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Continuously moderated effects of A,C, and E in the twin design

This article explores the moderation effects of A, C, and E in twin design, including the main effect of sex, sex by A and E interactions, and the inclusion of heterogeneity sources in the model.

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Continuously moderated effects of A,C, and E in the twin design

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  1. Continuously moderated effects of A,C, and E in the twin design Conor V Dolan & Michel Nivard Boulder Twin Workshop March, 2018

  2. Standard AE model s2Aor .5s2A A E A E s2E s2A s2A s2E 1 1 1 1 m pheno Twin 1 pheno Twin 2 m 1 1 Phi – m = Ai + Ei

  3. Standard AE model: Alt notation. VA =s2A VAor .5VA A E A E VE VA VA VE 1 1 1 1 m pheno Twin 1 pheno Twin 2 m 1 1 Phi – m = Ai + Ei

  4. Standard AE model 1 or .5 A E A E a e a e m pheno Twin 1 pheno Twin 2 m 1 1 Phi – m= a*Ai + e*Ei

  5. When we fit this model we assume that the modelholds in the population of interest (z=zygosity, mz or dz; rz=1 or .5) Sz = s2A + s2E rzs2A rzs2A s2A + s2E m = mm Phenotypic variable ~ N(m, s2A + s2E) normally distributed with mean m and variance s2A + s2E

  6. What if we have two populations of interest, e.g., males and females? * main effect of sex on phenotype mf ≠ mm? * sex by A, E interactions (sex as moderator of A,E variances): s2Af ≠ s2Am and / or s2Ef ≠ s2Em? If we ignore the source of “heterogeneity” the estimates of m, s2E, s2A are biased ... BAD!

  7. What to do? Include source of heterogeneity, moderator, in the model: Main effects of moderator mf ≠ mm? Interaction effects s2Af ≠ s2Am and / or s2Ef ≠ s2Em? If moderator is binary – multigroup model If moderator is continuous – moderation model

  8. Acta Genet Med GemelloI36:5·20 (1987) 1977 1987 (Open)Mx Michael C.Neale & team 1994 - 2018 2002 MANY PAPERS Substantive & Methods

  9. see https://genepi.qimr.edu.au/staff/classicpapers/ What is moderation? What is GxE interaction

  10. Environmental Dispersion Variance of E given G score Phenotypic scores Genetic level (score on A) Phi – m = a*Ai + e*Ei Homoskedastic model: e is constant over levels of A: environmental effects are the same given any A

  11. Environmental Dispersion Variance of E given G score Phenotypic scores Genetic level (score on A) G x E as “genetic control” of sensitivity to different environments: heteroskedasticity e=fe(A) or s2E =ge(A) Environmental effects (E) systematically vary with A

  12. Conditional variance of A given E Phenotypic scores Environmental level (score on E) G x E as “environmental control” of genetic effects: heteroskedasticity (a=fa(E) or s2A =ga(E)).

  13. Genetic variance or (and) Environmental variance given M Phenotypic scores Moderator level (score on moderator) Moderation of effects (A,C,E) by measured moderator M: heteroskedasticity (a=fa(M), c=fc(M), e=fe(M) or (s2A =ga(M), s2C =gc(M), s2E =ge(M)).).

  14. Sex X A interaction:Sex moderation of A effects Is the magnitude of genetic influences on ADHD the same in boys and girls? s2Af = s2Am ?

  15. 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: religious upbringing diminishes A effects on the personality trait of disinhibition (Boomsma et al., 1999). Binary moderator: multigroup approach

  16. Continuous moderation Age as a moderator of standardized A, C, Evariance components (Age x A,C,E interaction)

  17. Tucker-Drob & Bates (2015)

  18. A,C,E effects moderated by SES

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

  20. Standard ACE model 1 or .5 1 A C E A C E c c a e a e m m pheno Twin 1 pheno Twin 2 1 1

  21. Standard ACE model + Main effect on Means A C E A C E c c a e a e Pheno Twin 1 Pheno Twin 2 1 1 m+βM M1 m+βMM2

  22. res1 res1 m pheno Twin 1 1 pheno Twin 1 1 βM m+βM M1 M1 equivalent The regression of Phenotype on M ... pheno1 = m+ βM M1 + res1 What is left is the residual (res1), which is subject to (moderated) ACE modeling.

  23. Summary stats (no moderator) • Means vector • Covariance matrix (rz = r = 1 or ½) m m

  24. Allowing for a main effect of the moderator M • Means vector (conditional on M) • Covariance matrix (r = 1 or r=½)

  25. Standard ACE model + main effect and effect on A path A C E A C E c c e a+XM1 a+XM2 e m+ βM M1 m+ βM M2 Twin 1 Twin 1 Twin 2 Twin 2 1 1 M has main effect + moderation of A effect (a + bMM1)

  26. A C E c e a+XM1 m+ βM M1 Twin 1 Twin 1 1 If M is binary (0/1) (instead of continuous) M1=0 mean= m & A effect = a M1=1 mean= m+βM& A effect = a+X variance = (a+X)2

  27. Main Effect on phenotype (linear regression) • Effect path loadings: Moderation effects (A x M, C x M, E x M interaction)

  28. Expected variances Standard Twin Model: Var(P) = a2 + c2 + e2 Moderation Model: Var(P|M) = (a + βXM)2 + (c + βYM)2 + (e + βZM)2 P|M mean “the phenotype given a value on M” or “the phenotype conditional on M”

  29. Expected MZ / DZ covariances Cov(P1,P2|M)MZ = (a + βXM)2 + (c + βYM)2 Cov(P1,P2|M)DZ = 0.5*(a + βXM)2 + (c + βYM)2

  30. Var (P|M) = (a + βXM)2 + (c + βYM)2 + (e + βZM)2 hst2|M = (a + βXM)2/ Var(P|M) cst2|M =(c + βYM)2/ Var(P|M) est2|M =(e + βZM)2/ Var(P|M) (hst2|M + cst2|M + est2|M) = 1 Standardized conditional on value of M.. WARNING hst2|M can vary with M while βX = 0!

  31. Turkheimer study SES Moderation of unstandardized variance components raw variance components good!

  32. 1/.5 1 But what have we assumed concerning M?

  33. The M is a measured variable A 1 a+bxM m twin C c+byM M bM e+bzM E M is environmental? (sociaal support, employment, marital status)

  34. “Environmental” measures display genetic variance See Kendler & Baker, 2006 See Plomin & Bergeman, 199

  35. Full bivariate model MZ = DZ = 1 MZ=1 / DZ=.5 MZ = DZ = 1 Eu Eu Ac Ac Cc Cc Cu Cu MZ=1 DZ=.5 Ec Ec Au Au eu eu cu cu ac ac au cc cc cm au cm ec ec em am am em M2 T2 M1 T1 var(T) = {ec2+cc2+ac2}+ {eu2+cu2+au2} unique to T shared with M

  36. M with its own ACE + ACE cross loadings. MZ = DZ = 1 MZ=1 / DZ=.5 MZ = DZ = 1 Eu Eu Ac Ac Cc Cc Cu Cu MZ=1 DZ=.5 Ec Ec Au Au eu + beu*M1 eu + beu*M2 cu + bcu*M2 cu + bcu*M1 ac ac cc cc cm au + bau*M2 cm au + bau*M1 ec ec em am am em M2 T2 M1 T1

  37. Can we treat M in this manner? 1/.5 1 Not generally....

  38. OK if #1: r(M1,M2) = 0 MZ = DZ = 1 Eu Eu Cu Cu MZ=1 DZ=.5 Ec Ec Au Au eu + beu*M2 eu + beu*M1 cu + bcu*M2 cu + bcu*M1 au + bau*M1 au + bau*M2 ec ec em em M2 T2 T1 M1

  39. MZ = DZ = 1 MZ = DZ = 1 Eu Eu Cc Cc Cu Cu MZ=1 DZ=.5 Au Au eu + beu*M2 eu + beu*M1 cu + bcu*M2 cu + bcu*M1 cc cc au + bau*M1 au + bau*M2 cm cm M2 T2 M1 T1 Ok if #2: r(M1,M2) = 1

  40. MZ = DZ = 1 MZ=1 / DZ=.5 MZ = DZ = 1 Eu Eu Ac Ac Cc Cc Cu Cu MZ=1 DZ=.5 Ec Ec Au Au eu + beu*M2 eu + beu*M1 cu + bcu*M2 cu + bcu*M1 ac ac cc cc au + bau*M1 au + bau*M2 cm cm ec ec em am am em M2 T2 M1 T1 What to do otherwise? Fit model as shown:

  41. MZ = DZ = 1 MZ=1 / DZ=.5 MZ = DZ = 1 Eu Eu Ac Ac Cc Cc Cu Cu MZ=1 DZ=.5 Ec Ec Au Au eu + beu*M2 eu + beu*M1 cu + bcu*M2 cu + bcu*M1 ac ac cc cc au + bau*M1 au + bau*M2 cm cm ec ec em am am em M2 T2 M1 T1 But what about the common paths eccc& ac.... Why only moderation of effect unique to T?

  42. MZ = DZ = 1 MZ=1 / DZ=.5 MZ = DZ = 1 Eu Eu Ac Ac Cc Cc Cu Cu MZ=1 DZ=.5 Ec Ec Au Au eu + beu*M2 eu + beu*M1 ac + bac*M1 cu + bcu*M2 cu + bcu*M1 ac + bac*M2 cc + bcc*M2 cc + bcc*M1 au + bau*M1 au + bau*M2 cm cm ec + bec*M2 ec + bec*M1 em am am em M2 T2 M1 T1 umx function available!

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

  44. Non linear moderation? Extend the model from linear to linear + quadratic. E.g.,ec+ bec1*M1 + bec2*M12

  45. What about >1 number of moderators? Extend the model accordingly ec + bec1*SES + bec2*AGE With possible interaction: ec + bec1*SES + bec2*AGE +bec3*(AGE *SES)

  46. Michel Nivard 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 yearsold

  47. MZ = DZ = 1 MZ = DZ = 1 Eu Eu Cc Cc Cu Cu MZ=1 DZ=.5 Au Au eu + beu*M2 eu + beu*M1 cu + bcu*M2 cu + bcu*M1 cc cc au + bau*M1 au + bau*M2 cm cm M2 T2 M1 T1 Ok if #2: r(M1,M2) = 1 SES = M

  48. unx function available

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