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

2. 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

3. CRS 1 Adp 1 CRS 0 Adp 0 U XX X X

4. 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

5. X Y U X Y Z  zx  xy xx yy  zy Is the model identified ???

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

7. CC AA CRS 0 Adp 0 U X

8. 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

9. 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

10. 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)

11. 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)

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

13. 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

14. 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)

15. 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)    

16. * 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

17. 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

19. 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

20. 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]

21. 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]

22. 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]

24. 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

25. 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.

30. CRS 1 Adp 1 CRS 0 Adp 0 U XX X X

31. CRS 1 Adp 1 CRS 0 Adp 0 U XX X X