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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? - PowerPoint PPT Presentation
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Twin Analysis 104October 2010
• 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?
Bivariate Questions I
Two Traits
• 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
Bivariate Questions II
individual twin
within between
trait
withinwithin-twin within-trait co(variance)
(cross-twin within-trait) covariance
between
(within-twin cross-trait) covariance
cross-twin cross-trait covariance
Bivariate Twin Data
Bivariate Twin Covariance Matrix
X1 Y1 X2 Y2
X1 VX1 CX1Y1 CX1X2 CX1Y2
Y1 CY1X1 VY1 CY1X2 CY1Y2
X2 CX2X1 CX2Y1 VX2 CX2Y2
Y2 CY2X1 CY2Y1 CY2X2 VY2
Genetic Correlation
Alternative Representations
Cholesky Decomposition
Bivariate AE Model
X1 Y1 X2 Y2
X1
Y1
X2
Y2
a112
+e112
a222+a21
2+e22
2+e212
a21*a11+e21*e11
a222+a21
2
a112
a21*a11
MZ Twin Covariance Matrix
X1 Y1 X2 Y2
X1
Y1
X2
Y2
a112
+e112
a222+a21
2+e22
2+e212
a21*a11+e21*e11
.5a222+
.5a212
.5a112
.5a21*a11
DZ Twin Covariance Matrix
Within-Twin Covariances [Mx]
Within-Twin Covariances
Cross-Twin 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
Cross-Trait Covariances
• Dataset: MCV-CVT Study
• 1983-1993
• BMI, skinfolds (bic,tri,calf,sil,ssc)
• Longitudinal: 11 years
• N MZF: 107, DZF: 60
Practical Example I
Bivariate
• Saturated Model
• equality of means/variances
• Genetic Models (ACE)
• bivariate -> Cholesky Decomposition
> 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]
Multivariate
• Saturated Model
• equality of means/variances
• Genetic Models (ACE)
• multivariate -> Cholesky Decomposition
• Independent Pathway
• Common Pathway
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
Cholesky
TextText
F1 F2 F3 F4
P1 f11
P2 f21
P3 f31
P4 f41
F1 F2 F3 F4
P1 f11 0P2 f21 f22
P3 f31 f32
P4 f41 f42
F1 F2 F3 F4
P1 f11 0 0P2 f21 f22 0P3 f31 f32 f33
P4 f41 f42 f43
F1 F2 F3 F4
P1 f11 0 0
0P2 f21 f22 0
0P3 f31 f32 f33
0P4 f41 f42 f43
f44
Phenotypic
Cholesky Decomposition
F1 F2 F3 F4
P1 f11 0 0
0P2 f21 f22 0
0P3 f31 f32 f33
0P4 f41 f42 f43
f44
F1 F2 F3 F4
P1 f11 f21 f31 f41
P2 0 f22 f32
f42
P3 0 0 f33
f43
P4 0 0 0
f44
%*%
F %*% t(F)
Estimate covariance matrix, fully saturated
Genetic
& Environmental
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]
• IP
Common Factor
F1 P1 f11
P2 f21
P3 f31
P4 f41
F1 F2 F3 F4
P1 f11 f21 f31 f41
%*%
F %*% t(F)
Residuals
E1 E2 E3 E4
P1 f11 0 0
0P2 0 f22 0 0P3 0 0 f33
0P4 0 0 0
f44
%*%
E %*% t(E)
E1 E2 E3 E4
P1 f11 0 0
0P2 0 f22 0 0P3 0 0 f33
0P4 0 0 0
f44
F %*% t(F) + E %*% t(E)
with Twin Data
twin 1
GeneticCommon Factor
Shared environmentalCommon Factor
Unique environmental Common Factor
Genetic Residuals& C & E Residuals
A1 P1 a11
P2 a21
P3 a31
P4 a41
P1 P2 P3 P4
A1 a11 a21 a31 a41
%*%
ac %*% t(ac)
A1 A2 A3 A4
P1 a11 0 0
0P2 0 a22 0 0P3 0 0 a33
0P4 0 0 0
a44
%*%
as %*% t(as)
P1 P2 P3 P4
A1 a11 0 0
0A2 0 a22 0
0A3 0 0 a33
0A4 0 0 0
a44
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]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]
Independent Pathway
• Biometric model
• Different covariance structure for A, C and E
IP Model
Variance Componen
ta2 c2 e2
Common Factors
acnv x 1
ccnv x 1
ecnv x 1
Residual Factors
asnv x nv
csnv x nv
esnv x nv
• CP
Latent Phenotype
F1 P1 f11
P2 f21
P3 f31
P4 f41
F1 F2 F3 F4
P1 f11 f21 f31 f41
%*%
F %*% t(F)
ACE Model
F1 P1 f11
P2 f21
P3 f31
P4 f41
F1 F2 F3 F4
P1 f11 f21 f31 f41
%*% al11 %*% t(al11) %*%
fl %*% al %*% t(al) %*% t(fl)
fl %x% al %*% t(al)
Same Residuals
model:ACE, matrix:al [,1][1,] [NA]model:ACE, matrix:cl [,1][1,] [NA]model:ACE, matrix:el [,1][1,] [NA]model:ACE, matrix:fl [,1] [1,] [fl11][2,] [fl21][3,] [fl31][4,] [fl41][5,] [fl51]
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]
Common Pathway
• Psychometric model
• Same covariance structure for A, C and E
CP Model
Variance Compon
enta2 c2 e2
Common
Factors
al1 x 1
cl1 x 1
el1 x 1
flnv x 1
Residual Factors
asnv x nv
csnv x nv
esnv x nv
