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Gene-Environment
Interaction & Correlation
Danielle Dick & Tim York
VCU workshop, Fall 2010
Gene-Environment Interaction
• Genetic control of sensitivity to the
environment
• Environmental control of gene expression
• Bottom line: nature of genetic effects
differs among environments
Standard Univariate Model
1.0 (MZ) / .5 (DZ)
1.0
1.0
1.0
A1
a
C1
c
P1
1.0
1.0
E1
e
P=A+ C + E
Var(P) = a2+c2+e2
1.0
1.0
A2
a
C2
c
E2
e
P2
Contributions of Genetic, Shared Environment,
Genotype x Environment Interaction Effects
to Twin/Sib Resemblance
Shared
Environment
Additive
Genetic Effects
Genotype x Shared
Environment
Interaction
MZ Pairs
1
1
1x1=1
DZ Pairs/Full Sibs
1
½
1x½=½
Contributions of Genetic, Shared Environment,
Genotype x Shared Environment Interaction
Effects to Twin/Sib Resemblance
Shared
Environment
Additive
Genetic Effects
Genotype x Shared
Environment
Interaction
MZ Pairs
1
1
1x1=1
DZ Pairs/Full Sibs
1
½
1x½=½
In other words—if gene-(shared) environment interaction is not explicitly
modeled, it will be subsumed into the A term in the classic twin model.
Contributions of Genetic, Unshared
Environment, Genotype x Unshared
Environment Interaction Effects to Twin/Sib
Resemblance
Unshared
(Unique)
Environment
Additive
Genetic Effects
Genotype x
Unshared
Environment
Interaction
MZ Pairs
0
1
0x1=0
DZ Pairs/Full Sibs
0
½
0x½=0
If gene-(unshared) environment interaction is not explicitly modeled,
it will be subsumed into the E term in the classic twin model.
Ways to Model Gene-Environment
Interaction in Twin Data
• Multiple Group Models
– (parallel to testing for sex effects using
multiple groups)
Sex Effects
Females
1.0 (MZ) / .5 (DZ)
A1
aF
C1
cF
P1
E1
eF
Males
1.0
1.0 (MZ) / .5 (DZ)
A2
aF
C2
cF
P2
E2
eF
A1
aM
C1
cM
P1
E1
eM
1.0
A2
aM
C2
cM
P2
E2
eM
Sex Effects
Females
1.0 (MZ) / .5 (DZ)
A1
aF
C1
cF
E1
eF
P1
aF = aM ?
Males
1.0
1.0 (MZ) / .5 (DZ)
A2
aF
C2
cF
E2
A1
eF
P2
aM
C1
cM
E1
eM
1.0
A2
aM
P1
cF = cM ?
C2
cM
P2
eF = eM ?
E2
eM
GxE Effects
Urban
1.0 (MZ) / .5 (DZ)
A1
aF
C1
cF
E1
eF
P1
aU = aR ?
Rural
1.0
1.0 (MZ) / .5 (DZ)
A2
aF
C2
cF
E2
A1
eF
P2
aM
C1
cM
E1
eM
1.0
A2
aM
P1
cU = cR ?
C2
cM
P2
eU = eR ?
E2
eM
Practical:
Using Multiple Group Models to
test for GxE
Problem:
• Many environments of interest do not fall
into groups
– Regional alcohol sales
– Parental warmth
– Parental monitoring
– Socioeconomic status
• Grouping these variables into high/low categories
loses a lot of information
‘Definition variables’ in Mx
• General definition: Definition variables
are variables that may vary per subject
and that are not dependent variables
• In Mx: The specific value of the def var
for a specific individual is read into a
matrix in Mx when analyzing the data of
that particular individual
‘Definition variables’ in Mx
create dynamic var/cov structure
• Common uses:
1. To model changes in variance
components as function of some
variable (e.g., age, SES, etc)
2. As covariates/effects on the means (e.g.
age and sex)
Standard model
• Means vector
m
m
• Covariance matrix
a  c  e

 Za 2  c 2

2
2
2


2
2
2
a c e 
Model-fitting approach to GxE
A
C
a
c
E
e
A
C
a
c
E
e
m
M
m
Twin 1
Twin 2
M
Model-fitting approach to GxE
A
a+XM
m
M
C
c
E
e
Twin 1
A
a+XM
C
c
E
e
Twin 2
m
M
Individual specific moderators
A
a+XM1
C
c
E
e
A
a+XM2
C
c
E
e
m
M
m
Twin 1
Twin 2
M
E x E interactions
A
a+XM1
C
E
c+YM1
e+ZM1
A
a+XM2
C
c+YM2
e+ZM2
m
M
E
m
Twin 1
Twin 2
M
1.0 (MZ) / .5 (DZ)
• Classic Twin Model:
Var (T) = a2 + c2 + e2
A1
a
C1
c
E1
e
1.0
A2
C2
a
c
Twin 1
E2
e
Twin 2
• Moderation Model:
Var (T) =
(a + βXM)2 + (c + βYM)2 + (e + βZM)2
A
a + βXM
C
e + βZM
c + βyM
 + βMM
Purcell 2002,
Twin Research
E
T
Var (T) = (a + βXM)2 + (c + βYM)2 (e + βZM)2
Where M is the value of the moderator and
Significance of βX indicates genetic moderation
Significance of βY indicates common environmental
moderation
Significance of βZ indicates unique environmental moderation
BM indicates a main effect of the
moderator on the mean
A
a + βXM
C
E
e + βZM
c + βyM
 + βMM
T
Matrix Letters as Specified in Mx
Script
A
C
E
a+XM1 c+YM1
a +aM*D1
c+cM*D1 e+ M
Z 1
Twin 1
C
E
c+YM2
e+eM*D1
M
A
a+XM2 c+cM*D2 e+ M
Z 2
a+aM*D2
e+eM*D2
Twin 2
M
m
m
mu
mu
Practical:
Using Definition Variables
to test for GxE
Additional Things to Consider
• Unstandardized versus standardized
effects
Additional Things to Consider
Unstandardized versus standardized effects
ENVIRONMENT 1
ENVIRONMENT 2
Unstandardized
Variance
Standardized
Variance
Unstandardized
Variance
Standardized
Variance
Genetic
60
0.60
60
0.30
Common
environmental
35
0.35
70
0.35
Unique
environmental
5
0.05
70
0.05
Total variance
100
200
‘Definition variables’ in Mx
create dynamic var/cov structure
• Common uses:
1. To model changes in variance
components as function of some
variable (e.g., age, SES, etc)
2. As covariates/effects on the means (e.g.
age and sex)
Definition variables used as
covariates
General model with age and sex as covariates:
yi =  + 1(agei) + 2 (sexi) + 
Where yi is the observed score of individual i, 
is the intercept or grand mean, 1 is the
regression weight of age, agei is the age of
individual i, 2 is the deviation of males (if sex
is coded 0= female; 1=male), sexi is the sex of
individual i, and  is the residual that is not
explained by the covariates (and can be
decomposed further into ACE etc).
Allowing for a main effect of X
• Means vector
m  X1i
m  X 2 i 
• Covariance matrix
a  c  e

 Za 2  c 2

2
2
2


2
2
2
a c e 
Common uses of definition
variables in the means model
• Incorporating covariates (sex, age, etc)
• Testing the effect of SNPs (association)
• In the context of GxE, controlling for rGE
Gene-environment Interaction
• Genetic control of sensitivity to the environment
• Environmental control of gene expression
Gene-environment Correlation
• Genetic control of exposure to the environment
• Environmental control of gene frequency
This complicates interpretation of
GxE effects
• If there is a correlation between the moderator
(environment) of interest and the outcome,
and you find a GxE effect, it’s not clear if:
– The environment is moderating the effects of genes
or
– Trait-influencing genes are simply more likely to be
present in that environment
Adding Covariates to Means Model
A
C
a
c
E
e
A
C
a
c
E
e
m+MM1
M
m+MM2
Twin 1
Twin 2
M
Matrix Letters as Specified in Mx
Script
A
C
E
a+XM1 c+YM1
a +aM*D1
e+eM*D1
Twin 1
C
a+XM2 c+cM*D2 e+ M
Z 2
a+aM*D2
e+eM*D2
Twin 2
M
m+MM2
m+MM1
mu+b*D1
E
c+YM2
c+cM*D1 e+ M
Z 1
M
A
Main effects and moderating effects
mu+b*D2
Ways to deal with rGE
• Limit study to moderators that aren’t correlated with outcome
– Pro: easy
– Con: not very satisfying
• Moderator in means model will remove from the covariance genetic
effects shared by trait and moderator
– Pro: Any interaction detected will be moderation of the trait specific
genetic effects
– Con: Will fail to detect GxE interaction if the moderated genetic
component is shared by the outcome and moderator
• Explicitly model rGE using a bivariate framework
– Pro: explicitly models rGE
– Con: Power to detect BXU decreases with increasing rGE; difficulty
converging
AS
aM
AU
aS + βXSM
aU + βXUM
βXS indicates moderation
of shared genetic effects
M
T
BXU indicates moderation
of unique genetic effects
on trait of interest