Continuously moderated effects of A,C, and E in the twin design Conor V Dolan & Michel Nivard Boulder Twin Workshop March, 208
Standard AE model s 2 A or.5s 2 A A E A E s 2 A s 2 E s 2 A s 2 E m pheno Twin pheno Twin 2 m Ph i m = A i + E i
Standard AE model: Alt notation. V A = s 2 A V A or.5v A A E A E V A V E V A V E m pheno Twin pheno Twin 2 m Ph i m = A i + E i
Standard AE model or.5 A E A E a e a e m pheno Twin pheno Twin 2 m Ph i m = a*a i + e*e i
When we fit this model we assume that the model holds in the population of interest (z=zygosity, mz or dz; r z = or.5) S z = s 2 + s A 2 E r z s 2 A r z s 2 A s 2 + s A 2 E m = m m Phenotypic variable ~ N(m, s 2 + s A 2 E) normally distributed with mean m and variance s 2 + s A 2 E
What if we have two populations of interest, e.g., males and females? * main effect of sex on phenotype m f m m? * sex by A, E interactions (sex as moderator of A,E variances): s 2 Af s 2 Am and / or s 2 Ef s 2 Em? If we ignore the source of heterogeneity the estimates of m, s 2 E, s 2 A are biased... BAD!
What to do? Include source of heterogeneity, moderator, in the model: Main effects of moderator m f m m? Interaction effects s 2 Af s 2 Am and / or s 2 Ef s 2 Em? If moderator is binary multigroup model If moderator is continuous moderation model
977 Acta Genet Med GemelloI36:5 20 (987) 987 (Open)Mx Michael C.Neale & team 994-208 2002 MANY PAPERS Substantive & Methods
see https://genepi.qimr.edu.au/staff/classicpapers/ What is moderation? What is GxE interaction
Environmental Dispersion Phenotypic scores Variance of E given G score Genetic level (score on A) Ph i m = a*a i + e*e i Homoskedastic model: e is constant over levels of A: environmental effects are the same given any A
Phenotypic y scores -4-2 0 2 4 6 Environmental Dispersion Variance of E given G score -4-3 -2-0 2 3 Genetic level (score on A) x G x E as genetic control of sensitivity to different environments: heteroskedasticity e=f e (A) or s 2 E =g e (A) Environmental effects (E) systematically vary with A
yphenotypic scores -4-2 0 2 4 6 Conditional variance of A given E -4-3 -2-0 2 3 x Environmental level (score on E) G x E as environmental control of genetic effects: heteroskedasticity (a=f a (E) or s 2 A =g a (E)).
Phenotypic scores y -4-2 0 2 4 6 Genetic variance or (and) Environmental variance given M -4-3 -2-0 2 3 x Moderator level (score on moderator) Moderation of effects (A,C,E) by measured moderator M: heteroskedasticity (a=f a (M), c=f c (M), e=f e (M) or (s 2 A =g a (M), s 2 C =g c (M), s 2 E =g e (M)).).
Is the magnitude of genetic influences on ADHD the same in boys and girls? s 2 Af = s 2 Am? Sex X A interaction: Sex moderation of A effects
Other examples binary moderators A effects moderated by marital status: Unmarried women show greater levels of genetic influence on depression (Heath et al., 998). A effects moderated by religious upbringing: religious upbringing diminishes A effects on the personality trait of disinhibition (Boomsma et al., 999). Binary moderator: multigroup approach
Continuous moderation Age as a moderator of standardized A, C, E variance components (Age x A,C,E interaction)
0.7 0.6 0.5 Genes (A) Shared Environment (C) Nonshared Environment (E) Variance 0.4 0.3 0.2 0. 0.0 2 4 6 8 0 2 4 6 8 Age (Years) Tucker-Drob & Bates (205)
A,C,E effects moderated by SES
Continuous Moderators not amenable to multigroup approach treat the moderator as continuous (OpenMx and umx)
Standard ACE model or.5 A C E A C E a c e a c e m pheno Twin pheno Twin 2 m
Standard ACE model + Main effect on Means A C E A C E a c e a c e Pheno Twin Pheno Twin 2 m+ β M M m+β M M2
res res pheno Twin m pheno Twin m+ β M M M β M equivalent The regression of Phenotype on M... pheno = m+ β M M + res What is left is the residual (res), which is subject to (moderated) ACE modeling.
Summary stats (no moderator) Means vector Covariance matrix (r z = r = or ½) 2 2 2 2 2 2 2 2 * e c a c a r e c a m m
Allowing for a main effect of the moderator M Means vector (conditional on M) Covariance matrix (r = or r=½) i M i M m M m 2 M 2 2 2 2 2 2 2 2 * e c a c a r e c a
Standard ACE model + main effect and effect on A path A C E A C E a+ X M c e a+ X M 2 c e m+ β M M m+ β M M2 Twin Twin 2 M has main effect + moderation of A effect (a + M M )
A C E a+ X M c e m+ β M M Twin If M is binary (0/) (instead of continuous) M =0 mean= m & A effect = a M = mean= m+ β M & A effect = a+ X variance = (a+ X ) 2
A C E A C E a+ X M c+ Y M e+ Z M a+ X M 2 c+ Y M 2 e+ Z M 2 m+ M M Twin Twin 2 m+ M M2 Main Effect on phenotype (linear regression) Effect path loadings: Moderation effects (A x M, C x M, E x M interaction)
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 P M mean the phenotype given a value on M or the phenotype conditional on M
Expected MZ / DZ covariances Cov(P,P 2 M) MZ = (a + β X M) 2 + (c + β Y M) 2 Cov(P,P 2 M) DZ = 0.5*(a + β X M) 2 + (c + β Y M) 2
Var (P M) = (a + β X M) 2 + (c + β Y M) 2 + (e + β Z M) 2 h st2 M = (a + β X M) 2 / Var (P M) c st2 M = (c + β Y M) 2 / Var (P M) e st2 M = (e + β Z M) 2 / Var (P M) (h st2 M + c st2 M + e st2 M) = Standardized conditional on value of M.. WARNING h st2 M can vary with M while β X = 0!
Turkheimer study SES Moderation of unstandardized variance components raw variance components good!
/.5 A C E A C E a+ X M c+ Y M e+ Z M a+ X M 2 c+ Y M 2 e+ Z M 2 m+ M M Twin Twin 2 m+ M M2 But what have we assumed concerning M?
The M is a measured variable M m M twin a+ x M c+ y M e+ z M A C E M is environmental? (sociaal support, employment, marital status)
Environmental measures display genetic variance See Kendler & Baker, 2006 See Plomin & Bergeman, 99
Full bivariate model MZ = DZ = MZ= / DZ=.5 MZ = DZ = C c A c E u C u MZ= DZ=.5 C u E u A c C c E c e u A u A u e u E c c u a u c u c m e m a m a u a m c m e m M T T2 M2 var(t) = {e c2 +c c2 +a c2 } + {e u2 +c u2 +a u2 } shared with M unique to T
M with its own ACE + ACE cross loadings. MZ = DZ = MZ= / DZ=.5 MZ = DZ = C c A c E u C u MZ= DZ=.5 C u E u A c C c e u + b eu *M A u A u e u + b eu *M2 E c E c c u + b cu *M c u + b cu *M2 c m a u + b au *M a u + b au *M2 c m e m a m a m e m M T T2 M2
Can we treat M in this manner? /.5 A C E A C E a+ X M c+ Y M e+ Z M a+ X M 2 c+ Y M 2 e+ Z M 2 m+ M M Twin Twin 2 m+ M M2 Not generally...
OK if #: r(m,m2) = 0 MZ = DZ = E u MZ= DZ=.5 C u E u E c C u A u e u + b eu *M A u e u + b eu *M2 E c c u + b cu *M c u + b cu *M2 a u + b au *M a u + b au *M2 e m e m M T T2 M2
Ok if #2: r(m,m2) = MZ = DZ = MZ = DZ = C c E u C u MZ= DZ=.5 C u E u C c e u + b eu *M A u A u e u + b eu *M2 c u + b cu *M c u + b cu *M2 c m a u + b au *M a u + b au *M2 c m M T T2 M2
What to do otherwise? Fit model as shown: MZ = DZ = MZ= / DZ=.5 MZ = DZ = C c A c E u C u MZ= DZ=.5 C u E u A c C c E c e u + b eu *M A u A u e u + b eu *M2 E c c u + b cu *M c u + b cu *M2 c m a u + b au *M a u + b au *M2 c m e m a m a m e m M T T2 M2
MZ = DZ = MZ= / DZ=.5 MZ = DZ = C c A c E u C u MZ= DZ=.5 C u E u A c C c E c e u + b eu *M A u A u e u + b eu *M2 E c c u + b cu *M c u + b cu *M2 c m a u + b au *M a u + b au *M2 c m e m a m a m e m M T T2 M2 But what about the common paths e c c c & a c... Why only moderation of effect unique to T?
MZ = DZ = MZ= / DZ=.5 MZ = DZ = C c A c E u C u MZ= DZ=.5 C u E u A c C c E c e u + b eu *M A u A u e u + b eu *M2 E c c u + b cu *M c u + b cu *M2 c m a u + b au *M a u + b au *M2 c m e m a m a m e m M T T2 M2 umx function available!
Continuous data categorical data Moderation of means and variances Ordinal data Moderation of thresholds and variances See Medland et al. 2009
Non linear moderation? Extend the model from linear to linear + quadratic. E.g., e c + b ec *M + b ec2 *M 2
What about > number of moderators? Extend the model accordingly e c + b ec *SES + b ec2 *AGE With possible interaction: e c + b ec *SES + b ec2 *AGE + b ec3 *(AGE *SES)
Michel Nivard Practical Replicate findings from Turkheimer et al. with twin data from NTR Phenotype: FSIQ Moderator: SES in children cor(m,m2) = Data: 205 MZ and 225 DZ twin pairs 5 years old
Ok if #2: r(m,m2) = MZ = DZ = MZ = DZ = C c E u C u MZ= DZ=.5 C u E u C c e u + b eu *M A u A u e u + b eu *M2 c u + b cu *M c u + b cu *M2 c m a u + b au *M a u + b au *M2 c m M T T2 M2 SES = M
A C E A C E a+ X M c+ Y M e+ Z M a+ X M 2 c+ Y M 2 e+ Z M 2 m+ M M Twin Twin 2 m+ M M2 unx function available