1. Gene-Environment Interaction & Correlation
Danielle Dick & Tim York
VCU workshop, Fall 2010

2. Gene-Environment Interaction
Genetic control of sensitivity to the environment
Environmental control of gene expression
Bottom line: nature of genetic effects differs among environments

3. Standard Univariate Model
P = A + C + E
Var(P) = a2+c2+e2
1.0 (MZ) / .5 (DZ)
1.0
1.0
1.0
1.0
1.0
1.0
1.0 A1 C1 E1 A2 C2 E2 c c a a e e P1 P2

4. 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 x 1 = 1
DZ Pairs/Full Sibs
1 x ½ = ½

5. 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 x 1 = 1
DZ Pairs/Full Sibs
1 x ½ = 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.

6. 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 1
0 x 1 = 0
DZ Pairs/Full Sibs
0 x ½ = 0
If gene-(unshared) environment interaction is not explicitly modeled,
it will be subsumed into the E term in the classic twin model.

7. Ways to Model Gene-Environment Interaction in Twin Data
Multiple Group Models
(parallel to testing for sex effects using multiple groups)

8. Sex Effects
Females
Males

9. Sex Effects
Females
Males
aF = aM ? cF = cM ? eF = eM ?

10. GxE Effects
Urban
Rural
aU = aR ? cU = cR ? eU = eR ?

11. Practical: Using Multiple Group Models to test for GxE

12. 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

13. ‘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

14. ‘Definition variables’ in Mx
create dynamic var/cov structure
Common uses:
To model changes in variance components as function of some variable (e.g., age, SES, etc)
As covariates/effects on the means (e.g. age and sex)

15. Standard model
Means vector
Covariance matrix

16. Model-fitting approach to GxE
Twin 1
Twin 2 M M

17. Model-fitting approach to GxE
a+XM c e
a+XM c e m m
Twin 1
Twin 1
Twin 2
Twin 2 M M

18. Individual specific moderators
a+XM1 c e
a+XM2 c e m m
Twin 1
Twin 1
Twin 2
Twin 2 M M

19. E x E interactions A C E A C E c+YM1 c+YM2 a+XM1 a+XM2 e+ZM1
Twin 1
Twin 1
Twin 2
Twin 2 M M

20. Classic Twin Model: Var (T) = a2 + c2 + e2 Moderation Model: Var (T) =
(a + βXM)2 + (c + βYM)2 + (e + βZM)2
Purcell 2002,
Twin Research

21. 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

22. Matrix Letters as Specified in Mx Script
c+YM1
c+YM2
a+XM1
a+XM2
c+cM*D1
c+cM*D2
e+ZM1
e+ZM2
a +aM*D1
a+aM*D2
e+eM*D2
e+eM*D1
Mediation versus moderation
If a variable mediates an effect,
significant in the means model, M > 0
significant increase in a, c or e when M is fixed to 0
If a variable moderates an effect
X>0, Y>0, or Z > 0
irrespective of M or any reduction in a, c or e.
Twin 1
Twin 2 M M m m mu mu

23. Practical: Using Definition Variables to test for GxE

24. Additional Things to Consider
Unstandardized versus standardized effects

25. Additional Things to Consider
ENVIRONMENT 1
ENVIRONMENT 2
Unstandardized Variance
Standardized Variance
Genetic 60
0.60
0.30
Common environmental 35
0.35 70
Unique environmental 5
0.05
Total variance
100
200
Unstandardized versus standardized effects

26. ‘Definition variables’ in Mx
create dynamic var/cov structure
Common uses:
To model changes in variance components as function of some variable (e.g., age, SES, etc)
As covariates/effects on the means (e.g. age and sex)

27. 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).

28. Allowing for a main effect of X
Means vector
Covariance matrix

29. 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

30. 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

31. 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

32. Adding Covariates to Means Model
m+MM1
m+MM2
Twin 1
Twin 2 M M

33. Matrix Letters as Specified in Mx Script
c+YM1
c+YM2
a+XM1
a+XM2
c+cM*D1
c+cM*D2
e+ZM1
e+ZM2
a +aM*D1
a+aM*D2
e+eM*D2
e+eM*D1
Mediation versus moderation
If a variable mediates an effect,
significant in the means model, M > 0
significant increase in a, c or e when M is fixed to 0
If a variable moderates an effect
X>0, Y>0, or Z > 0
irrespective of M or any reduction in a, c or e.
Twin 1
Twin 2 M M
m+MM2
m+MM1
mu+b*D2
mu+b*D1
Main effects and moderating effects

34. 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

35. AS AU M T aM aS + βXSM aU + βXUM βXS indicates moderation
of shared genetic effects
BXU indicates moderation
of unique genetic effects
on trait of interest M T