Michel Chavance
INSERM U1018, CESP, BiostatistiqueUse of Structural Equation Models to estimate longitudinal relationships
Restrained Eating and weight gainRestrained eating = tendency to consciously restrain food intake to control body weight or promote weight loss
Positive association between Restrained Eating and fat mass
Paradoxical hypothesis : induction of weight gain through frequent episodes of loss of control and dishinibited eating
CRS1Adp1CRS0Adp0UXXX
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XYUXYIn both cases, we observe
In 1) it is not a structural equation, because E[Y|do(X=x)] a+bx While in 2) it is a structural equation becauseE[Y|do(X=x)] = a+bx
A structural equation is true when the right side variables are observed AND when they are manipulated
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XYUXY
ZazxbxyexeyazyIs the model identified ???
Cross-sectional and longitudinal effectsCross-sectional model (time 0)
Model for changes
(changes are negatively correlated with baseline values)Longitudinal extension
DCDACRS0Adp0UX
FLVS II studyFleurbaix Laventie Ville Sant Study (risk factors for weight and adiposity changes)293/394 families recruited on a voluntary basis2 measurements (1999 and 2001)4 anthropometric measurementsBMI = weight / height2WC = Waist CircumferenceSSM = Sum of Skinfold Thicknesses (4 measurements)PBF = % body fat (foot to foot bioimpedance analyzer)Cognitive restrained scale
Structural Equation and Latent Variable modelsLatent variable : several observed variables are imperfect measurements of a single latent concept (e.g. for subject i, 4 indicators Iik of adiposity Ai)The measurement model
postulates relationships between the unobserved value of adiposity A for subject i and its 4 observed measurements Ik, and thus between the observed measurements
Measurement model and factor analysis
Identification problem: the parameters depend on the measurement scale of the latent variable AUsual solution : constraint l1=1 (i.e. same scale for A and its 1st observed measurement)
Estimation and testsAim = modeling the covariance structureMaximum likelihood estimator (assuming normal distributions)
with S(q) the predicted and S the observed covariance matrixLikelihood ratio test of compared to saturated model (deviance)
Estimation and tests Variance of the estimator
Confidence intervals and Walds tests
Overal model fitNormed fit index (Bentler and Bonett, 1980) relative change when comparing deviances of model 1 (D1) and model 0 assuming independence (D0)
RMSEA=Root Mean Squared Error Approximation measures a distance between the true and the model covariance matrices at the population level
Studied population in 1999mean (standard deviation)
** sex difference (p
Measurement model
1) 4 separate analyses by sex and time 2) 2 separate analyses (identical loadings at each time) 3) all subjects togetherAdp%BFlog(BMI)Log(SST)Log(WC)l1=1l4l3l2
* model with equality constraints
The same measurement model holds for both years, but not for both sexes
MalesFemales19992001Both Years*19992001Both YearsNFI0.9990.9970.960.9880.9960.96
Measurement model for changesMeasurement model at time j
n,4 n,1 1,4 n,4Because the loadings are identical at both times, the same measurement model holds for the changes
Estimated Loadings of the Global Measurement Model (Females)
Standardized coefficients
EstimateEstimateStandard Dev.Standardized EstimatesBaselineChange%BF1.000-0.9550.603log(BMI)0.0240.00070.9560.996log(SST)0.0550.00210.8790.558log(WC)0.0190.00060.9380.647
Structural Equation Model:Regression Coefficients (Females)Baseline Adiposity
covariatesbsdCI95Age0.2540.096[0.07, 0.44]Baseline CRS0.0510.020[.012, .090]
Structural Equation Model:Regression Coefficients (Females)Adiposity Change
covariatesbsdCI95Adiposity0-0.0240.021[-0.07, 0.02]
Age00.0380.030[-0.02, 0.10]CRS0-0.0100.007[-.04, 0.02]CRS change-0.0140.010[-0.03, 0.01]
Structural Equation Model:Regression Coefficients (Females)CRS Change
covariatesbsdCI95Adiposity00.4380.134[0.17, 0.70]
Age00.0230.200[-0.37, 0.42]CRS0-0.2860.042[-0.37, -0.20]
Direct and Indirect Effects of Baseline CRS on Adiposity change
standard errors obtained by bootstrapping the sample 1,000 times
Estimatesd1: direct-0.00960.00692: indirect through CRS change0.00400.00313: indirect through baseline adiposity-0.00120.00111+2 (partial)-0.00560.00641+2+3 (total)-0.00680.0064
Often useful to model the changes rather than the successive outcomes.
Structural equation modeling = translation of a DAG, but some models are not identified.
We still need to assume that all confounders of the effect of interest are observed.
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