Genetic Analysis 2 Marleen de Moor KeesJan Kan amp Nick Martin March 7 2012 1 M de Moor Twin Workshop Boulder March 7 2012 M de Moor Twin Workshop Boulder 2 Outline 11001230 ID: 617599
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Slide1
Introduction to MultivariateGenetic Analysis (2)Marleen de Moor, Kees-Jan Kan & Nick Martin
March 7, 2012
1
M. de Moor, Twin Workshop BoulderSlide2
March 7, 2012M. de Moor, Twin Workshop Boulder2Outline11.00-12.30Lecture Bivariate Cholesky
Decomposition
Practical
Bivariate
analysis
of IQ and
attention
problems
12.30-13.30 LUNCH
13.30-15.00
Lecture
Multivariate
Cholesky
Decomposition
PCA versus
Cholesky
Practical Tri- and
Four-variate
analysis
of IQ,
educational
attainment
and
attention
problemsSlide3
March 7, 2012M. de Moor, Twin Workshop Boulder3Outline11.00-12.30Lecture Bivariate Cholesky
Decomposition
Practical
Bivariate
analysis
of IQ and
attention
problems
12.30-13.30 LUNCH
13.30-15.00
Lecture
Multivariate
Cholesky
Decomposition
PCA versus
Cholesky
Practical Tri- and
Four-variate
analysis
of IQ,
educational
attainment
and
attention
problemsSlide4
Bivariate CholeskyMarch 7, 2012M. de Moor, Twin Workshop Boulder4Twin 1Phenotype 1
A
1
A
2
E
1
E
2
a
11
a
21
a
22
e
11
e
21
e
22
1
1
1
Twin 1
Phenotype 2
C
1
C
2
c
11
c
21
c
22
1
1
P1
P2
a1
a2
P1
P2
c
1
c
2
P1
P2
e1
e2Slide5
Adding more phenotypes…Twin 1Phenotype 1A1
A
2
E
1
E
2
a
11
a
21
a
22
e
11
e
21
e
22
1
1
1
Twin 1
Phenotype 2
C
1
C
2
c
11
c
21
c
22
1
1
Twin 1
Phenotype 3
E
3
e
33
e
31
e
32
C
3
c
33
1
A
3
a
33
c
32
a
31
c
31
a
32
1
1
P1
P2
a1
a2
P1
P2
c
1
c
2
P1
P2
e1
e2
a3
c3
e3
P3
P3
P3Slide6
Adding more phenotypes…Twin 1Phenotype 1A1
A
2
E
1
E
2
a
11
a
21
a
22
e
11
e
21
e
22
1
1
1
Twin 1
Phenotype 2
C
1
C
2
c
11
c
21
c
22
1
1
Twin 1
Phenotype 3
E
3
e
33
e
31
e
32
C
3
c
33
1
A
3
a
33
c
32
a
31
c
31
a
32
1
1
Twin 1
Phenotype
4
C
4
A
4
1
c
44
a
44
E
4
e
44
1
e
4
1
e
42
e
43
P1
P2
a1
a2
P1
P2
c
1
c
2
P1
P2
e1
e2
a3
c3
e3
P3
P3
P3
a4
c4
e4
P4
P4
P4Slide7
Trivariate CholeskyTwin 1Phenotype 1A1
A
2
E
1
E
2
a
11
a
21
a
22
e
11
e
21
e
22
1
1
1
1
Twin 1
Phenotype 2
C
1
C
2
c
11
c
21
c
22
1
1
1/0.5
1/0.5
1
1
Twin 1
Phenotype 3
E
3
e
33
e
31
e
32
C
3
c
33
1
A
3
a
33
c
32
a
31
c
31
a
32
1
1
Twin 2
Phenotype 1
A
1
A
2
E
1
E
2
a
11
a
21
a
22
e
11
e
21
e
22
1
1
1
1
Twin 2
Phenotype 2
C
1
C
2
c
11
c
21
c
22
1
1
Twin 2
Phenotype 3
E
3
e
33
e
31
e
32
C
3
c
33
1
A
3
a
33
c
32
a
31
c
31
a
32
1
1
1/0.5
1Slide8
Vars <- c(’varx', ’vary’, ‘varz’)nv <- 3# or
, even more efficiently
: nv <- length
(
Vars
)
…
#
Matrices a, c, and e to store a, c, and e path coefficients
mxMatrix
(
type
=
"Lower"
,
nrow
=
nv
,
ncol
=
nv
,
free
=
TRUE
,
values
=
.6,
name="a" ),
mxMatrix( type
="Lower",
nrow=nv,
ncol=nv,
free=TRUE,
values=.6,
name="c" ),
mxMatrix(
type="Lower",
nrow=nv
, ncol=
nv, free=TRUE
, values=.6
, name="e"
),
OpenMx
What to change in
OpenMx
script?Slide9
Standardized solution – 3 pheno’sMarch 7, 2012M. de Moor, Twin Workshop Boulder9Twin 1Phenotype 1
A
1
A
2
E
1
E
2
a
11
a
22
e
11
e
22
1
1
1
Twin 1
Phenotype 2
C
1
C
2
c
11
c
22
1
1
1/0.5
1/0.5
1
1
Twin 1
Phenotype 3
E
3
e
33
C
3
c
33
1
A
3
a
33
1
1
Twin 2
Phenotype 1
A
1
A
2
E
1
E
2
a
11
a
22
e
11
e
22
1
1
1
1
Twin 2
Phenotype 2
C
1
C
2
c
11
c
22
1
1
Twin 2
Phenotype 3
E
3
e
33
C
3
c
33
1
A
3
a
33
1
1
1/0.5
1Slide10
Genetic correlationsMarch 7, 2012M. de Moor, Twin Workshop Boulder10
corA
<-
mxAlgebra
(name ="
rA
", expression = solve(
sqrt
(I*A))%*%A%*%solve(
sqrt
(I*A)))
OpenMx
2x2
3x3Slide11
The order of variablesOrder of variables does not matter for the solution!Fit is identical, just different parameterizationStandardized solutions are identical in terms of fit and parameter estimates!But
interpretation of A/C/E variance
components is different!Where A2 refers
to
those
genetic
factors
that
are
not
shared
with
phenotype
1
Sometimes
there
is
natural
ordering:Temporal ordering (IQ at 2 time points
)Neuroticism and MDD symptomsMarch 7, 2012
M. de Moor, Twin Workshop Boulder
11Slide12
Cholesky decomposition is not a model…No constraints on covariance matricesJust reparameterization……But very useful to explore the data!Observed statistics = Number
of parameters
March 7, 2012M. de Moor, Twin Workshop Boulder
12Slide13
Cholesky decomposition is not a model…Bivariate constrained saturated model:2 variances, 1 within-twin covariance MZ=DZ2 within-trait cross-twin covariances MZ1 cross-trait
cross-twin covariance MZ
2 within-trait cross-twin covariances
DZ
1
cross-trait
cross-twin
covariance
DZ
Bivariate
Cholesky
decomposition
a11, a21, a22
c11, c21, c22
e11, e21, e22
March 7, 2012
M. de Moor, Twin Workshop Boulder
13
9
observed
statistics
9 parametersSlide14
Comparison with other modelsMarch 7, 2012M. de Moor, Twin Workshop Boulder14Cholesky
decomposition models
Principal
component
analysis
Sanja,
now
Confirmatory
factor models
Dorret, Sanja, Michel,
this
morning
Genetic
factor models
Hermine,
after
coffee
breakSlide15
Further readingThree classic papers:Martin NG, Eaves LJ: The genetical analysis of covariance structure. Heredity 38:79-95, 1977Carey, G. Inference About Genetic Correlations, BG, 1988Loehlin, J. The Cholesky Approach: A Cautionary Note, BG, 1996Carey, G. Cholesky Problems, BG, 2005SEE ALSO:http://genepi.qimr.edu.au/staff/classicpapers/March 7, 2012
M. de Moor, Twin Workshop Boulder
15Slide16
March 7, 2012M. de Moor, Twin Workshop Boulder16Outline11.00-12.30Lecture Bivariate Cholesky
Decomposition
Practical
Bivariate
analysis
of IQ and
attention
problems
12.30-13.30 LUNCH
13.30-15.00
Lecture
Multivariate
Cholesky
Decomposition
PCA versus
Cholesky
Practical Tri- and
Four-variate
analysis
of IQ,
educational
attainment and attention problemsSlide17
PracticalTrivariate ACE Cholesky model126 MZ and 126 DZ twin pairs from Netherlands Twin RegisterAge 12Educational achievement (EA)FSIQAttention Problems (AP) [mother-report]
March 7, 2012
M. de Moor, Twin Workshop Boulder
17Slide18
PracticalScript CholeskyTrivariate.RDataset Cholesky.datMarch 7, 2012M. de Moor, Twin Workshop Boulder18Slide19
ExerciseAdd Educational Achievement as the first of the 3 variablesRun the saturated model, ACE model and AE modelQuestion: Can we drop C?March 7, 2012M. de Moor, Twin Workshop Boulder
19
-2LL
df
chi2
∆
df
P-value
ACE
model
-
-
-
AE modelSlide20
ExerciseRun 4 submodelsSubmodel 1: drop rg between EA and APSubmodel 2: drop rg between FSIQ and APSubmodel 3: drop re between EA and APSubmodel 4: drop re between FSIQ and APCompare fit of each submodel with full AE model
March 7, 2012
M. de Moor, Twin Workshop Boulder
20Slide21
ExerciseQuestions: Can we drop rg between EA and AP?Can we drop rg between FSIQ and AP?Can we drop re between EA and AP?Can we drop re between FSIQ and AP?March 7, 2012
M. de Moor, Twin Workshop Boulder
21
-2LL
df
chi2
∆
df
P-value
AE
model
-
-
-
No a31
No a32
No e31
No e32Slide22
March 7, 2012M. de Moor, Twin Workshop Boulder22Slide23
Extra exerciseReplace FSIQ by VIQ and PIQ, and run a fourvariate Cholesky model.Questions:Is AP differentially related to VIQ and PIQ, phenotypically and genotypically?March 7, 2012
M. de Moor, Twin Workshop Boulder
23