Transcript bivarate2

Bivariate Model: traits A and B measured in twins
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Questions
• Do the genes that influence trait B also influence trait A?
• Are there genes that are unique to trait A?
• Is the phenotypic correlation caused by genetic correlation?
• What is the genetic correlation? (ie the genetic covariance
/genetic SD(A)*genetic SD(B)
• The same questions apply to environmental (shared and uqniue)
influences.
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What is the variance for traits A and B?
What is the covariance between A and B?
What are the MZ and DZ twin covariances for A and B?
What are the twin cross-covariances (A in twin 1 with B in twin 2)?
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What is the variance for traits A and B?
A: h2a + h2ca + e2a and B : h2cb+ e2b
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What is the covariance for MZ and DZ twins for traits A and B?
MZ Covar(A) : h2ca + h2a and MZ Covar(B) : h2cb
DZ Covar(A) : 0.5 h2ca + 0.5 h2a and DZ Covar(B) : 0.5 h2cb
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What is the covariance between traits A and B?
Covar(A,B) : hca * hcb
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What is the cross-covariance for MZ and DZ twins (A1 with B2)?
MZ Covar(A1, B2) : hca * hcb
DZ Covar(A1, B2) : 0.5(hca * hcb)
Kendler & Eaves (1986; see also Eaves 1982)
describe 3 models for how genes and environment
may jointly influence a phenotype:
-Contribution of genes and environment is additive
-Genes and environments are
correlated: genes
:
alter exposure to the relevant environmental factors
-Genes and environment interact: Genes control
sensitivity to the environment; or: the environment
controls gene expression
Eaves LJ (1984) Genetic Epidemiology , 215-228
Kendler KS, Eaves LJ (1986) American J Psychiatry, 279-289
-Contribution of genes and environment is additive
:
P=G+E
(P = hG + eE)
Var (P) = Var G + Var (E) (Var (P) = h2 + e2)
Genes and environments are correlated:
P=G+E
Var (P) = Var (G) + Var (E) + 2 Cov (G,E)
Genotype-environment interaction I:
Influences of genotype and environment on a trait can be
estimated conditional on environmental exposure
(e.g. mariatal status; religious upbringing, SES).
*No GxE interaction: influence of genes and environment
is the same for subjects with different degrees of exposure.
*GxE interaction: genetic effects are modified by exposure:
heritabilities differ between exposure-positive and
exposure-negative groups.
Genotype-environment interaction II:
Evidence for GxE interaction based on
differences in heritabilities does not tell us if
the same genes are expressed in different
groups.
To address this issue, data from twins discordant
for environmental exposure are required, or
longitudinal data from twins measured under
different environmental conditions.
Genotype-environment interaction III:
Longitudinal data available: GxE interaction can
then be detected from the genetic correlation
between traits. If the genetic correlation is high,
then trait values in the two environments are
determined by the same genes. If the genetic
correlation is low, then the trait is influenced by
different genes in different environments.
Falconer DS (1989) Introduction to Quantitative Genetics (3rd
ed.), Longman, London.
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Suppose A and B are the same trait measured on the same subjects
under different (experimental) conditions; e.g. depression assessed
before and after Ss participated in an exercise program.
How many twins would we need to measure to test if the same
genes are expressed before and after exercise?
Mx exercise 1
• Simulate bivariate data in twins
• Fit the correct model to the simulated data
• Fit the wrong model to the data:
– no genetic covariance (different genes)
– genetic correlation = 1 (same genes)
Mx exercise 2
• Analyze bivariate data in twins: brain size
(MRI) and IQ
• Fit the model to raw data
– for IQ 233 pairs
– for MRI 111 pairs
– in addition to data from twins there are also data
from their siblings
Mx exercise 1: F:\dorret\2002
• Simulate bivariate data : bivarSimulajob
• Fit the correct model: bivarTruejob
• Fit the wrong model to the data:
– no genetic covariance: change bivarTruejob
– genetic correlation = 1: bivarConstrainjob
– genetic correlation = 0: change bivarConstrainjob
• NB no data file because data are simulated in the first job
Mx exercise 1: F:\dorret\2002
1) bivarTruejob: chi2 = 0 (df=11)
2) no genetic covariance: chi2 = 3.17 (df=12)
3) genetic correlation = 1: chi2 = 2.97 (df=15),
power is 41%
4) genetic correlation = 0: chi2 = 3.17 (df=15),
power is 42%
ANALYSES 2 AND 4 GIVE THE SAME
ANSWER WHY ARE THE DF DIFFERENT?
MRI-IQ dataset
Phenotypic correlations
Working memory – gray matter volume
0.27
Working memory – white matter volume
0.28
Posthuma et al. Nature Neuroscience, Feb 2002
MRI-IQ dataset
Twin and sibling correlations (npairs)
WM
BBGM
BBWM
MZ
0.72 (102) 0.86 (54)
0.89 (54)
DZ
0.27 (131) 0.45 (57)
0.34 (57)
Twin1-sib
0.13 (129) 0.53 (30)
0.58 (30)
Twin2-sib
0.39 (129) 0.43 (31)
0.58 (31)
MRI-IQ dataset
WM = working memory
BBGM = gray matter volume of the cerebrum
BBWM = white matter volume of the cerebrum
(Variables have already been corrected for the
effects of age and sex)
Mx job: MRI_IQx.mx (f:\dorrret\2002)
Data file: mri3.dat (f:\dorrret\2002)
IQ-MRI
• What is the genetic correlation between brain
size (MRI) and IQ?
• What is the correlation of common E factors?
• What is the correlation of unique E factors?
• What are h2 c2 and e2 ?
• What determines the phenotypic correlation?
IQ-MRI
• genetic correlation: 0.29
• correlation of common E factors: .99
• correlation of unique E factors: .05
MRI
•
•
•
•
IQ
h2 =
.65
.87
c2 =
.00
.00
e2 =
.35
.13
What determines the phenotypic correlation?
Other bivariate models: Causal Model
Causality: for example the association of
regular physical exercise and depression
Regular exercise
psychological well-being
Regular exercise
psychological well-being
Regular exercise
psychological well-being
Regular exercise
psychological well-being
Environment
Heredity
Genetic & environmental correlation
Rc
Ere
Cre
ere
Rg
Are
cre are
Regular Exercise
Epw
Cpw
Apw
epw cpw apw
Psychological Well-being
Estimate Rc, Rg and path coefficients
from MZ and DZ (cross) correlations
Cross-sectional approach to
causality in twins
If A  B: r(A1, B2) =
(ha rg ha ha )* i:
i.e. the cross-correlation in twins
Is a function of the heritability of A
If BA then …
Twin 1
Twin 2
Cross-sectional modeling of causality in twins
Twin 1
Twin 2
A1
A2
B1
B2
A1
A2
B1
B2
1
ha 2
i
i ha 2
1
i ha 2
i
1
hb2+i2ha2
A1
A2
B1
B2
1
.5ha2
i
.5i ha2
1
.5i ha2
1
i
.5hb2+.5i2ha2 1
A1
A2
B1
B2
1
ha2+j2hb2 1
j
j h b2
j h b2
j
1
hb 2
A1
A2
B1
B2
1
.5ha2+.5j2hb2 1
j
.5j hb2
.5j hb2
j
1
.5hb2
MZ
1
1
1
DZ
MZ
DZ
Bivariate models and direction of causality.
The casual model is nested under the general bivariate
model; however, the issue of measurement error is a
very critial one in establishing the direction of causality
(or reciprocal interaction).