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 = 1.5 + 0.95 η jt + ǫ j1t y j2t = 0 + 1 η jt + ǫ j2t E(η jt ) and var(η jt ) vary over time
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.
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 9.31 3.61 Slope 5.58 0.88 log(hhg) log(bhg) i 2 η hg Covariates s Mercury effect β p Intercept 0.01 0.99 Slope 0.43 0.10 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. (4 0.43 = 1.72)
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
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: 1986-87 1993-94 2000-2001 Children: 1022 917 856 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 = 0 + 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 = 0 + 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)
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, 371-402 Larsen, K. (2001). Analysis of questionnaire data from longitudinal studies. Ph.D thesis Department of Biostatistics, University of Copenhagen.