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Michel Chavance INSERM U1018, CESP, Biostatistique Use of Structural Equation Models to estimate longitudinal relationships

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Restrained Eating and weight gain Restrained 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

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CRS 1 Adp 1 CRS 0 Adp 0 U XX X X

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X Y U X Y In both cases, we observe In 1) it is not a structural equation, because E[Y|do(X=x)] ≠ + x While in 2) it is a structural equation because E[Y|do(X=x)] = + x A structural equation is true when the right side variables are observed AND when they are manipulated 12 U

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X Y U X Y Z zx xy xx yy zy Is the model identified ???

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Cross-sectional and longitudinal effects Cross-sectional model (time 0) Model for changes (changes are negatively correlated with baseline values) Longitudinal extension

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CC AA CRS 0 Adp 0 U X

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FLVS II study Fleurbaix Laventie Ville Santé Study (risk factors for weight and adiposity changes) 293/394 families recruited on a voluntary basis 2 measurements (1999 and 2001) 4 anthropometric measurements –BMI = weight / height 2 –WC = Waist Circumference –SSM = Sum of Skinfold Thicknesses (4 measurements) –PBF = % body fat (foot to foot bioimpedance analyzer) Cognitive restrained scale

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Structural Equation and Latent Variable models Latent variable : several observed variables are imperfect measurements of a single latent concept (e.g. for subject i, 4 indicators I i k of adiposity A i ) The measurement model postulates relationships between the unobserved value of adiposity A for subject i and its 4 observed measurements I k, and thus between the observed measurements

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Measurement model and factor analysis Identification problem: the parameters depend on the measurement scale of the latent variable A Usual solution : constraint l 1 =1 (i.e. same scale for A and its 1 st observed measurement)

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Estimation and tests Aim = modeling the covariance structure Maximum likelihood estimator (assuming normal distributions) with the predicted and S the observed covariance matrix Likelihood ratio test of compared to saturated model (deviance)

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Estimation and tests Variance of the estimator Confidence intervals and Wald’s tests

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Overal model fit Normed fit index (Bentler and Bonett, 1980) relative change when comparing deviances of model 1 (D 1 ) and model 0 assuming independence (D 0 ) RMSEA=Root Mean Squared Error Approximation measures a « distance » between the true and the model covariance matrices at the population level

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Studied population in 1999 mean (standard deviation) ** sex difference (p<0.01) *** sex difference (p<0.001) Similar findings in 2001 Beware the sign of the differences …….. Males (n=201)Females (n=256) Age***44.0 (4.9)42.4 (4.5) % body fat***23.0 (6.2)33.2 (7.1) BMI**25.7 (3.4)24.7 (4.6) Skinfold thickness***58.6 (25.2)75.0 (32.2) Waist circumference***91.6 (10.4)79.4 (11.7) CRS***26.9 (19.7)40.4 (21.3)

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Measurement model 1) 4 separate analyses by sex and time 2) 2 separate analyses (identical loadings at each time) 3) all subjects together Adp %BF log(BMI)Log(SST)Log(WC)

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* model with equality constraints The same measurement model holds for both years, but not for both sexes MalesFemales 19992001Both Years*19992001Both Years NFI0.9990.9970.960.9880.9960.96

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Measurement model for changes Measurement model at time j n,4 n,1 1,4 n,4 Because the loadings are identical at both times, the same measurement model holds for the changes

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Estimated Loadings of the Global Measurement Model (Females) Standardized coefficients Estimate Standard Dev.Standardized Estimates BaselineChange %BF1.000-0.9550.603 log(BMI)0.0240.00070.9560.996 log(SST)0.0550.00210.8790.558 log(WC)0.0190.00060.9380.647

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Structural Equation Model: Regression Coefficients (Females) Baseline Adiposity covariates sdCI 95 Age0.2540.096[0.07, 0.44] Baseline CRS 0.0510.020[.012,.090]

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Structural Equation Model: Regression Coefficients (Females) Adiposity Change covariates sdCI 95 Adiposity 0 -0.0240.021[-0.07, 0.02] Age 0 0.0380.030[-0.02, 0.10] CRS 0 -0.0100.007[-.04, 0.02] CRS change-0.0140.010[-0.03, 0.01]

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Structural Equation Model: Regression Coefficients (Females) CRS Change covariates sdCI 95 Adiposity 0 0.4380.134[0.17, 0.70] Age 0 0.0230.200[-0.37, 0.42] CRS 0 -0.2860.042[-0.37, -0.20]

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Direct and Indirect Effects of Baseline CRS on Adiposity change standard errors obtained by bootstrapping the sample 1,000 times Estimatesd 1: direct-0.00960.0069 2: indirect through CRS change 0.00400.0031 3: indirect through baseline adiposity -0.00120.0011 1+2 (partial)-0.00560.0064 1+2+3 (total)-0.00680.0064

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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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CRS 1 Adp 1 CRS 0 Adp 0 U XX X X

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CRS 1 Adp 1 CRS 0 Adp 0 U XX X X

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