1. Gene x Environment Interactions Brad Verhulst (With lots of help from slides written by Hermine and Liz) September 30, 2014

2. What does a GxE interaction in a twin model really mean? Univariate Analysis: What are the contributions of A, C/D & E to the variance? Heterogeneity Analysis: Are the contributions of genetic and environmental factors equal across different groups, such as sex, race, ethnicity, SES, environmental exposure, etc.? Moderation Analysis: Are the contributions of genetic and environmental factors to the variance constant across the range of a second (moderator) variable?

3. Gene-Environment Interaction GxE genetic control of sensitivity to the environment environmental control of gene expression – (environmental modulation of non-genetic paths) Examples: Does heritability of IQ depend on SES? Does heritability of ADHD depend on age? Does the role parental monitoring depend on genotype?

4. Gene-Environment Correlation rGE genetic control of exposure to the environment environmental control of gene frequency Examples: Active rGE: Children with high IQ read more books Passive rGE: High IQ parents give their children books Reactive/Evocative rGE: Children with ADHD are treated differently by their parents

5. Moderating Variables Almost any variable can be used as a moderator… … but be careful as not all variables make sense as moderators (or are easy to interpret) If a variable has a genetic component (A > 0) interpreting the GxE path is complicated by the fact that the moderator is a function of both G & E. Is it a GxG or a GxE interaction?

6. Heterogeneity Moderation An easy (but much less powerful) method of conducting GxE For categorical variables, estimate separate parameters for each group. – Sex Limitation is a classic case of GxE where separate parameters are estimated for each group – This can be extended to any number of categories (but quickly gets tedious and difficult to interpret) This approach would not work for continuous variables (as there are no discrete categories) – Age – Factor Scores of X, Y & Z Grouping these variables into categories loses a lot of information and power

7. GxE Model & Theory Purcell 2002 Twin Research

8. GxE Application Turkheimer et al. 2003 Psychological Science

10. Definition Variables in OpenMx General definition: Definition variables are variables that may vary per subject/pair and are not dependent variables In OpenMx: Specific values of definition variables for a specific individual/pair is read into mxMatrix when analyzing data of that particular individual/pair Common Use of Definition Variables To model main effects of on the means (e.g. age and sex) To model changes in variance components as function of some moderator variable (e.g. age, SES)

11. Cautionary Note about Definition Variables Definition variables should not be missing if dependent variable is not missing Definition variables should not have the same missing values as dependent variable (e.g. use - 2.00 for definition variable and -1.00 for dependent variable) It is helpful to have very large values for missing definition variables (so that if things go wrong the results are unmistakably funky)

12. Definition Variables as Main Effects General model with age and sex as main effects: y i = α + β 1 (Age i ) + β 2 (Sex i ) + ε i Where: y i is the observed score of individial i α is the intercept or grand mean β 1 is the regression weight of age Age i is the age of individual i β 2 is the deviation of females (if sex coded 0:males, 1:females) Sex i is the sex of individual i ε i is the residual not explained by definition vars (and can be decomposed further into ACE etc.)

13. Allowing for Main Effect M + Xβ Means Vector a 2 + c 2 + e 2 H * a 2 + c 2 a 2 + c 2 + e 2 Covariance Matrix

14. Allowing for Moderation M + Xβ Means Vector (a + Xϒ a ) 2 + (c + Xϒ c ) 2 + (e+ Xϒ e ) 2 H * (a + Xϒ a ) 2 + (c + Xϒ c ) 2 H * (a + Xϒ a ) 2 + (c + Xϒ c ) 2 (a + Xϒ a ) 2 + (c + Xϒ c ) 2 + (e+ Xϒ e ) 2 Covariance Matrix

15. Existing Gene-Environment Interaction Models Classical Twin Design P t1 P t2 1 A C E μ E C A 1 MZ=1 DZ = ½ 1 1 1 1 1 1 a + β a Ma e+ β e M c + β c M c e+ β e M + Mβ m μ Purcell (2002) Means Moderation Model Basic Means and Variances

16. Example: Turkheimer Study Moderation of unstandardized variance components Moderation of standardized variance components

17. Cautions about interpreting the Parameters Unstandardized (UV) vs Standardized (SV) Environment 1Environment 2 Unstandardized Variance Standardized Variance Unstandardized Variance Standardized Variance Genetic 60.6060.30 Common Environment 35.3570.35 Unique Environment 5.0570.35 Total Variance 1001.002001.00

18. Cautions about interpreting the Parameters Parameters are Conditional The estimated values of a, c & e in a Purcell model depend on the value of the intercept (or the mean). If the mean is 0, the interpretation of the direct effect of a (or c) on the phenotype is the genetic (or common environment) variance at the mean. If the mean is not 0, the interpretation of the direct effect of a (or c) on the phenotype is the genetic (or common environment) variance is more complicated. Therefore, it is always suggested that the variance components are plotted across the range of the moderator.

19. GxE in context of rGE If there is a correlation between moderator (environment) and outcome, and you find a GxE effect, it is 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

20. Ways to deal with rGE Limit study to moderators not correlated with outcome Put moderator in means model to remove covariance genetic effects shared by trait and moderator Explicitly model rGE in bivariate framework