 or .5E1E2e11e221 1e21E1E2e11e221 1e21MZ Twin Covariance MatrixY2X2Y1X1Y2X2Y1X1a112+e112a222+a212+e222+e212a21*a11+e21*e11a222+a212a112a21*a118DZ Twin Covariance MatrixY2X2Y1X1Y2X2Y1X1a112+e112a222+a212+e222+e212a21*a11+e21*e11.5a222+.5a212.5a112.5a21*a11Within-Twin Covariances [Mx]A1A2a11a22X11 1Y1a21a11a220a21A1A2X1Y1X Lower 2 2a112a222+a212a21*a11a11*a21=A=X*X'a11a220a21a11a220a21*EA=9Within-Twin Covariancesa112a222+a212a21*a11a11*a21EA=e112e222+e212e21*e11e11*e21EE=a112+ e112a222+a212 +e222+e212a11*a21 + e11*e21EP= EA+EE =a21*a11 + e21*e11Cross-Twin Covariancesa112a222+a212a21*a11a11*a21EA=MZ.5a112.5a222+.5a212.5a21*a11.5a11*a21.5@EA=DZ10Cross-Trait Covariances Within-twin cross-trait covariances implycommon etiological influences Cross-twin cross-trait covariances implyfamilial common etiological influences MZ/DZ ratio of cross-twin cross-traitcovariances reflects whether commonetiological influences are genetic orenvironmentalUnivariate Expected Covariancesa2+c2+e2 .5a2+c2.5a2+c2 a2+c2+e2E DZ =a2+c2+e2 a2+c2a2+c2 a2+c2+e2E MZ = 2 x 22 x 211Univariate Expected Covariances IIE DZ =EA+E C+EE .5@EA+EC.5@EA+EC EA+EC+EEEA+E C+EE EA+ECEA+E C EA+EC+EEE MZ = 2 x 22 x 2Bivariate Expected CovariancesE DZ =EA+E C+EE .5@EA+EC.5@EA+EC EA+EC+EEEA+E C+EE EA+ECEA+E C EA+EC+EEE MZ = 4 x 44 x 412Practical Example I Dataset: MCV-CVT Study 1983-1993 BMI, skinfolds (bic,tri,calf,sil,ssc) Longitudinal: 11 years N MZF: 107, DZF: 60Practical Example II Dataset: NL MRI Study 1990’s Working Memory, Gray & White Matter N MZFY: 68, DZF: 2113! Bivariate ACE model! NL mri data I #NGroups 4 #define nvar 2 ! N dependent variables per twin G1: Model Parameters Calculation Begin matrices; X Lower nvar nvar Free ! additive genetic path coefficient Y Lower nvar nvar Free ! common environmental path coefficient Z Lower nvar nvar Free ! unique environmental path coefficient H Full 1 1 ! G Full 1 nvar Free ! means End matrices; Matrix H .5 Start .5 X 1 1 1 Y 1 1 1 Z 1 1 1 Start .7 X 1 2 2 Y 1 2 2 Z 1 2 2 Matrix G 6 7 Begin algebra; A= X*X'; ! additive genetic variance C= Y*Y'; ! common environmental variance E= Z*Z'; ! unique environmental variance V= A+C+E; ! total variance S= A%V | C%V | E%V ; ! standardized variance components End algebra; Labels Row V WM BBGM Labels Column V A1 A2 C1 C2 E1 E2 Endnlmribiv.mx! Bivariate ACE model! NL mri data II G2: MZ twins Data NInputvars=8 ! N inputvars per family Missing=-2.0000 ! missing values ='-2.0000' Rectangular File=mri.rec Labels fam zyg mem1 gm1 wm1 mem2 . . Select if zyg =1 ; Select gm1 wm1 gm2 wm2 ; Begin Matrices = Group 1; Means G| G; ! model for means, assuming mean t1=t2 Covariances ! model for MZ variance/covariances A+C+E | A+C _ A+C | A+C+E ; Options RSiduals End G3: DZ twins Data NInputvars=8 Missing=-2.0000 Rectangular File=mri.rec Labels fam zyg mem1 gm1 wm1 mem2 . . Select if zyg =2 ; Select gm1 wm1 gm2 wm2 ; Begin Matrices = Group 1; Means G| G; ! model for means, assuming mean t1=t2 Covariances ! model for DZ variance/covariances A+C+E | H@A+C _ H@A+C | A+C+E ; Options RSiduals Endnlmribiv.mx14! Bivariate ACE model! NL mri data III G4: summary of relevant statistics Calculation Begin Matrices = Group 1 Begin Algebra ; R= \stnd(A)| \stnd(C)| \stnd(E); ! calculates rg|rc|re End Algebra ; Interval @95 S 1 1 1 S 1 1 3 S 1 1 5 ! CI's on A,C,E for first phenotype Interval @95 S 1 2 2 S 1 2 4 S 1 2 6 ! CI's on A,C,E for second phenotype Interval @95 R 4 2 1 R 4 2 3 R 4 2 5 ! CI's on rg, rc, re
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