Standard genetic simplex models in the classical twin design with phenotype to E transmission Conor Dolan & Janneke de Kort Biological Psychology, VU 1

2 Two general approaches to longitudinal modeling Markov models: (Vector) autoregressive models for continuous data (Hidden) Markov transition models discrete data (Mixtures thereof) Growth curve models (Brad Verhulst, Lindon Eaves): Focus on linear and non-linear growth curves Typically multilevel or random effects model (Mixtures thereof) Which to use? Use the model that fit the theory / data / hypotheses

y1 y2y3y4 x1x2x3x4 xx xx xx e1e1 e4e4 e3e3 e2e b2,1 b3,2b4,3 First order autoregression model. A (quasi) simplex model (var(e)>0). 1 b 01 b 02 b 03 b 04

y1 y2y3y4 x1x2x3x4 xx xx xx e1e1 e4e4 e3e3 e2e b2,1 b3,2b4,3 y ti = b 0t + x ti + e ti x 1i = x 1i =  x 1i x ti = b t-1,t x t-1i +  x ti var(y 1 ) = var(x 1 ) + var(e 1 ) var(x t ) = b t-1,t 2 var(x t ) + var(  x t ) cov(x t,x t-1 ) = b t-1,t var(x t ) First order autoregression model. A quasi simplex model (var(e)>0).

y1 y2y3y4 x1x2x3x4 xx xx xx e1e1 e4e4 e3e3 e2e b2,1 b3,2b4,3 Identification issue: var(e 1 ) and var(e t ) are not identified. Solution set to zero, or equate var(e 1 ) = var(e 2 ), var(e 3 ) = var(e 4 ) First order autoregression model. A quasi simplex model (var(e)>0).

var(y 1 ) = var(x 1 ) + var(e 1 ) var(x t ) = b t-1,t 2 var(x t ) + var(  x t ) cov(x t,x t-1 ) = b t-1,t var(x t ) Reliability at each t, rel(t) : rel(x 1 ) = var(x 1 ) / {var(x 1 ) + var(e 1 )} Interpretation: % of variance in y at t due to latent x at t Standardized stats I:

var(y 1 ) = var(x 1 ) + var(e 1 ) var(x t ) = b t-1,t 2 var(x t ) + var(  x t ) cov(x t,x t-1 ) = b t-1,t var(x t ) Stability at each t,t-1, stab(t,t-1): b t-1,t 2 var(x t ) / {b t-1,t 2 var(x t ) + var(  x t )} Interpretation: % of the variance in x at t explained by regression on x at t-1 (latent level!) Standardized stats II:

var(y 1 ) = var(x 1 ) + var(e 1 ) var(x t ) = b t-1,t 2 var(x t ) + var(  x t ) cov(x t,x t-1 ) = b t-1,t var(x t ) Correlation t,t-1, cor(t,t-1): b t-1,t 2 var(x t ) / {sd(y t-1 ) * sd(y t )} sd(y t ) = sqrt(var(x t ) + var(e t )) var(x t ) = b t-1,t 2 var(x t ) + var(  x t ) Interpretation: strength of linear relationship Standardized stats III:

y1 y2y3y4 x1x2x3x4 xx xx xx e1e1 e4e4 e3e3 e2e

y1 y2y3y4 x1x2x3x4 e1e1 e4e4 e3e3 e2e b2,1 b3,2b4,3 Special case: factor model var(  x t ) (t=2,3,4) = 0 y1 y2y3y4 x1 e1e1 e4e4 e3e3 e2e b 2,1 b 2,1 b 3,2 b 2,1 b 3,2 b 4,3

var(  x t ) (t=2,3,4) = 0

12  ph =  A +  C +  E Multivariate decomposition of phenotypic covariance matrix:  ph1  ph12  ph2  =  A +  C +  E r  A +  C +  E  A +  C +  E

13  ph =  A +  C +  E Estimate  A using a Cholesky-decomp  A =  A  A t  A =                       

14  ph =  A +  C +  E Model  A a simplex model  A =  A  A  A t  A

15  A =  A  A  A  t  A  A = 0000 b A b A b A43 0

16  A =  A  A  A  t  A  A = var(A 1 )000 0var(  A2 )00 00 var(  A3 )0 000var(  A4 )

17  A =  A  A  A  t  A  A = var(a 1 )000 0var(a 2 )00 00 var(a 3 )0 000var(a 4 )

18 A1 A2A3A4 A1A2A3A4 A2A2   A4A4 a1a4 a3a b A2,1 b A3,2 b A4,3 The genetic simplex (note my scaling)

y11 E1 1 y21 E2 1 C1 y12 E1 2 y22 E2 2 C2 y13 E1 3 y23 E2 3 C3 y14 E1 4 y24 E2 4 C4 A1 1 A1 2 A1 3 A1 4 A2 1 A2 2 A2 3 A2 4 AA AA AA AA AA AA EE EE EE EE EE EE CC CC CC

AA y12 E1 2 y22 E2 2 C2 A1 2 A2 2 AA EE EE CC a2 c2 e2 a2 c2 e2 Occasion specific effects

21 Question: h 2, c 2, and e 2 at each time point? var(y t ) = {var(A t ) + var(a t )} + {var(C t ) + var(c t )} + {var(E t ) + var(e t )} h 2 = {var(A t ) + var(a t )} / var(y t ) c 2 = {var(C t ) + var(c t )} / var(y t ) e 2 = {var(E t ) + var(e t )} / var(y t ) y12 E1 2 C2 A1 2 AA EE CC a2 c2 e2

22 y13 E1 3 C3 y14 E1 4 C4 A1 3 A1 4 AA AA EE EE CC CC Question: contributions to stability t-1 to t b At-1,t 2 var(A t ) / {b At-1,t 2 var(A t ) + var(  A t )+var(a t )} b Ct-1,t 2 var(C t ) / {b Ct-1,t 2 var(C t ) + var(  C t )+var(c t } b Et-1,t 2 var(E t ) / {b Et-1,t 2 var(E t ) + var(  E t )+var(e t )}

23 Question: contributions to stability t-1 to t {b At-1,t 2 var(A t )+ b Ct-1,t 2 var(C t )+ b Et-1,t 2 var(E t )} [{b At-1,t 2 var(A t ) + var(  A t )+var(a t )} + {b Ct-1,t 2 var(C t ) + var(  C t )+var(c t } + {b Et-1,t 2 var(E t ) + var(  E t )+var(e t )}] Decompose the phenotypic covariance into A,C,E components

24 Hottenga, etal. Twin Research and Human Genetics, 2005

Birley et al. Behav Genet 2005 (alternative: growth curve modeling)

26 Anx/dep stability due to A and E from 3y to 63 years Nivard et al, 2014

27 Variations on the theme

y11 E1 1 y21 E2 1 C1 y12 E1 2 y22 E2 2 C2 y13 E1 3 y23 E2 3 C3 y14 E1 4 y24 E2 4 C4 A1 1 A1 2 A1 3 A1 4 A2 1 A2 2 A2 3 A2 4 AA AA AA AA AA AA EE EE EE EE EE EE CC CC CC Phenotype-to-phenotype transmission (Eaves etal. 1986)

y11 E1 1 y21 E2 1 C1 y12 E1 2 y22 E2 2 C2 y13 E1 3 y23 E2 3 C3 y14 E1 4 y24 E2 4 C4 A1 1 A1 2 A1 3 A1 4 A2 1 A2 2 A2 3 A2 4 AA AA AA AA AA AA EE EE EE EE EE EE CC CC CC Sibling “interaction” mutual direct phenotypic influence (Eaves, 1976; Carey, 1986)

y E y E C y E y E C y E y E C y E y E C AAAA AAAA aaa aaa e e ee ee ccc 30 Niching picking (Eaves et al., 1977 )

31 Niching picking (Eaves et al., 1977 )

32 “Niche-picking” During development children seek out and create and are furnished surrounding (E) that fit their phenotype. A smart child growing up will pick the niche that fits her/her phenotypic intelligence. A anxious child growing up may pick out the niche that least aggrevates his / her phenotypic anxiety. Phenotype of twin 1 at time t -> environment of twin 1 at time t+1 (parameters  t )

33 Mutual influences During development children’s behavior may contribute to the environment of their siblings. A smart child growing up will pick the niche that fits her/her phenotypic intelligence and in so doing may influence (contriibute to) the environment of his or her sibling. A behavior of an anxious child may be a source of stress for his or her siblings. Phenotype of twin 1 at time t -> environment of twin 2 at time t+1 (parameters  t )

ACE simplex T=4 Identification in ACE simplex with no additional constraints. Except: Occasion specific residual variance decomposition: Var[y(t)|A(t), E(t), C(t)] = var[  (t)] Var[  (t)] = var[a(t)]+var[e(t)], t=1,…,4 34

Identification #1 T=4 ph ti  E (t+1)j (i=j) ph ti  E (t+1)j (i≠j)  1,  2,  3  1,  2,  3 (not ID)  1,  2 =  3  1,  2 =  3  k =b 0  +(k-1)*b 1   k =b 0  +(k-1)*b 1  (k=1,2,3) but  1 =  2,  3  1 =  2,  3 (not ID) Presence of C not relevant to this results Good…..? 35

36

N required given plausible values ACE  1,  2 =  3 &  1,  2 =  3  k  k ~N (power=.80) Hypothesis:  1 =  2 =  3 =0 &  1 =  2 =  3 =0 (4df) 37 Good....?

Not Good!

N required given plausible values AE  1,  2 =  3 &  1,  2 =  3  k  k ~N (power=.80) Hypothesis:  1 =  2 =  3 =0 &  1 =  2 =  3 =0 (4df) Good...?

True AE+  k &  k (Nmz=Ndz=1000)  k b k ACE simplex df= (approximate model equivalence) Good...?

y*11 E1 1 y*21 E2 1 y*12 E1 2 y*22 E2 2 y*13 E1 3 y*23 E2 3 y*14 E1 4 y*24 E2 4 A1 1 A1 2 A1 3 A1 4 A2 1 A2 2 A2 3 A2 4 AA AA AA AA AA AA EE EE EE EE EE EE 1 1 11 22 33 33 22 11 11 11 22 33 33 22 1 1

43 The proportions of observed FSIQ data 0.812, 0.295, 0.490, (MZ twin 1) 0.812, 0.295, 0.490, (MZ twin 2) 0.774, 0.379, 0.598, (DZ twin1) 0.774, 0.379, 0.598, (DZ twin 2) 261 MZ and 301 DZ twin pairs. Full scale IQ. mean (std) ages 5.5y (.30), 6.8y (.19), 9.7y (.43), and 12.2y (.24).

44 5.5y6.8y9.7y12.2y MZ FSIQ correlation DZ FSIQ correlation Red: MZ stdevs Green: DZ stdevs

45 Fitted standard simplex Var(a t ) = zero time specific A zero Var(  A3 ) = var(  A4 ) = 0A inno at t=3,4 zero Chi2(63) = 76.8, p= 0.11

46 FSIQ 5.5y6.8y9.7y12.2y A variance h 2 =.27h 2 =.48h 2 =.60h 2 =.54

47 E variance e 2 =.23 e 2 =.29e 2 =.20e 2 =.22

48 C variance c 2 =.50c 2 =.23c 2 =.20c 2 =.24

49 OPEN ACCESS Dolan, C.V, Janneke M. de Kort, Kees-Jan Kan, C. E. M. van Beijsterveldt, Meike Bartels and Dorret I. Boomsma (2014). Can GE-covariance originating in phenotype to environment transmission account for the Flynn effect? Journal of Intelligence. Submitted. Dolan, C.V., Johanna M. de Kort, Toos C.E.M. van Beijsterveldt, Meike Bartels, & Dorret I. Boomsma (2014). GE Covariance through phenotype to environment transmission: an assessment in longitudinal twin data and application to childhood anxiety. Beh. Gen. In Press.

y E y E y E y E y E y E y E y E AAAA AAAA AAA AAA E E EE EE 1 1 11 22 33 33 22 11 11 11 22 33 33 22 1 1

Anxious depression Twins aged 3,7,10,12 Netherlands Twin Register (NTR), which includes the Young NTR (YNTR; van Beijsterveldt, Groen-Blokhuis, Hottenga, et al., 2013) ASEBA CBCL instruments (Achenbach), maternal ratings Observed 89%, 54%, 45%, 37% (NMZ=3480) Observed 89%, 50%, 39%, 32% (NDZ=3145) 51

Phenotypic correlations FIML phenotypic twin correlations MZ:.71 (3),.58 (7),.58 (10), and.63 (12). DZ:.31 (3),.36 (7),.35 (10), and.40 (12). From 7y onwards looks like ACE model 52

Model loglnparAICBIC 1 ACE AE AE +  k,  k AE +  k ACE ph1->ph2 (2) ACE ph->ph (1)

y E y E y E y E y E y E y E y E AAAA AAAA aaa aaa e e ee ee 1 1 1 22 11 22 22 22  1 =0.123 (s.e..041) and  2 =  3 = (s.e..027)

55 Parameter estimates in the analysis of Anxiety from 3y to 12y. ML estimates and robust standard errors in parentheses in the AE simplex with parameter  k, logl = ). t=1 (3y)t=2 (7y)t=3 (10y)t=4 (12y) b At,t (.062)0.886 (.099)0.790 (.100)  At  b Et,t (.182)1.017 (.232)0.799 (.125)  Et   a t   e t   1   2  3 

56 A1E A E A Correlated E – But not C! E G-E covariance

Identification #2 T=4 b A2,1 =b A3,2 =b A4,3 OR b C2,1 =b C3,2 =b C4,3 OR b E2,1 =b E3,2 =b E4,3 OR (of course) b C2,1 =b C3,2 =b C4,3 = b E2,1 =b E3,2 =b E4,3 ph ti  E (t+1)j (i=j) ph ti  E (t+1)j (i≠j)  1,  2,  3  1,  2,  3 57