Michel Chavance

INSERM U1018, CESP, Biostatistique

Use of Structural Equation Models to estimate longitudinal relationships

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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In 1) it is not a structural equation, because E[Y|do(X=x)] ≠ +x While in 2) it is a structural equation becauseE[Y|do(X=x)] = +x

A structural equation is true when the right side variables are observed AND when they are manipulated

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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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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 / height2

– WC = Waist Circumference– SSM = Sum of Skinfold Thicknesses (4 measurements)– PBF = % body fat (foot to foot bioimpedance analyzer)

• Cognitive restrained scale

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 Ii

k 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

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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 l1=1 (i.e. same scale for A and its 1st 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 (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

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Studied population in 1999mean (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)

Measurement model

1) 4 separate analyses by sex and time

2) 2 separate analyses (identical loadings at each time)

3) all subjects together

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%BF log(BMI) Log(SST) Log(WC)

* model with equality constraints

The same measurement model holds for both years, but not for both sexes

Males Females

1999 2001 Both Years*

1999 2001 Both Years

NFI 0.999 0.997 0.96 0.988 0.996 0.96

Measurement model for changes

• Measurement model at time j

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• 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 Estimate Standard Dev. Standardized Estimates

Baseline Change

%BF 1.000 - 0.955 0.603

log(BMI) 0.024 0.0007 0.956 0.996

log(SST) 0.055 0.0021 0.879 0.558

log(WC) 0.019 0.0006 0.938 0.647

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Structural Equation Model:Regression Coefficients (Females)

Baseline Adiposity

covariates sd CI95

Age 0.254 0.096 [0.07, 0.44]

Baseline CRS

0.051 0.020 [.012, .090]

Structural Equation Model:Regression Coefficients (Females)

Adiposity Change

covariates sd CI95

Adiposity0 -0.024 0.021 [-0.07, 0.02]

Age0 0.038 0.030 [-0.02, 0.10]

CRS0 -0.010 0.007 [-.04, 0.02]

CRS change -0.014 0.010 [-0.03, 0.01]

Structural Equation Model:Regression Coefficients (Females)

CRS Change

covariates sd CI95

Adiposity0 0.438 0.134 [0.17, 0.70]

Age0 0.023 0.200 [-0.37, 0.42]

CRS0 -0.286 0.042 [-0.37, -0.20]

Direct and Indirect Effects of Baseline CRS on Adiposity change

standard errors obtained by bootstrapping the sample 1,000 times

Estimate sd

1: direct -0.0096 0.0069

2: indirect through CRS change

0.0040 0.0031

3: indirect through baseline adiposity

-0.0012 0.0011

1+2 (partial) -0.0056 0.0064

1+2+3 (total) -0.0068 0.0064

• 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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