Genetic simplex model in the classical twin design. Conor Dolan & Sanja Franic. Boulder Workshop 2016
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1 Genetic simplex model in the classical twin design Conor Dolan & Sanja Franic Boulder Workshop 206 boulder 206 dolan & franic simplex model
2 Two general approaches to longitudinal modeling (not mutually exclusive) Markov models: (Vector) autoregressive models for continuous data (Hidden) Markov transition models discrete data Growth curve models: Focus on linear and non linear growth curves Typically multilevel or random effects model Which to use? Use the model that fit the theory / data / hypotheses boulder 206 dolan & franic simplex model 2
3 Growth curve modeling? If you re interested in growth trajectories. Linear or non linear: Autoregressive modeling? If you re mainly interested in stability. Can be combined (this afternoon) boulder 206 dolan & franic simplex model 3
4 First order autoregression model. A (quasi) simplex model (var(e)>0). x x x b2, b3,2 b4,3 x x2 x3 x4 y y2 y3 y4 e e2 e3 e4 b 0 b 02 b 03 b 04 boulder 206 dolan & franic simplex model 4
5 First order autoregression model. A quasi simplex model (var(e)>0). var(x ) true score var(e ) error y ti = b 0t + x ti + e ti x ti = b t,t x t i + x ti var(y t ) = var(x t ) + var(e t ) var(x t ) = b t,t2 var(x t ) + var( x t ) cov(x t,x t ) = b t,t var(x t ) cov(y t,y t ) = b t,t var(x t ) boulder 206 dolan & franic simplex model 5
6 First order autoregression model. var(e ) var(e 4 ) Identification issue: var(e ) and var(e t ) are not identified. Solution set to zero, or equate var(e ) = var(e 2 ), var(e 3 ) = var(e 4 ) boulder 206 dolan & franic simplex model 6
7 var(y t ) = var(x t ) + var(e t ) var(x t ) = b t,t2 var(x t ) + var( x t ) cov(x t,x t ) = b t,t var(x t ) cov(y t,y t ) = b t,t var(x t ) Standardized stats part I: Reliability at each t, rel(y t ) : rel(y t ) = var(x t ) / {var(x t ) + var(e t )} Interpretation: % of variance in y t due to latent x t boulder 206 dolan & franic simplex model 7
8 var(y t ) = var(x t ) + var(e t ) var(x t ) = b t,t2 var(x t ) + var( x t ) cov(x t,x t ) = b t,t var(x t ) cov(y t,y t ) = b t,t var(x t ) Standardized stats part II: Stability at level of X, stab(x t,x t ): b t,t2 var(x t ) / {b t,t2 var(x t ) + var( x t )} Interpretation: % of the variance in x t due to x t boulder 206 dolan & franic simplex model 8
9 var(y t ) = var(x t ) + var(e t ) var(x t ) = b t,t2 var(x t ) + var( x t ) cov(x t,x t ) = b t,t var(x t ) cov(y t,y t ) = b t,t var(x t ) Standardized stats part III: Correlation t,t, cor(t,t ): b t,t var(x t ) / {sd(y t ) * sd(y t )} sd(y t ) = (var(x t ) + var(e t )) var(x t ) = b t,t var(x t ) + var( x t ) Interpretation: strength of linear relationship boulder 206 dolan & franic simplex model 9
10 x x x x x2 x3 x4 y y2 y3 y4.25 e e e 3 e 4 Covariance matrix Correlation matrix boulder 206 dolan & franic simplex model 0
11 reliability: rel(x t ) = var(x t ) / {var(x t ) + var(e t )} = / (+.25) = /.25 =.8 R 2 : b t,t2 var(x t ) / {b t,t2 var(x t ) + var( x t )} = {.7 2 * } / (.7 2 * +.5) =.49/ =.49 cor(t,t+) : b t,t var(x t ) / {sd(y t ) * sd(y t )} ={.7 * } / {.25*.25} =.56 boulder 206 dolan & franic simplex model
12 0 0 0 x2 x3 x4 x x2 x3 x4 y y2 y3 y4 e e2 e3 e4 What happens if var( xt) = 0? boulder 206 dolan & franic simplex model 2
13 Special case: factor model var( x t ) (t=2,3,4) = 0 b 2, b 3,2 b 4,3 x x2 x3 x4 y y2 y3 y4 e e2 e3 e4 x y y2 y3 y4 e e2 e3 e4 boulder 206 dolan & franic simplex model 3
14 Multivariate decomposition of phenotypic covariance matrix (TxT, say T=4): ph = A + C + ph ph2 ph2 ph2 = A + C + r A + C + r A + C + A + C + (r= or.5) boulder 206 dolan & franic simplex model 4
15 ph = A + C + stimate A using a Cholesky decomp A = A A t A = boulder 206 dolan & franic simplex model 5
16 ph = A + C + Model A using a simplex model A = A A A t A boulder 206 dolan & franic simplex model 6
17 A = A A A t A A = b A b A b A43 0 boulder 206 dolan & franic simplex model 7
18 A = A A A t A A = var(a ) var( A2 ) var( A3 ) var( A4 ) boulder 206 dolan & franic simplex model 8
19 A = A A A A A = var(a ) var(a 2 ) var(a 3 ) var(a 4 ) required: var(a) = var(a2) var(a3) = var(a4) boulder 206 dolan & franic simplex model 9
20 The genetic A simplex boulder 206 dolan & franic simplex model 20
21 boulder 206 dolan & franic simplex model 2
22 Occasion specific effects required: var(a) = var(a2) var(a3) = var(a4) var(e) = var(e2) var(e3) = var(e4) var(c) = var(c2) var(c3) = var(c4) boulder 206 dolan & franic simplex model 22
23 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( 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( t ) + var(e t )} / var(y t ) boulder 206 dolan & franic simplex model 23
24 contributions to stability (A,C,) t to t b At,t2 var(a t ) / {b At,t2 var(a t ) + var( A t )} b Ct,t2 var(c t ) / {b Ct,t2 var(c t ) + var( C t )} b t,t2 var( t ) / {b t,t2 var( t ) + var( t )} boulder 206 dolan & franic simplex model 24
25 contributions of A to Phenotypic stability t to t b At,t2 var(a t ) {b At,t2 var(a t ) + var( A t )} + {b Ct,t2 var(c t ) + var( C t )} + {b t,t2 var( t ) + var( t )} boulder 206 dolan & franic simplex model 25
26 A = A = y = A + A = h 2 at t=? answer:.2 (e 2 =.8) h 2 at t=2? answer:.2 (r 2 =.8) correlation between A and A2?.84 / (.2*.2) =.92 correlation between and 2?. 4 / (.8*.8) =.50 covariance between Y and Y2?.584 contribution of A to covariance Y and Y2?.84/.584 =.35 contribution of to covariance Y and Y2?.4/.584 =.685 boulder 206 dolan & franic simplex model 26
27 Nivard et al, 204 Anx/dep stability due to A and from 3y to 63 years boulder 206 dolan & franic simplex model 27
28 Sanja s Practical: the genetic simplex model applied to FSIQ at 4 occasions. But first... Variations on the theme boulder 206 dolan & franic simplex model 28
29 Hottenga, etal. Twin Research and Human Genetics, 2005 boulder 206 dolan & franic simplex model 29
30 Birley et al. Behav boulder 206 Genet dolan & franic 2005 simplex model 30
31 Niche picking During development children seek out and create and are furnished surrounding () 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 at time t > environment of twin at time t+ boulder 206 dolan & franic simplex model 3
32 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 at time t > environment of twin 2 at time t+ boulder 206 dolan & franic simplex model 32
33 a a a A A A A y e y e y e y c c c C C C C e e e y y y y A A A A a a a A, C, uncorrelated boulder 206 dolan & franic simplex model 33
34 A A A A A A A y y y y y y y y A A A A A A A boulder 206 dolan & franic simplex model 34
35 boulder 206 dolan & franic simplex model 35
36 nvironments selected by genotypes (Scarr & McCartny, 983; Plomin, DeFries & Loehlin, 977) Sibling effects (Carey, 986, Behav Genet 6:39 34) boulder 206 dolan & franic simplex model 36
37 boulder 206 dolan & franic simplex model 37
38 boulder 206 dolan & franic simplex model 38
39 boulder 206 dolan & franic simplex model 39
40 boulder 206 dolan & franic simplex model 40
41 Mi chael boulder 206 dolan & franic simplex model 4
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