1. GxE and GxG interactions
Michael C Neale
Boulder Workshop March

2. Background I Various kinds of E and G Latent/Observed GxG interaction
Observed x Observed
Latent x Latent
Latent x Observed
Same for GxE

3. Background II Latent Genotype x Measured Environment Interaction
Kendler, KS & Eaves, LJ 1986 Models for the joint effect of genotype and environment on liability to psychiatric illness. Am J Psychiat 143:279-89
Heath, AC, Cates, R, Martin, NG, Meyer, JM, Hewitt, JK, Neale, MC & Eaves, LJ (1993) Genetic contributions to smoking initiation: Comparisons across cohorts and across cultures. Journal of Substance Abuse 5:
Neale, MC & Cardon, LR (1992) Methodology for Genetic Studies of Twins & Families. Dordrecht, NL: Kluwer.
Models based on multiple-group approach: binary/nominal classifications “Heterogeneity”

4. Background III 1994: Definition variables in Mx
Enables ‘continuous’ version of multiple group analysis
Purcell, S (2002). Variance components models for gene- environment interaction in twin analysis. Twin Res, 5(6):
Useful framework: do A/C/E change as a function of a measured ‘environmental’ variable?
Often regress out main effects of moderator on trait of interest

5. ? G x E interaction The environment modifies the effect of a gene
Gene-environment interaction
Gene effect
Environmental effect
G x E interaction
The environment modifies the effect of a gene
A gene modifies the effect of an environment
S.Purcell ©

6. Gene × gene interaction
Epistasis
Gene effect
Gene effect
Gene × gene interaction
Epistasis: one gene modifies the effect of another
S.Purcell ©

7. Biometrical G × E model No interaction M Interaction β 1 -β M
Equivalently… 2β 1 β
Genotype Mean a -a
Environment
Environment
Environment M AA Aa aa

8. Standard ACE model for twin resemblance
1(MZ) .5(DZ) 1 1 1 1 1 1 1 A C E A C E a c e a c e
Twin 1 ∝ ∝
Twin 2 1

9. Moderated ACE model for twin resemblance
1(MZ) .5(DZ) 1 1 1 1 1 1 1 A C E A C E
a + βXM1 c e
a + βXM2 c e
Twin 1
∝ + βmM1
∝ + βmM2
Twin 2 1

11. Within-person correlations across age Single-factor model (r=1)
Red = 1.0 Green = -1.0

12. Growth Curve Model VL VS r L S 1 1 1 2 1 3 1 4 Trait Age 1 Trait Age 2

13. Enhanced Moderated AE model
1(MZ) .5(DZ) 1 1 r 1 1 1 r 1 1 AS AL E AS AL E
aSM1 aL e
aSM2 aL e
Twin 1
µ + βmM1
µ + βmM2
Twin 2 1

14. Enhanced Moderated AE model
Two-factor Moderated AE model r=.0 : .9
Enhanced Moderated AE model

15. Within-person correlations across age
r=1.0 r=.9 r=.6 r=.3

16. Extension for correlated moderator model: 1 Twin
Perhaps best to integrate across AC, CC or other components of variance
Volume controls on Factor or on T are coherent

17. Bivariate Moderated AE model: 1 Twin
1(MZ) .5(DZ) 1 1 1 1 1 1 1 r AM CM EM
AST
ALT ET
eTM aM cM eM
cTM
aSM M
aLT eT
aTM
Moderator
Trait µM µT 1

18. Research Design Considerations
No variation in Age(M) at measurement
Estimation of parameters via mixture modeling (but could be due to anything)
Variation in age, but twins measured once at same time
Know variance, MZ and DZ different ages
Don’t know if same genes/envt at different ages (assume r)
Variation in age, twins measured once at different times
Know 2.1 plus MZ & DZ covariances as f(Age difference), i.e., have pairs concordant & discordant wrt Age
Can get A & C factors split into level & slope
Longitudinal data, twins possibly measured at different times
Can get E factor split (rE) as well

19. Greg Carey Article
Genotype-Environment Interaction Problems and a Multivariate Solution
“Please do not cite until I’m certain I got the whole thing right.”
Definition variables
Conditional Distributions
New Mx script

20. Necessary background
“For example, if the ACE model in the general population assumes that A, C, and E are uncorrelated, then A, C, and E within the poor environmental group will be usually be correlated when home environment is correlated with ASB”
Conditional distributions
Distribution of Phenotype conditional on Moderator
Pearson-Aitken selection formulae

21. Pearson Aitken Selection

22. Pearson Aitken Selection

23. Careful with measurement!
Example item response probability curves in white
0.5
Response
Probability N
0.4 1
0.3
.75
0.2 .5
0.1
.25 .0 -4 -3 -2 -1 1 2 3 4
Low Info High Info