1. Continuously moderated effects of A,C, and E in the twin design Conor V Dolan & Sanja Franić (BioPsy VU) Boulder Twin Workshop March 4, 2014 Based on PPTs by Sophie van der Sluis & Marleen de Moor Thanks to Neale family, Sarah Medland, & Brad Verhulst

2. Twin Research Volume 5 Number 6 pp. 554- 571 Acta Genet Med GemelloI36:5·20 (1987) Mx MCNeale et al

3. Phenotypic scores Genetic level (score on A) Environmental Dispersion Variance of E given G score G x E as “genetic control” of sensitivity to different environments: heteroskedasticity Not all heteroskedasticity is GxE!

4. Phenotypic scores Environmental level (score on E) Genetic Dispersion Conditional A variance G x E as “environmental control” of genetic effects: heteroskedasticity.

5. Not all heteroskedasticity is moderation! Phenotypic scores Moderator level (score on moderator) Genetic Dispersion and / or Environmental dispersion Moderation of effects (A,C,E) by measured moderator M: heteroskedasticity.

6. Is the magnitude of genetic influences on ADHD the same in boys and girls? Do different genetic factors influence ADHD in boys and girls? (DZOS) Moderation of A effects by binary variable with the bonus of the information provided by DZ opposite sex twins Sex X A interaction: Moderation of A by sex

7. Other examples binary moderators “A” effects moderated by marital status: Unmarried women show greater levels of genetic influence on depression (Heath et al., 1998). “A” effects moderated by religious upbringing: A religious upbringing diminishes A effects on the personality trait of disinhibition (Boomsma et al., 1999). Binary moderator: multigroup approach

8. Continuous moderation Age as a moderator

9. A,C,E effects moderated by SES SES ordinal or continuous

10. Continuous Moderators not amenable to multigroup approach … treat the moderator as continuous (OpenMx)

11. Standard ACE model pheno Twin 1 ACE pheno Twin 2 ACE a c e c e a 1 b0 1 1 1 or.5 Regression on unit: just a way to estimate the mean.

12. Standard ACE model + Main effect on Means Pheno Twin 1 ACE Pheno Twin 2 ACE a c e c e a 1 b0+ β M M 1 1 b0+β M M 2

13. pheno Twin 1 1 b0+ β M M 1 pheno Twin 1 1 βMβM b0 M1M1 Actually: regression of Phenotype on M... What is left is the residual, subject to ACE modeling. Why a triangle? Fixed regressors res1 equivalent

14. Summary stats Means vector Covariance matrix (r = 1 or r=1/2) mean1 mean2

15. Allowing for a main effect of X Means vector Covariance matrix (r = 1 or r=1/2)

16. Standard ACE model + Effect on Means and a path Twin 1 ACE Twin 2 ACE a+  X M 1 c e ce a+XM2a+XM2 Twin 1Twin 2 1 m+ β M M 1 1 m+ β M M 2 M has main effect on mean + moderation of A effect

17. Twin 1 ACE a+  X M 1 c e Twin 1 1 m+ β M M 1 M1=0 -> mean= m & A effect = a M1=1 -> mean= m+ β M M 1 & A effect = a+  X M 1

18. Standard ACE model + Effect on Means and a,c, & e paths Twin 1 ACE Twin 2 ACE a+  X M 1 c+  Y M 1 e+  Z M 1 a+  X M 2 c+  Y M 2 e+  Z M 2 Twin 1Twin 2 1 m+MM1m+MM1 1 m+MM2m+MM2

19. Effect on means:  Main effects (regression of phenol on M) Effect on a/c/e path loadings:  Moderation effects (A x M, C x M, E x M interaction)

20. Expected variances Standard Twin Model: Var (P) = a 2 + c 2 + e 2 Moderation Model: Var (P|M) = (a + β X M) 2 + (c + β Y M) 2 + (e + β Z M) 2

21. Expected MZ / DZ covariances Cov(P 1,P 2 |M) MZ = (a + β X M) 2 + (c + β Y M) 2 Cov(P 1,P 2 |M) DZ = 0.5*(a + β X M) 2 + (c + β Y M) 2

22. Var (P|M) = (a + β X M) 2 + (c + β Y M) 2 + (e + β Z M) 2 h 2 |M = (a + β X M) 2 / Var (P|M) c 2 |M = (c + β y M) 2 / Var (P|M) e 2 |M = (e + β y M) 2 / Var (P|M) (h 2 |M + c 2 |M + e 2 |M ) = 1 Standardized conditional on value of M

23. Turkheimer study SES Moderation of unstandardized variance components (top) Moderation of standardized variance components (bottom)

24. 1/.51 But what have assumed concerning M?

25. Fixed regressor x y  Assumption: y|x*~N(m y|x, s y|x ) m y|x* = β 0 + β 1 x* s y|x* = s  β1β1 1 β0β0

26. Fixed regressor x y  Assumption: y|x*~N(m y|x, s y|x ) m y|x* = β 0 + β 1 x* s y|x* = s  β1β1 1 β0β0 A C E

27. Random regressor y Assumption: (y,x)~N( ,  ) x 1 β0β0 β1β1

28. “Random regressor not to be confused with Random effects regression (multilevel modeling)”

29. Bivariate distribution of X and Y Assumption: (y,x)~N( ,  ) x ACE y ACE Regression of Y on X in terms of latent variables

30. M1,M2 (co)variance is not included in the model. M is a “fixed regressor”. Phenotype = BMI Moderator = Age.... A fixed regressor Phenotype = BMI Moderator = Intelligence.... Can we treat Intelligence as a fixed fixed In this context? Depends... on...

31. EcEc EuEu T1 a u + b au *M1 CcCc M1 amam acac AcAc CuCu AuAu c ecec cmcm emem c u + b cu *M1 e u + b eu *M1 T2 AuAu CuCu EuEu e u + b eu *M2 c u + b cu *M2 a u + b au *M2 M2 CcCc EcEc AcAc amam emem cmcm ecec c acac MZ=1 / DZ=.5 MZ=1 DZ=.5 MZ = DZ = 1 M as a random regressor, with its own ACE + ACE cross loadings.

32. Can we treat M as a fixed effect in this manner? 1/.51 Not generally

33. EuEu T1 a u + b au *M1 CcCc M1 amam acac EcEc AcAc CuCu AuAu c ecec cmcm emem c u + b cu *M1 e u + b eu *M1 T2 AuAu CuCu EuEu e u + b eu *M2 c u + b cu *M2 a u + b au *M2 M2 CcCc EcEc AcAc amam emem cmcm ecec c acac MZ=1 / DZ=.5 MZ=1 DZ=.5 MZ = DZ = 1 OK if #1: a c /a m = c c /c m = e c / e m

34. EuEu T1 a u + b au *M1 M1 EcEc CuCu AuAu ecec emem c u + b cu *M1 e u + b eu *M1 T2 AuAu CuCu EuEu e u + b eu *M2 c u + b cu *M2 a u + b au *M2 M2 EcEc emem ecec MZ=1 DZ=.5 MZ = DZ = 1 OK if #2: r(M1,M2) = 0

35. EuEu T1 a u + b au *M1 CcCc M1 CuCu AuAu c cmcm c u + b cu *M1 e u + b eu *M1 T2 AuAu CuCu EuEu e u + b eu *M2 c u + b cu *M2 a u + b au *M2 M2 CcCc cmcm c MZ=1 DZ=.5 MZ = DZ = 1 Ok if #3: r(M1,M2) = 1

36. EuEu T1 a u + b au *M1 CcCc M1 amam acac EcEc AcAc CuCu AuAu c ecec cmcm emem c u + b cu *M1 e u + b eu *M1 T2 AuAu CuCu EuEu e u + b eu *M2 c u + b cu *M2 a u + b au *M2 M2 CcCc EcEc AcAc amam emem cmcm ecec c acac MZ=1 / DZ=.5 MZ=1 DZ=.5 MZ = DZ = 1 What to do otherwise?

37. MZ:T1=β 0,MZ + β 1,MZ *M 1 + β 2,MZ *M 2 T2=β 0,MZ + β 1,MZ *M 2 + β 2,MZ *M 1 DZ:T1=β 0,DZ + β 1,DZ *M 1 + β 2,DZ *M 2 T2=β 0,DZ + β 1,DZ *M 2 + β 2,DZ *M 1 E C A T1 e + β e *M1 a + β a *M1 c + β c *M1 A C E T2 a + β a *M2 e + β e *M2 c + β c *M2 1 /.5 1 1 β 0k + β 1k *M1 + β 2k *M2 1 β 0k + β 1k *M2 + β 2k *M1

38. But what have assumed concerning the covariance between M and T? E C A T1 e + β e *M1 a + β a *M1 c + β c *M1 A C E T2 a + β a *M2 e + β e *M2 c + β c *M2 1 /.5 1 1 β 0k + β 1k *M1 + β 2k *M2 1 β 0k + β 1k *M2 + β 2k *M1

39. EuEu T1 a u + b au *M1 CcCc M1 amam acac EcEc AcAc CuCu AuAu c ecec cmcm emem c u + b cu *M1 e u + b eu *M1 T2 AuAu CuCu EuEu e u + b eu *M2 c u + b cu *M2 a u + b au *M2 M2 CcCc EcEc AcAc amam emem cmcm ecec c acac MZ=1 / DZ=.5 MZ=1 DZ=.5 MZ = DZ = 1

40. EuEu T1 a u + b au *M1 CcCc M1 amam a c + b ac *M1 EcEc AcAc CuCu AuAu c c + b cc *M1 e c + b ec *M1 cmcm emem c u + b cu *M1 e u + b eu *M1 T2 AuAu CuCu EuEu e u + b eu *M2 c u + b cu *M2 a u + b au *M2 M2 CcCc EcEc AcAc amam emem cmcm e c + b ec *M2 c c + b cc *M2 a c + b ac *M2 MZ=1 / DZ=.5 MZ=1 DZ=.5 MZ = DZ = 1

41. EuEu T1 a u + b au *M1 CcCc M1 amam a c + b ac *M1 EcEc AcAc CuCu AuAu c c + b cc *M1 e c + b ec *M1 cmcm emem c u + b cu *M1 e u + b eu *M1 T2 AuAu CuCu EuEu e u + b eu *M2 c u + b cu *M2 a u + b au *M2 M2 CcCc EcEc AcAc amam emem cmcm e c + b ec *M2 c c + b cc *M2 a c + b ac *M2 MZ=1 / DZ=.5 MZ=1 DZ=.5 MZ = DZ = 1 M has to be treated as random and modeled appropriately (as shown)!

42. Conclusion: Use standard fixed regression approach if Cov(M,T) equally due to A,C,E Cov(M1,M2) = zero Cov(M1,M2) = one Use extended fixed regression approach if Cross paths are not moderated Otherwise use full model.

43. A simpler conclusion: ALWAYS USE FULL MODEL UNLESS Cor(M1,M2) = 1.

44. categorical data Continuous data –Moderation of means and variances Ordinal data –Moderation of thresholds and variances –See Medland et al. 2009

45. Non linear moderation? Extend the model from linear to Linear + quadratic? e c + b ec1 *M1 + b ec2 *M1 2

46. What about >1 number of moderators Extend the model accordingly e c + b ec1 *SES + b ec2 *AGE

47. What about >1 number of moderators and interaction Effects (sex X age) Extend the model accordingly... e c + b ec1 *SES + b ec2 *AGE + b ec2 *(AGE *SES)

48. What about power given such extensions? Do power calculation.... Exactly, by simulation, by exact simulation ?

49. Practical Replicate findings from Turkheimer et al. with twin data from NTR Phenotype: FSIQ Moderator: SES in children (cor(M1,M2) = 1 Data: 205 MZ and 225 DZ twin pairs 5 years old

50. Gene-Environment Correlation rGE: Genetic control of exposure to the environment Examples: Active rGE: Children with high IQ read more books Passive rGE: Parents of children with high IQ take their children more often to the library

51. Genetic control of exposure to the environment? “short hand” Chain of causality? A “causes” high IQ high IQ “causes” interest in astronomy join the astronomy club study astronomy