The Faroese Cohort 2. Structural Equation Models Latent growth models. Multiple indicator growth modeling. Response profiles. Number of children: 182

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1 The Faroese Cohort 2 Structural Equation Models Latent growth models Exposure: Blood Hg Hair Hg Age: Birth 3.5 years 4.5 years 5.5 years 7.5 years Number of children: 182 Multiple indicator growth modeling Response profiles General measurement model y j1t = ν 1t + λ 1t η jt + ǫ j1t y j2t = ν 2t + λ 2t η jt + ǫ j2t Measurement invariance y j1t = ν 1 + λ 1 η jt + ǫ j1t y j2t = ν 2 + λ 2 η jt + ǫ j2t y j1t = η jt + ǫ j1t y j2t = η jt + ǫ j2t E(η jt ) and var(η jt ) vary over time

2 Model for the joint distribution of the ηs Random coefficient approach Structural model j = i j + ζ 3.5j j = i j + 1s j + ζ 4.5j j = i j + 2s j + ζ 5.5j j = i j + 4s j + ζ 7.5j Each child has his own line Use well-known univariate models for longitudinal data, accounting for within subject correlation i j : j child s intercept (level at 3.5 years) N(α i, ψ i ) s j : j child s slope N(α s, ψ s ) cov(i j, s j ) = ψ is Note: Coefficients of latent variable i j and s j are known. Random coefficient model for Faroese data i Note: factor loadings on (i) and (s) are fixed. Loadings on random intercept (i) fixed at 1. s Random coefficient approach - effects of covariates The child s intercept and slope may depend on (time invariant) covariates (z j ) Structural model j = i j + ζ 3.5j j = i j + 1s j + ζ 4.5j j = i j + 2s j + ζ 5.5j j = i j + 4s j + ζ 7.5j i j = α i + γ i z j + ζ ij s j = α s + γ s z j + ζ sj Loadings on random slope (s) fixed respective at 0, 1, 2, 4.

3 Effects of time dependent covariates Random coefficient model for Faroese data Time varying covariates (w tj ) may influence the latent variable directly Structural model i Covariates s j = i j + + γ 3.5 w 3.5j + ζ 3.5j j = i j + 1s j + γ 4.5 w 4.5j + ζ 4.5j j = i j + 2s j + γ 5.5 w 5.5j + ζ 5.5j j = i j + 4s j + γ 7.5 w 7.5j + ζ 7.5j i j = α i + γ i z j + ζ ij s j = α s + γ s z j + ζ sj Random coefficient model - Results Adjustment for exposure error Random coefficients MEAN STD Intercept Slope log(hhg) log(bhg) i 2 η hg Covariates s Mercury effect β p Intercept Slope Effect of 10-fold increase Mercury has no effect on cognitive level at 3.5 years. Mercury affects learning curve. A higher exposure leads to a weaker slope. ( = 1.72)

4 Alternative approach: Autoregressive model Simpler example: bone mineral density in girls A total of 112 girls at age 12 were randomized to calcium treatment or placebo. The response was bone mineral density (bmd) in g/cm 2. The girls were examined 5 times in approximately 6 months intervals. For the purposes of this exercise we regard the visits as equally spaced. Thus, we will assumed that the girls were measured at time 0, 1, 2, 3 and 4. Scatter plots show that response profiles for treated and control subjects can be assumed to be linear. The model: random intercept and slope Measurement part: bmd1 j = η 0j + ǫ 1j bmd2 j = η 0j + 1 η sj + ǫ 2j bmd3 j = η 0j + 2 η sj + ǫ 3j bmd4 j = η 0j + 3 η sj + ǫ 4j bmd5 j = η 0j + 4 η sj + ǫ 5j Structural part: η 0j = α 0 + γ 0 group j + ζ 0j η sj = α 1 + γ 1 group j + ζ 1j data(bmd) bmd$bmd1<-bmd$bmd1*10 bmd$bmd2<-bmd$bmd2*10 bmd$bmd3<-bmd$bmd3*10 bmd$bmd4<-bmd$bmd4*10 bmd$bmd5<-bmd$bmd5*10 In lava m0 <- lvm() regression(m0) <- c(bmd1,bmd2,bmd3,bmd4,bmd5)~eta0 regression(m0) <- c(bmd2,bmd3,bmd4,bmd5)~etas latent(m0) <- ~ETA0+ETAS intercept(m0, ~bmd1+bmd2+bmd3+bmd4+bmd5)<-0 regression(m0, bmd2~etas)<-1... Do exercise 3.3

5 The Faroese Cohort 1 Path diagram illustrating longitudinal model for motor outcomes Age: EXPOSURE: 1. Cord Blood Mercury 2. Maternal Hair Mercury 3. Maternal Seafood Intake Birth RESPONSE: Neuropsychological Tests 7 Years 14 Years ǫ H Hg log(h-hg) ǫ log(b-hg) B Hg η 0 ζ0 Covariates HEC ǫ ζ HEC 7 F T 1 ǫ F T 1 7 η 7 3 F T 2 ǫf T 2 η 14 ζ 14 F T 3 ǫf T 3 F T 1 ǫ F T 1 F T 2 ǫ F T 2 F T 3 ǫ F T 3 CATSYS1 ǫ CATSYS1 CATSYS2 ǫ CATSYS2 Calendar: Children: Modeling the covariance of the error terms HEC ǫ HEC ζ 7 FT 1 ǫft1 η 7 3 FT 2 ǫft2 FT 3 ǫft3 FT 1 ǫft1 FT 2 ǫft2 η 14 FT 3 ǫft3 ζ 14 CATSYS1 ǫ CATSYS1 CATSYS2 ǫ CATSYS2 } local dependence correlation in same outcome at different time points Measurement invariance: Selected equations - Measurement part F T 1 = η 7 +ǫ F T 1 F T 2 = ν F T 2 + λ F T 2 η 7 +ǫ F T 2 F T 3 = ν F T 3 + λ F T 3 η 7 +ǫ F T 3 F T 1 = η 14 + ǫ F T 1 F T 2 = ν F T 2 + λ F T 2 η 14 + ǫ F T 2 F T 3 = ν F T 3 + λ F T 3 η 14 + ǫ F T 3 λ F T 2 = λ F T 2, λ F T 3 = λ F T 3 and ν F T 2 = ν F T 2, ν F T 3 = ν F T 3 Error distribution: ǫ i N(0, Ω)

6 References Latent growth model Muthén, B, Curran PJ. (1997). General Longitudinal Modeling of Individual Differences in Experimental Designs: A Latent Variable Framework for Analysis and Power Estimation. Psychological Methods, Vol. 2, No. 4, Larsen, K. (2001). Analysis of questionnaire data from longitudinal studies. Ph.D thesis Department of Biostatistics, University of Copenhagen.