Developmental models

1
Developmental models
2
Multivariate analysis

• choleski models
• factor models
• y ? f u
• genetic factor models
• Pj h Gj c Cj e Ej
• common pathway
• independent pathway

3
Repeated measures

• developmental models
• simplex models
• growth models

4
Structural Equation Model

• measurement model
• relations between latent and observed variables
• y ? f u
• structural equation model
• relations among the latent variables
• f B f z

5
Longitudinal simplex pattern

• correlations fall away as time between occasions
  increase
• time 1 2 3 4 5
• time 1 1
• time 2 .69 1
• time 3 .44 .74 1
• time 4 .35 .61 .76 1
• time 5 .25 .44 .57 .73 1

6
Simplex model

• first-order auto-regressive process
• each latent variable is influenced by the
  preceding latent variable
• ft bt ft-1 zt
• b autoregressive coefficient
• z innovation

7
Path diagram

• units of measurement in the latent variables are
  the same as in the observed variables
• yi fi ui and fi bi fi-1 zi

8
First order autoregression(B matrix)

• regression coefficients (b2 to b4)
• 0 0 0 0
• b2 0 0 0
• 0 b3 0 0
• 0 0 b4 0
• no regression coefficient for time 1 (b1)

9
Translation into Mx

• measurement model Y F U
• structural model F B F Z
• F B F Z, therefore
• F - B F Z, therefore
• (I - B) F Z, therefore
• F (I - B)-1 Z
• Y (I - B)-1 Z U

10
Covariance structure

• Y (I - B)-1 Z U
• Covariance (Y)
• (I-B)-1 var(Z) ((I-B)-1 ) var(U)
• B contains autoregressive coefficients
• Z contains innovations
• U contains error variances
• I is identity matrix

11
Mx Longitudinal IQ data (4 ages)

• in baal\simplex
• 6 rectangular datafiles (raw data)
• iq4all.rec N 209 pairs
• iq4mzm/dzm/mzf/dzf/dos.rec
• missing -1
• 8 variables
• t1iq5 t1iq7 t1iq10 t1iq12 t2iq5 t2iq7 t2iq10
  t2iq12

12
Dutch IQ data (209 twin pairs)

• tested at ages 5, 7, 10 and 12 years
• not all twins participated at every occasion!
• missing values fit models to raw data
• stability (subdiagtwin1, superdiagtwin2)
• 5 7 10 12
• 5 1 .57 .62 .53
• 7 .70 1 .70 .61
• 10 .68 .73 1 .69
• 12 .65 .67 .76 1

13
2 Mx jobs

• 1factor Pfact4iq.mx
• simplex Psimp4iq.mx
• input file iq4all.rec

14
output variances

• ft bt ft-1 zt
• var(f t) bt2 var(f t-1 ) var(zt )
• variance of latent factor (twin 1)
• time 1 var(f1) var(z1) 11.62 134.1
• time 2 var(f2) 1.12134.1 4.62 162.2
• variance of observed variable (twin 1)
• time 1 var(f1) var(u1) 134.1 50.6 184.7
• time 2 var(f2) var(u2) 162.2 50.6 212.8

15
output covariances

• cov(f t , f t-1) btvar(f t-1 )
• cov (f2,f3) 1.0 162.2 162.2
• cov (f3,f4) 0.8 189.0 151.2
• cov(f t , f t-2) bt bt-1 var(f t-1 )
• cov (f1,f3) 1.0 1.1 134.1 147.5

16
output correlations

• cor(f t , f t-1) cov(f t ,f t-1)/?var(f t
  )?var(f t-1)
• cor (f2,f3) 162.2 / ?162.2 ?189.0 0.93
• cor (f3,f4) 151.2 / ?189.0 ?115.8 1.00
• cor (f1,f3) 147.5 / ?134.1 ?189.0 0.93

17
Extension to genetic designs

• For each source of variation (G, E and C (not
  shown here)) a simplex structure is specified

18
parameters of the model

• variances of latent factors (G, C and E) at t0
• variances of innovations at tgt0
• autoregressive coefficients (transmission)
• variances of measurement errors

19
Genetic simplex model

• G(t) bg(t) G(t-1) zg(t)
• E(t) be(t) E(t-1) ze(t)
• C(t) bc(t) C(t-1) zc(t)
• Variances of latent variables
• var(G(t)) bg(t)2 var(G(t-1)) var(zg(t))
• var(E(t)) be(t)2 var(E(t-1)) var(ze(t))
• var(C(t)) bc(t)2 var(C(t-1)) var(zc(t))

20
specifications

• var(G1) var(zg(1)) (innovation)
• var(G(t)) bg(t)2 var(G(t-1)) var(zg(t))
• cov(G(t),G(t-1)) bg(t)var(G(t-1))

21
Covariance matrix

• (t3, no measurement error)
• Genetic covariance
• Var(G1)
• bg2var(G1) Var(G2)
• bg2bg3var(G1) bg3var(G2) Var(G3)
• Environmental covariance
• Var(E1)
• be2var(E1) Var(E2)
• be2be3var(E1) be3var(E2) Var(E3)

22
To Mx!

• in baal\simplex 5 datafiles iq4mzm/dzm/mzf/dzf/
  dos.rec
• 8 variables (missing -1)
• t1iq5 t1iq7 t1iq10 t1iq12 t2iq5 t2iq7 t2iq10
  t2iq12
• Mx job guess.mx incomplete!
• For ages 5, 7 and 10 years only
• Full of mistakes
• fix, run for 3 ages, then adjust and run for 4
  ages.

23
Twin correlations (N)

• 5 7
  10 12
• MZM .77 (42) .56 (37) .73 (38) .87
  (30)
• DZM .53 (43) .41 (41) .54 (41) .62
  (33)
• MZF .77 (47) .78 (42) .87 (43) .87
  (36)
• DZF .73 (37) .50 (34) .46 (37) .60
  (31)
• DOS .65 (39) .56 (38) .50 (37) .24
  (34)

24
Structure MX job

• 3 groups to specify the genetic, unique
  environmental and common environmental simplex
  structures
• 2 data groups to describe the MZ and DZ
  covariance matrices
• MZ GCE GC _ GC GCE /
• DZ GCE 0.5_at_GC _ 0.5_at_GC GCE /