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
CRS1
Adp1
CRS0
Adp0
UXXXX
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 becauseE[Y|do(X=x)] = +x
A structural equation is true when the right side variables are observed AND when they are manipulated
2,0 NXY
1 2
U
X
Y
U
X
Y
. variablesexogenous
,C , variablesendogenous Y
XTwith
Y
X
Y
X
Y
X
11
0
0
Z
BIABI
CZTAZBIBAZ
XZY
ZX
yxyzyy
xzxx
Z
zx
xy
x
y
zy
Is the model identified ???
Cross-sectional and longitudinal effects
• Cross-sectional model (time 0)
• Model for changes
(changes are negatively correlated with baseline values)
• Longitudinal extension
000 XY C
jjLCj XXXY 00
0100 XXYY jLj
C
A
CRS0
Adp0
U X
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
0)cov
),0 iid , 2
(A,
N(AI kikikikk
k
i
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)
lklk
kkikk
iiAikik
ikikik
AVII
AVIVIE
AN(AN(
AI
AI
,cov
0
0 ,cov , ),0 iid , ),0 iid
1ncol(A))nline( :factor 1 centered
22
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)
ˆˆlog2
ˆ 1 StrN
L
)dim(logˆˆlog1ˆ
log2 1 SSStrNL
L
s
Estimation and tests
Variance of the estimator
Confidence intervals and Wald’s tests
2
2
ˆ
L
V
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
0
10
D
DDNFI
0,1
1max
ndf
F
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
Adp
%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
n,4 n,1 1,4 n,4
• Because the loadings are identical at both times, the same measurement model holds for the changes
2,0 jjjjj NAI
2
0101
01
01
0,N
A
j
A
AA
II
AA
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
ikik
k
ik
A
i
k
Ak
k
ikikikik
A
AIAI
*
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.
CRS1
Adp1
CRS0
Adp0
UXXXX
CRS1
Adp1
CRS0
Adp0
UXXXX