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

Conor V Dolan & Sanja Franić (BioPsy VU)

Boulder Twin WorkshopMarch 4, 2014

Based on PPTs by Sophie van der Sluis & Marleen de Moor

Thanks to Neale family, Sarah Medland, & Brad Verhulst

Twin Research Volume 5 Number 6 pp. 554- 571

Acta Genet Med GemelloI36:5·20 (1987)

MxMCNeale et al

-4 -3 -2 -1 0 1 2 3

-4-2

02

46

x

yPhenotypicscores

Genetic level (score on A)

EnvironmentalDispersion

Variance ofE given G score

G x E as “genetic control” of sensitivity to different environments: heteroskedasticity

Not all heteroskedasticity is GxE!

Not all heteroskedasticity is GxE!

-4 -3 -2 -1 0 1 2 3

-4-2

02

46

x

yPhenotypicscores

Environmental level (score on E)

GeneticDispersion

ConditionalA variance

G x E as “environmental control” of genetic effects: heteroskedasticity.

Not all heteroskedasticity is moderation!

-4 -3 -2 -1 0 1 2 3

-4-2

02

46

x

yPhenotypicscores

Moderator level (score on moderator)

GeneticDispersion

and / or

Environmental dispersion

Moderation of effects (A,C,E) by measured moderator M: heteroskedasticity.

• 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

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

Continuous moderation Continuous moderation

Age as a moderator

A,C,E effects moderated by SES

SES ordinal or continuous

Continuous Moderators not amenable to multigroup approach

…

treat the moderator as continuous(OpenMx)

Standard ACE model

phenoTwin 1

A C E

phenoTwin 2

A C E

a ce

cea

1

b0

1

b0

11 or .5

Regression on unit: just a way to estimate the mean.

Standard ACE model + Main effect on Means

PhenoTwin 1

A C E

PhenoTwin 2

A C E

a ce

cea

1

b0+ βM M1

1

b0+βMM2

phenoTwin 11

b0+ βM M1

phenoTwin 11

βM

b0

M1

Actually: regression of Phenotype on M ...What is left is the residual, subject to ACE modeling.

Why a triangle? Fixed regressors

res1 res1

equivalent

Summary stats

• Means vector

• Covariance matrix (r = 1 or r=1/2)

22222

222

* ecacar

eca

mean1 mean2

Allowing for a main effect of X

• Means vector

• Covariance matrix (r = 1 or r=1/2)

iMi XmXm 21M

22222

222

* ecacar

eca

Standard ACE model + Effect on Means and a path

Twin 1

A C E

Twin 2

A C E

a+XM1c e c ea+XM2

Twin 1 Twin 21

m+ βM M1

1

m+ βM M2

M has main effect on mean + moderation of A effect

Twin 1

A C E

a+XM1c e

Twin 11

m+ βM M1

M1=0 -> mean= m & A effect = aM1=1 -> mean= m+ βM M1 & A effect = a+XM1

Standard ACE model + Effect on Means and a,c, & e paths

Twin 1

A C E

Twin 2

A C E

a+XM1

c+YM1

e+ZM1a+XM2

c+YM2

e+ZM2

Twin 1 Twin 21

m+MM1

1

m+MM2

• 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)

Twin 1

A C E

Twin 2

A C E

a+XM1

c+YM1

e+ZM1a+XM2

c+YM2

e+ZM2

Twin 1 Twin 21

m+MM1

1

m+MM2

Expected variances

Standard Twin Model: Var (P) = a2 + c2 + e2

Moderation Model: Var (P|M) = (a + βXM)2 + (c + βYM)2 + (e + βZM)2

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

Var (P|M) = (a + βXM)2 + (c + βYM)2 + (e + βZM)2

h2 |M = (a + βXM)2 / Var (P|M)

c2 |M = (c + βyM)2 / Var (P|M)

e2 |M = (e + βyM)2 / Var (P|M)

(h2|M + c2|M + e2|M ) = 1 Standardized conditional on value of M

Turkheimer study SESModeration of unstandardized variance components (top)

Moderation of standardized variance components (bottom)

Twin 1

A C E

Twin 2

A C E

a+XM1

c+YM1

e+ZM1a+XM2

c+YM2

e+ZM2

Twin 1 Twin 21

m+MM1

1

m+MM2

1/.5 1

But what have assumed concerning M?

Fixed regressor

x y e

Assumption: y|x*~N(my|x, sy|x)my|x* = β0 + β1x*sy|x* = se

β1

1β0

Fixed regressor

x y e

Assumption: y|x*~N(my|x, sy|x)my|x* = β0 + β1x*sy|x* = se

β1

1β0

A

C

E

Random regressor

y

Assumption: (y,x)~N(m, S)

x

1β0

β1

“Random regressor not to be confused withRandom effects regression (multilevel modeling)”

Bivariate distribution of X and Y

Assumption: (y,x)~N(m, S)

x

A C E

y

A C E

Regression of Y on X in terms of latent variables

M1,M2 (co)variance is not included in the model. M is a “fixed regressor”.

Phenotype = BMIModerator = Age .... A fixed regressor

Phenotype = BMIModerator = Intelligence ....Can we treat Intelligence as a fixed fixedIn this context? Depends... on ...

Ec

Eu

T1

au + bau*M1

Cc

M1

am

ac

AcCu

Au

cc

ec

cmem

cu + bcu*M1

eu + beu*M1

T2

Au

Cu

Eu

eu + beu*M2

cu + bcu*M2

au + bau*M2

M2

Cc

Ec

Ac

am em

cm

e c

c c

a c

MZ=1 / DZ=.5

MZ=1 DZ=.5

MZ = DZ = 1

MZ = DZ = 1

M as a random regressor, with its own ACE + ACE cross loadings.

Can we treat M as a fixed effect in this manner?

Twin 1

A C E

Twin 2

A C E

a+XM1

c+YM1

e+ZM1a+XM2

c+YM2

e+ZM2

Twin 1 Twin 21

m+MM1

1

m+MM2

1/.5 1

Not generally

Eu

T1

au + bau*M1

Cc

M1

am

ac

Ec

AcCu

Au

cc

ec

cm

em

cu + bcu*M1

eu + beu*M1

T2

Au

Cu

Eu

eu + beu*M2

cu + bcu*M2

au + bau*M2

M2

Cc

Ec

Ac

am em

cm

e c

c c

a c

MZ=1 / DZ=.5

MZ=1 DZ=.5

MZ = DZ = 1

MZ = DZ = 1

OK if #1: ac/am = cc /cm = ec / em

Eu

T1

au + bau*M1

M1

Ec

Cu

Au

ec em

cu + bcu*M1

eu + beu*M1

T2

Au

Cu

Eu

eu + beu*M2

cu + bcu*M2

au + bau*M2

M2

Ec

eme c

MZ=1 DZ=.5

MZ = DZ = 1

OK if #2: r(M1,M2) = 0

Eu

T1

au + bau*M1

Cc

M1

Cu

Au

cccm

cu + bcu*M1

eu + beu*M1

T2

Au

Cu

Eu

eu + beu*M2

cu + bcu*M2

au + bau*M2

M2

Cc

cm

c c

MZ=1 DZ=.5

MZ = DZ = 1

MZ = DZ = 1

Ok if #3: r(M1,M2) = 1

Eu

T1

au + bau*M1

Cc

M1

am

ac

Ec

AcCu

Au

cc

ec

cm

em

cu + bcu*M1

eu + beu*M1

T2

Au

Cu

Eu

eu + beu*M2

cu + bcu*M2

au + bau*M2

M2

Cc

Ec

Ac

am em

cme c

c c

a c

MZ=1 / DZ=.5

MZ=1 DZ=.5

MZ = DZ = 1

MZ = DZ = 1

What to do otherwise?

MZ: T1=β0,MZ + β1,MZ*M1 + β2,MZ*M2

T2=β0,MZ + β1,MZ*M2 + β2,MZ*M1

DZ: T1=β0,DZ + β1,DZ*M1 + β2,DZ*M2

T2=β0,DZ + β1,DZ*M2 + β2,DZ*M1

E C A

T1

e + βe*M1a + βa*M1

c + βc*M1

A C E

T2

a + βa*M2 e + βe*M2

c + βc*M2

1 / .5

1

1

β0k + β1k*M1 + β2k*M2

1

β0k + β1k*M2 + β2k*M1

But what have assumed concerning the covariance between M and T?

E C A

T1

e + βe*M1a + βa*M1

c + βc*M1

A C E

T2

a + βa*M2 e + βe*M2

c + βc*M2

1 / .5

1

1

β0k + β1k*M1 + β2k*M2

1

β0k + β1k*M2 + β2k*M1

Eu

T1

au + bau*M1

Cc

M1

am

ac

Ec

AcCu

Au

cc

ec

cm

em

cu + bcu*M1

eu + beu*M1

T2

Au

Cu

Eu

eu + beu*M2

cu + bcu*M2

au + bau*M2

M2

Cc

Ec

Ac

am em

cme c

c c

a c

MZ=1 / DZ=.5

MZ=1 DZ=.5

MZ = DZ = 1

MZ = DZ = 1

Eu

T1

au + bau*M1

Cc

M1

am

ac + b

ac *M1

Ec

AcCu

Au

cc + b

cc *M1ec + b

ec*M1cm

em

cu + bcu*M1

eu + beu*M1

T2

Au

Cu

Eu

eu + beu*M2

cu + bcu*M2

au + bau*M2

M2

Cc

Ec

Ac

am em

cm

e c + b ec

*M2c c + b cc

*M2

a c +

b ac*M

2

MZ=1 / DZ=.5

MZ=1 DZ=.5

MZ = DZ = 1

MZ = DZ = 1

Eu

T1

au + bau*M1

Cc

M1

am

ac + b

ac *M1

Ec

AcCu

Au

cc + b

cc *M1ec + b

ec*M1cm

em

cu + bcu*M1

eu + beu*M1

T2

Au

Cu

Eu

eu + beu*M2

cu + bcu*M2

au + bau*M2

M2

Cc

Ec

Ac

am em

cm

e c + b ec

*M2c c + b cc

*M2

a c +

b ac*M

2

MZ=1 / DZ=.5

MZ=1 DZ=.5

MZ = DZ = 1

MZ = DZ = 1

M has to be treated as random and modeled appropriately (as shown)!

Conclusion: Use standard fixed regression approach ifCov(M,T) equally due to A,C,ECov(M1,M2) = zeroCov(M1,M2) = one Use extended fixed regression approach ifCross paths are not moderated Otherwise use full model.

A simpler conclusion:

ALWAYS USE FULL MODEL UNLESSCor(M1,M2) = 1.

categorical data• Continuous 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?

ec + bec1*M1 + bec2*M12

What about >1 number of moderatorsExtend the model accordingly

ec + bec1*SES + bec2*AGE

What about >1 number of moderators and interaction Effects (sex X age)Extend the model accordingly...

ec + bec1*SES + bec2*AGE +

bec2*(AGE *SES)

What about power given such extensions?

Do power calculation.... Exactly, by simulation, by exact simulation ?

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

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

Genetic control of exposure to the environment? “short hand”

Chain of causality?

A “causes” high IQ

high IQ “causes” interest in astronomyjoin the astronomy clubstudy astronomy