A bridge between PLS path
modeling and ULS-SEM
Michel Tenenhaus
1
A SEM tree
SEM
Component-based SEM
(Score computation)
Herman Wold
NIPALS (1966)
PLS approach (1975)
Svante Wold
Harald Martens
PLS regression
(1983)
SIMCA-P
The Unscrambler
Covariance-based SEM (CSA)
(Model validation)
H. Hwang
Y. Takane
Generalized
Structured
GSCA (2004)
K. Joreskog
(LISREL, 1970)
Component Analysis (ALS)
J.-B. Lohmöller
LVPLS 1.8 (1984)
W. Chin
PLS-Graph
H. Hwang
VisualGSCA 1.0
(2007)
C. Ringle
SMART-PLS
Chatelin-Esposito Vinzi
Fahmy-Jäger-Tenenhaus
XLSTAT-PLSPM (2008)
R. McDonald(1996)
M. Tenenhaus (2001)
2
Covariance-based
Structural Equation Modeling
Latent Variables :
 1 
η   
m 
1 
ξ   
 k 
Endogenous LV
Exogenous LV
Structural model (Inner model):
η = Bη + Γξ + ζ
3
Structural Equation Modeling
Measurement model (outer model) :
 y j1    jy1 
  j1 

 

 
yj = 



 
 j  
 y   y 
 
jp
jp
 j  j
 jp j 
λ yj
y = Λy η + ε
MV
LV
Endogenous
 x 1    x1 
 1 
  



x = 
  

 x p    xp 
 p 
  



λx
εj
δ
x = Λ xξ + δ
MV
LV
Exogenous
4
Structural Equation Modeling
Mixing Structural and measurement models :
1
- η = Bη + Γξ + ζ  η = (I - B) (Γξ + ζ)
- x = Λ xξ + δ
1
- y = Λy η + ε  y = Λy (I - B) (Γξ + ζ) + ε




= Cov() = E(’)
= Cov() = E(’)
= Cov() = E(’)
= Cov() = E(’)
Residual variances
are diagonal matrices
5
Structural Equation Modeling
Covariance matrix for manifest variables :
Outer
model
Inner
model
Exogen.
LV Cov.
Structural
Residual
variance
 Σ xx
Σ( Λ x , Λ y , B, Γ, Φ, Ψ, Θε , Θδ )  
 Σ yx
Measurement
Residual
variance
Σ xy 
Σ yy 
1
 Λ xΦΛ x ' + Θδ

Λ xΦΓ ' (I - B) ' Λ y '


1
1
1
 Λ y  (I - B) ΓΦΛ x ' Λ y [(I - B) (ΓΦΓ' + Ψ )][(I - B ) '] Λ y ' + Θ ε 
6
Covariance-based SEM
ULS algorithm :
S = Observed covariance matrix for MV
  Λ x , Λ y , B, Γ,Φ, Ψ, Θε , Θδ 
Minimize
PCA Generalization
S  Σ( Λ x , Λ y , B, Γ, Φ, Ψ, Θε , Θδ )
2
Λx , Λy ,B,Γ ,Φ, Ψ ,Θε ,Θδ
Goodness-of-fit Index (Jöreskog & Sorbum):
GFI  1 
ˆ ,Λ
ˆ , Bˆ , Γ,Φ,
ˆ ,Θ
ˆ ,Θ
ˆ )
ˆ ˆ Ψ
S  Σ( Λ
x
y
ε
δ
S
2
2
7
Path model describing causes and consequences
of Customer Satisfaction: The inner model
Image
Loyalty
Customer
Expectation
Perceived
value
Customer
satisfaction
.
Perceived
quality
Complaints
8
Use of
AMOS 6.0
e2
e1
1
1
e5
e4
e3
1
1
e6
1
PQ2
CE1
CE2
CE3
1
e8
1
PQ3
PQ1
e7
PQ4
e9
1
1
PQ5
PQ6
e10
1
1
1
PQ7
CUST_EXP
Méthod = ULS
1
PER_QUALI
d1
1
e11
e12
1
1
PV1
PV2
d5
1
PER_VALUE
1
1
ima1
e19
e20
d2
1
1
d7
ima2
d3
1
1
1
ima3
e21
CUS_SAT
IMAGE
1
e22
1
e13
CSI1
ima4
1
First Roderick McDonald’s idea (1996)
1
e23
ima5
Measurement residual variances are canceled:
ˆ =Θ
ˆ =0
Θ
ε
δ
1
e14
CSI2
1
CSI3
e15
1
d4
CUST_ LOY
COMPLAINTS
1
0.8
This is a computational trick:
Residual variances are passed to errors
and can always be computed afterwards.
CL1
1
e16
CL2
1
e17
CL3
1
e18
1
Complaints
e24
9
Covariance-based SEM
ULS algorithm with the McDonald’s constraints:
S = Observed covariance matrix for MV
Ω  ( Λ x , Λ y , B, Γ, Φ, Ψ, Θε  0, Θδ  0)
2
Minimize S  Σ( Λ x , Λ y , B, Γ, Φ, Ψ , Θε  0, Θδ  0)
Λ x , Λ y ,B, Γ ,Φ , Ψ
Outer model
Inner model
Goodness-of-fit Index (Jöreskog & Sorbum):
GFI  1 
ˆ ,Λ
ˆ ,B
ˆ ,Θ
ˆ
ˆ
ˆ , Γ,Φ,
ˆ ˆ Ψ
S  Σ( Λ
x
y
ε , final , Θδ , final )
S
2
2
10
0
Use of
AMOS 6.0
1
0
e1
CE2
e4
1
CE3
CE1
1
PQ2
PQ3
PQ4
1
0
1
CUST_EXP
e8
0
0 PER_QUALI
0
1
1
PQ5
1
PQ1
- Méthod = ULS
- Measurement
residual
variances = 0
1
1
0
1
e7
e6
e5
e3
0
0
0
0
e2
1
e12
e11
1
PQ6
1
1
e9
d1
d5
PV2
PV1
PQ7
1
0
1
e10
0
1
PER_VALUE
ima1
e19
1
0
1
1
d2
ima2
e20
d3
1
0
1
1
e21
0
1
CSI1
e13
CUS_SAT
ima3
IMAGE
0
e22
0
1
1
e14
CSI2
ima4
0
1
e23
0
ima5
CSI3
1
e15
1
d4
CUST_ LOY
Complaints
1
1
d6
CL1
1
0
e16
CL2
1
0
e17
CL3
1
0
e18
11
.00
Results
1
.00
e1
e4
1
GFI = .869
CE1
CE3
1.03
1.11
1.00
PQ2
1
Outer LV Estimates:
1
.00
1
.98
ima3
1.18
.00
1
e10
18.51
d3
1
196.90
.40
.00
1
1.00
CSI1
ima4
e13
CUS_SAT
IMAGE
1.53
.00
1
1
.89
1.68
.20
.00
1
e23
PQ6
.57
.79
.00
e9
d2
ima2
1
e22
PQ7
1
285.53
.15
1.00
.00
e21
54.73 1.33
1
PER_VALUE
1.00  CSI1  1.53  CSI2  1.68  CSI3
CSI 
1  1.53  1.68
e20
.00
1.04
1.00 .95
.59
e8
d1
.51
-.60
1
PQ5
PV2
PV1
ima1
e19
1
e12
.68
2nd McDonald’s idea + Fornell’s
idea
.00
.00
.93
1
1
d5
PQ4
.00PER_QUALI
e11
29.10
PQ3
1.00
1.06
.00
1
1
.99 1.29 1.03
PQ1
CUST_EXP
1
1
.00
1
CE2
e7
e6
e5
e3
.00
.00
.00
.00
e2
1.40
e14
CSI2
1.52
.00
ima5
CSI3
325.29
1
e15
-.02
Complaints
PLS estimate of LV:
1
1.00
.95
.31
402.67
- Mode A
- LV inner estimate = theoretical LV
1
1
1
.00
.00
- LV inner estimate computation is useless. .00
d4
1
CUST_ LOY
d6
CL1
CL2
e16
e17
CL3
e18
12
Cross-validation
by bootstrap
PQ2
PQ1
CE1
PQ3
PQ4
1
1.03
CE2
1.11
Expectation
CE3
1.29
.99
1
1.03
Perceived
quality
1.06
PV1
.95
1
.59
.51
.57
Perceived
value
1
PQ7
d1
-.59
IM1
d2
R2=.36
IM3
1.53
.98
Satisfaction
.40
.79
R2=.82
Image
.59
IM5
CS2
1.68
1
CS3
d3
1.18
IM4
CS1
1
.15
1
IM2
PQ6
1.33
1
PV2
d5
.68
1.04
R2=.67
R2=.70
1
PQ5
.93
.61
.89
Loyalty
1
d4
-.02
Complaints
R2=.44
1
.31
R2=.37
1
.95
d6
CL1
CL2
CL3
13
Figure 2 : Estimation of the ECSI model
(Significant links in bold, non-significant dotted)
Cross-validation
by bootstrap
Parameter
Customer expectation <--- Image
Perceived quality
<--- Customer expectation
Perceived value
<--- Perceived quality
Perceived value
<--- Customer expectation
Customer satisfaction <--- Perceived value
Customer satisfaction
<--- Customer expectation
Customer satisfaction <--- Image
Customer satisfaction <--- Perceived quality
x6 (Complaints)
<--- Customer satisfaction
Customer loyalty
<--- Customer satisfaction
Customer loyalty
<--- Image
Customer loyalty
<--- x6 (Complaints)
x11
<--- Image
x12
<--- Image
x13
<--- Image
x14
<--- Image
x15
<--- Image
x21
<--- Customer expectation
x22
<--- Customer expectation
x23
<--- Customer expectation
x31
<--- Perceived quality
x32
<--- Perceived quality
x33
<--- Perceived quality
x34
<--- Perceived quality
x35
<--- Perceived quality
x36
<--- Perceived quality
x37
<--- Perceived quality
x41
<--- Perceived value
x42
<--- Perceived value
x51
<--- Customer satisfaction
x52
<--- Customer satisfaction
x53
<--- Customer satisfaction
x71
<--- Customer loyalty
x72
<--- Customer loyalty
x73
<--- Customer loyalty
Estimate
0.593
1.058
0.511
0.679
0.152
-0.595
0.402
0.568
1.519
1.405
0.201
-0.018
1
0.792
0.981
1.184
0.890
1
1.034
1.112
1
0.989
1.295
1.031
0.929
1.043
1.328
1
0.954
1
1.535
1.682
1
0.312
0.953
Inf (95%) Sup (95%) P
0.380
0.807
0.050
0.777
1.638
0.050
-0.195
0.979
0.170
0.000
1.883
0.100
0.059
0.246
0.050
-1.538
0.048
0.070
0.184
0.918
0.050
0.158
0.847
0.060
1.089
1.938
0.050
0.806
5.161
0.050
-2.296
0.558
0.520
-0.165
0.120
0.790
1
1
...
0.553
1.057
0.05
0.652
1.527
0.05
0.972
1.551
0.05
0.718
1.151
0.05
1
1
...
0.559
1.589
0.05
0.675
1.693
0.05
1
1
...
0.736
1.255
0.05
1.069
1.547
0.05
0.881
1.207
0.05
0.773
1.102
0.05
0.900
1.223
0.05
1.082
1.606
0.05
1
1
...
0.851
1.083
0.05
1
1
...
1.269
1.899
0.05
1.294
2.139
0.05
1
1
...
-0.069
0.662
0.11
0.808
1.127 14 0.05
Comparison between the Fornell-ULS
and Fornell-PLS standardized weights
Manifest variable
Image
Image
Image
Image
Image
Customer expectation
Customer expectation
Customer expectation
Perceived quality
Perceived quality
Perceived quality
Perceived quality
Perceived quality
Perceived quality
Perceived quality
Perceived value
Perceived value
Customer satisfaction
Customer satisfaction
Customer satisfaction
Customer loyalty
Customer loyalty
Customer loyalty
Fornell-ULS
standardized weight
0.206
0.163
0.202
0.244
0.184
0.318
0.329
0.353
0.131
0.130
0.170
0.135
0.122
0.137
0.174
0.512
0.488
0.237
0.364
0.399
0.441
0.138
0.421
Fornell-PLS
standardized weight
0.199
0.173
0.187
0.242
0.198
0.326
0.316
0.357
0.139
0.121
0.168
0.134
0.119
0.135
0.183
0.492
0.508
0.242
0.354
0.404
0.393
0.130
0.478
15
Comparison between the Fornell-ULS
and Fornell-PLS standardized weights
and all Cor(LVFornell-ULS , LVFornell-PLS) > .99
16
First particular case :
Factor Analysis
and
Principal Component Analysis
17
a posteriori
computation
First particular case : FA and PCA
.00
.02
1
Capacity
1
Capacity
e1
.00
.14
1
Power
.99
1.00
F1
e2
1
Speed
.87
e3
1.00
Weight
1
F1
e4
e2
1
Speed
.89
.21
e3
.00
.76
Weight
1
.42
e4
.80
.74
.00
.45
.73
Width
1
.80
e5
Width
1
e5
Length
Factor Analysis
Reflective mode
1
e6
.36
.00
.47
AVE = .69
.15
.00
.92
.53
.68
1
Power
.96
.25
.93
.08
e1
Length
1
Principal component Analysis
Formative mode
AVE = .74
18
e6
.36
FA vs PCA : Variance reconstruction
Covariances
Cylindrée
Cylindrée
Puissance
Vitesse
Poids
Largeur
Longueur
1
.954
.885
.692
.706
.664
1
.934
1
.529
.466
1
.730
.619
.477
.527
.578
.795
1
.591
1
Puissance
Vitesse
Poids
Largeur
Longueur
Implied covariances (FA)
Cylindrée
Puissance
Vitesse
Poids
Cylindrée
Puissance
Vitesse
Poids
Largeur
Longueur
1
.918
.860
.678
.737
.722
1
.804
1
.633
.593
1
.689
.645
.508
.674
.632
.498
1
.541
1
Largeur
Longueur
FA does not
care for variance
reconstruction
  sij  ˆij   .169
2
i j
Implied covariances (PCA)
FA (reflective
mode)
yields
to Largeur
better
covariances
PCA cares for
Cylindrée
Puissance
Vitesse
Poids
Longueur
variance
reconstruction
(formative
mode).
Cylindrée
.926
.889 than
.853PCA .728
.771
.765
Puissance
Vitesse
Poids
Largeur
Longueur
.853
.818
.785
.699
.671
.573
.740
.710
.606
.734
.705
.602
.642
.637
.632
reconstruction
s
i j
ij
 ˆ ij 19 .229
2
MIMIC mode (with this new approach)
(Multiple effect indicators for multiple causes)
.09
.00
1
e4
Capacity
Weight
1.00
.00
1
e5
.81
Width
1
.21
.89
F1
.81
e6
e1
.95
.77
.00
1
1
e2
Power
.83
.31
Length
Formative mode
(multiple cause)
Speed
1
e3
Reflective mode
(multiple effect)
20
MIMIC mode (usual in CSA)
?
.43
d
1.00
-.05
Weight
1
.27
.48
1.00
.59
?
1.00
Length
Formative mode
(multiple cause)
.09
1
e1
1.00
.07
.55
Width
.79
Capacity
.94
F1
1
e2
Power
.88
.20
Speed
1
e3
Reflective mode
(multiple effect)
21
MIMIC mode (usual in CSA)
.42
d
1.00
.00
Weight
.27
.48
1.00
Width
.79
.59
Capacity
1
1.00
Length
Formative mode
(multiple cause)
?
.09
1
e1
1.00
.07
.54
.95
F1
1
e2
Power
.89
.19
Speed
1
e3
Reflective mode
(multiple effect)
22
MIMIC mode (better)
 PCA oriented vs
the dependent block
e6
.46
d
.00
.00
1
Weight
1
.84
.00
e5
e4
e1
1.00
.07
1.00
1
.87
Width
1
.95
.75
F1
e2
Power
F2
.88
.00
1
1
Capacity
.89
.19
Length
Speed
Proposal: Compute a global score as




i
 
Cov( X i , F2 ) * X i  / 
 
Formative mode
(multiple cause)

i
1
e3

Cov( X i , F2 ) 

Reflective mode
(multiple effect)
(Same than before)
23
Second particular case :
Multi-block data analysis
24
3 Appellations
4 Soils
Sensory
analysis of 21
Loire Red Wines (J. Pagès)
4 blocks of variables
2el (Saumur),1
X1
X2
X3
X4
Smell intensity at rest
3.07
Aromatic quality at rest
3.00
Fruity note at rest
2.71
Floral note at rest
2.28
Spicy note at rest
1.96
Visual intensity
4.32
Shading (orange to purple)
4.00
Surface impression
3.27
Smell intensity after shaking
3.41
Smell quality after shaking
3.31
Fruity note after shaking
2.88
Floral note after shaking
2.32
Spicy note after shaking
1.84
Vegetable note after shaking
2.00
Phenolic note after shaking
1.65
Aromatic intensity in mouth
3.26
Aromatic persisitence in mouth
3.26
Aromatic quality in mouth
3.26
Intensity of attack
2.96
Acidity
2.11
Astringency
2.43
Alcohol
2.50
Balance (Acid., Astr., Alco.)
3.25
Mellowness
2.73
Bitterness
1.93
Ending intensity in mouth
2.86
Harmony
Illustrative3.14
Global quality
3.39
variable
1cha (Saumur),1
1fon (Bourgueil),1
1vau (Chinon),3
…
2.96
2.82
2.38
2.28
1.68
3.22
3.00
2.81
3.37
3.00
2.56
2.44
1.74
2.00
1.38
2.96
2.96
2.96
3.04
2.11
2.18
2.65
2.93
2.50
1.93
2.89
2.96
3.21
2.86
2.93
2.56
1.96
2.08
3.54
3.39
3.00
3.25
2.93
2.77
2.19
2.25
1.75
1.25
3.08
3.08
3.08
3.22
2.18
2.25
2.64
3.32
2.68
2.00
3.07
3.14
3.54
2.81
2.59
2.42
1.91
2.16
2.89
2.79
2.54
3.16
2.88
2.39
2.08
2.17
2.30
1.48
2.54
2.54
2.54
2.70
3.18
2.18
2.50
2.33
1.68
1.96
2.46
2.04
2.46
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
…
t1 (Saumur),4 t2 (Saumur),4
3.70
3.19
2.83
1.83
2.38
4.32
4.00
3.33
3.74
3.08
2.83
1.77
2.44
2.29
1.57
3.44
3.44
3.44
2.96
2.41
2.64
2.96
2.57
2.07
2.22
3.04
2.74
2.64
3.71
2.93
2.52
2.04
2.67
4.32
4.11
3.26
3.73
2.88
2.60
2.08
2.61
2.17
1.65
3.10
3.10
3.10
3.33
2.57
2.67
2.70
2.77
2.31
2.67
3.33
3.00
2.85
X1 = Smell at rest, X2 = View, X3 = Smell after shaking, X4 = Tasting
PCA of
each block:
Correlation
loadings
SMELL AT REST
VIEW
SMELL AFTER SHAKING
TASTING
1.0
1.0
Spicy note
0.8
Vegetable note
0.6
T2
0.8
Smell intensity
Phelonic note
T1
3EL
Aromatic persistency
in mouth
0.4
Aromatic intensity
mouth
0.2
PER1
4EL
1VAU
1TUR
-0.0
-0.2
2ING
1FON
1CHA
1POY 2DAM
1BOI
Fruity note
Smell quality
Aromatic quality
in mouth
Floral note
2ING
-0.0
0.2
0.4
Harmony
Balance
-0.6
-1.0
-0.2
Intensity of attack
-0.4
-1.0
-0.4
4EL
2BOU
2BEA
1ING
PER1
1ROC
1DAM
1BOI
1TUR
3EL
DOM1
1BEN
2DAM
1POY
1CHA 1FON
2EL
-0.2
-0.8
-0.6
Ending intensity
in mouth
0.2
-0.8
-0.8
T1
1VAU
Mellowness
-0.4
-1.0
Astringency
Alcohol
T2
0.4
-0.0
1DAM
Bitterness
0.6
2BEA in
DOM1 1ING
2EL
1ROC
2BOU
1BEN
-0.6
Acidity
0.6
0.8
1.0
-1.0
-0.8
-0.6
-0.4
-0.2
-0.0
0.2
0.4
0.6
0.8
1.0
PCA of
each block:
Correlation
loadings
e1
1
.00
e20
1
.85.91
rest4
.34
.00
view3
View
shaking1
1
shaking2
.47
.88
1
shaking3
.82
.33
1
e9
.00
1
e8
.00
1
e21
.00
shaking4
shaking6
1.00
Shaking 1
.09
-.64
.37
.89
shaking5
.88
.89
.00
shaking10
shaking7
1
.96
shaking8
.00
e24
1
shaking9
e23
1.00
tasting7
.90
.84
.77
tasting1
1
.00
tasting2
1
.00
tasting8
.77
1
e18
.00
e17
e26
tasting4
1
.00
e16
.00
1
e25
.00
1
tasting6
tasting5
tasting3
.00
.38
.94
e19
1
.97
Tasting 1
-.26
e27
tasting9
1
.00
1
e22
.98 .99
1.00
.95
.74
1
e12
.00
e10
.00
1
view1
view2
rest1
Rest 1
e13
.00
.00
1
1
1.00
.08
rest5
e7
e6
1
1
rset2
.00
.00
e5
e4
1
rest3
.00
.00
e3
e2
1
.00
e11
GFI = .301
.00
.00
e14
.00
1
e15
Multi-block data analysis = Confirmatory Factor Analysis
.00
.00
e1
1
.00
SMELL AT
REST
e20
1
.77.90
rest4
.50
.00
.00
.98
.87
shaking1
.83
1
shaking2
.59
.79
.72
.23
1
shaking4
shaking6
.74
1.00
Shaking 1
.25
-.54
.37
.88
shaking5
.89
.76
.91
e24
1
shaking9
e23
1.00
tasting7
.85
.80
.77
tasting1
1
.00
tasting2
1
.00
tasting8
.77
1
e18
.00
e17
.00
1
e25
.00
1
tasting6
tasting5
tasting3
.00
e26
.35
.85
e19
1
.92
Tasting 1
-.20
e27
.91
shaking8
.00
1
tasting9
1
.00
1
.00
.92
shaking10
shaking7
e22
VIEW
.89 .91
1.00
View
Rest 1
shaking3
e9
.00
1
e8
.00
1
e21
.00
view1
view2
.72
1
e10
.00
1
GFI = .849
view3
rest1
1
e12
.00
e11
1
1
1.00
.00
rest5
1
.59
e7
e6
1
rset2
.00
.00
e5
e4
1
rest3
e13
.00
SMELL AFTER
SHAKING
e3
e2
1
.00
.00
.00
e14
TASTING
.00
1
e15
tasting4
1
.00
e16
28
First
dimension
.00
.00
.00
e5
e3
e2
.84
.94
view1
view2
.90
.99 .88
.65
1.00
1.00
View
Rest 1
Using MV with
significant loadings
1
1
view3
rest2
rest3
e7
e6
1
1
1
.00
.00
.87
.00
.79
1
e12
shaking2
.84
.00
1
shaking3
e11
.69
1.00
.78
Shaking 1
.00
.87
1
shaking8
e22
.88
.84
.81
.00
1
e23
.90
shaking9
.00
e24
1
1.00
shaking10
.93
Tasting 1
tasting9
e27
.91
.85
.89
1
tasting8
.00
e26
.84
.76
.00
1
tasting1
tasting6
1
1
.00
e19
tasting4
1
.00
e16
tasting5
1
.00
e15
.00
e14
29
First global
score
.88
e7
e6
view1
view2
1.00
1.12.98
1.00
.00
.00
1
d1
Rest 1
1
View
.79
d2
.80
1
e12
shaking2
1.00
1.04
.00
1
.96
shaking3
e11
GFI = .973
1
1
view3
rest2
rest3
.00
.00
1
1
1
.00
e5
e3
e2
2nd order CFA
.00
.00
.00
Score 1
.78
Shaking 1
.00
1.08
1
1
shaking8
e22
1.09
.00
d3
1.00
.00
.89
1
e23
.88*rest3 + 1*rest2
Rest 1 
.88 + 1
shaking9
.00
e24
1
shaking10
.00 1
d4
1.00
Tasting 1
e27
.99
.91
.95
1
tasting8
Score 1
.79*Rest 1 + .80*View + .78*Shaking 1 + .89*Tasting 1

.79 + .80 + .78 + .89
.00
e26
.91
.82
.00
1
tasting9
tasting1
tasting6
1
1
.00
e19
tasting4
1
.00
e16
tasting5
1
.00
e15
.00
e14
30
Validation of the first dimension
Correlations
Rest1
View
Shaking1
Tasting1
Score1
Rest1
1
.621
.865
.682
.813
View
1
.762
.813
.920
Shaking1
Tasting1
1
.895
.942
1
.944
31
.00
.00
e20
1
e4
Second
dimension
1
rest5
rest1
.74
.91
1.00
Rest 2
.00
1
e13
.75
shaking1
.77
.90
1.00
.00
.74
1
Shaking 2
shaking5
e9
.57
.00
e21
.79
1
shaking7
1.00
Tasting 2
.87
tasting3
tasting7
1
1
.00
e17
.81
.00
e25
32
.00
.00
e20
1
e4
2nd
global
score
1
rest5
rest1
1.00
1.25
.00
1
Rest 2
d1
GFI = .905
.66
.00
1
e13
.00
shaking1
1
.00
e9
1.34
1
1.00
d3
1.62
Score 2
.52
Shaking 2
shaking5
.75
1.00
.00
Score 2
1
e21
shaking7
.66*Rest 2 + .52*Shaking 2 + .75*Tasting 2

.66 + .52 + .75
.00
Tasting 2
1.08
tasting3
d4
1.00
tasting7
1
1
.00
e17
1
.00
e25
33
Validation of the second dimension
Correlations
Rest2
Rest2
Shaking2
Tasting2
Score2
1
.789
.782
.944
Shaking2
Tasting2
1
.803
.904
1
.928
34
Mapping of the correlations with the global scores
Score 2 unrelated
with quality
Score 1
related with
quality
35
Correlation with global quality
Variables related to
dimension 1
Aromatic quality at rest
Fruity note at rest
Visual intensity
Shading (from orange to purple)
Surface impression
Smell quality
Aromatic intensity in mouth
Aromatic persistence in mouth
Aromatic quality in mouth
Intensity of attack
Alcohol
Balance (acidity, astringency, alcohol)
Mellowness
Ending intensity in mouth
Harmony
Variables related to
dimension 2
Smell intensity at rest
Spicy note at rest
Smell intensity after shaking
Spicy note after shaking
Phelonic note
Astringency
Bitterness
Global quality
0.62
0.50
0.54
0.51
0.67
0.76
0.61
0.68
0.85
0.77
0.52
0.95
0.92
0.80
0.88
New result.
Not obtained with other
multi-block data analysis
methods, nor with factor
analysis of the whole data.
Global quality
0.04
-0.31
0.17
-0.08
0.09
0.41
0.05
36
Wine visualization in the global score space
Wines marked by Appellation
37
Wine visualization in the global score space
Wines marked by Soil
38
Visualization of wine variability among the blocks
Star-plot of the “best wine” – 2DAM SAUMUR
3,0
DAM =
Dampierre-sur-Loire
2,8
2,6
2,4
2DAM
GLOBAL SCORE
Tasting
2,2
Smell after shaking
View
2,0
2,25
Smell at rest
2,50
2,75
3,00
3,25
3,50
Cuvée Lisagathe 1995
A soft, warm, blackberry nose. A good core of fruit on the palate
with quite well worked tannin and acidity on the finish; Good
length and a lot of potential.
DECANTER (mai 1997)
(DECANTER AWARD ***** : Outstanding quality, a virtually perfect example)
Third particular case : Analysis of covariance between
two blocks of binary variables
(with C. Guinot et E. Mauger (CERIES))
Data
 = Sun exposure
 = Sun protection
A = Gender (A1 = Men, A2 = Women)
Model
1 < 0?
2 < 3 ?
 = 0 + 1A1 + 2 *A1 + 3 *A2 + 
(1)
42
Theory: background
 = 0 + 1A1 + 2 *A1 + 3 *A2 + 


W
(1)

W
M


No gender effect
1 = 0,
2 = 3
M
Gender main effect
1  0,
2 = 3

Interaction
*gender
2  3
43
Theory: background
X1 Sun exposure during lifetime (4)
Score for
sun exposure
X2 Sun exposure during mountain sports (2)
X3 Sun exposure during nautical sports (2)
ˆ  Xw
X4 Sun exposure during hobbies (2)
X5 Practice of naturism (1)
Y Sun protection behavior over the
past year (6)
Score for
sun protection
ˆ  Yc

A Gender
44
Theory: background
 = 0 + 1A1 + 2 *A1 + 3 *A2 + 
(1)
Equation (1) is replaced by:
Yc = 0 + 1A1 + 2Xw*A1 + 3Xw*A2 + 
(2)
= 0 + 1A1 + 2(X*A1)w + 3(X*A2)w +  (3)
Question:
How to estimate and test the parameters
w, c, 0 , 1, 2, 3 ?
45
Theory: methods
Covariance based SEM with constraints
X1*A1
X2*A1
w1
w2
w3
X*A1
2
Y
X3*A1
3
X1*A2
w1
X2*A2 w2
w3
X3*A2
c1
c2
c3
1
Y1
Y2
Y3
A1
X*A2
No group effect on the
measurement model
46
Application: material
We have applied the ULS-SEM to a study on sunexposure behavior in 8,084 French adults.
Development of skin cancers
Premature skin ageing.
Data came from the
SU.VI.MAX study*
*Hercberg S. et al. Arch Intern Med. 2004;164:2325-42
47
0
1
e11
e10
0
e9
0
1
s15_1_h
1
e7
w1
s28_34_h
1
1
1
1
s25j2_h
0
1
e5
s26_h
0
e4
1
s26j2_h
0
s28_34_f
w4
s29_34_f
Sun
exposure (W)
1
w7
w7
W8
w8
1
1
W10
e1
1
0
e28
1
s27j2_f
e21
0
1
s21_f
e22
M en
Sun Protection
1
1
0
e20
1
d
s6_1
e19
0
1
s27_f
s21_h
e18
0
1
s26j2_f
s27j2_h
e150
0
1
1
0
0
e17
s26_f
w11
e14
e160
1
s25j2_f
w9
w9
w10
w11
1
s25_f
s27_h
0
e2
w2
w3
w5
Sun
Exposure (M )
e13
0
1
e3
1
s15_1_f
w5
s25_h
e12
0
w4
0
e6
s14z3_f
w1
w2
w3
s29_34_h
e8
0
Use of AMOS
method = ULS
s14z3_h
0
0
1
c2
s6quand2
1
0
e27
c3
s6regul
c5
c4
s6vis
c6
s6cps
1
0
1
0
1
0
e26
e25
e24
s6hors1
1
0
e23
48
.
Sun exposure of body
and face
Sun exposure
between 11am and 4 pm
Basking in the sun is
important or very important
Intensity of sun exposure
moderate or severe
Sun exposure during
practice of mountain sports
Nb of days of mountain sport
actvities > 200 days
AMOS results
2.82
2.82
GFI = .870
2.30
.
.
1.46
3.91
3.91
2.13
1.00
Sun
expo.
(M)
2.13
Sun
expo.
(W)
1.00
1.23
.64
Men
2.67
-.17
.62
.62
[-.20, -.14]
d
Several times while sun exposure
Sun exposure during practice
of nautical sports
Nb of days of practice of nautical
activites > 400 days
Sun exposure during
practice of hobbies
Nb of days of lifetime
hobbies > 900 days
.62
Product used for face with SPF > 15
.40
.82
While sun exposure
Nb of days of mountain sport
actvities > 200 days
Practice of naturism
during lifetime
Sun
Protection
While sun tanning
Sun exposure
between 11am and 4 pm
Basking in the sun is
important or very important
Intensity of sun exposure
moderate or severe
Sun exposure during
practice of mountain sports
1.35
1.38]
1.24 [1.12,
.75
1.00
.
2.30
1.46
Sun exposure during practice
1.23
of nautical sports
Confidence Interval
Nb of days of practice of nautical .64
activites > 400(Bootstrap)
days
2.67
Sun exposure during
practice of hobbies[.64, .89]
1.35
Nb of days of lifetime
hobbies > 900 days
Practice of naturism
during lifetime
Sun exposure of body
and face
.90
.45
Product used for body with SPF > 15
Product used besides voluntarily sun
exposure periods
Results: AMOS ULS
 = 0 + 1M + 2*M + 3*W + 
Coefficients
Estimate
Lower
Upper
2 Sun exposure (Men)

Sun Protection
.754
.647
891
3 Sun exposure (Women)

Sun Protection
1.237
1.123
1.385
1 Men

Sun Protection
-.166
-.198
-.139
Conclusion
1. Women tend to protect themselves from the sun more than
men (1 < 0).
2. This difference between men and women increases as
lifetime sun exposure increases (3 - 2 > 0)
50
LV estimation using PLS
(Mode A, Fornell’s normalisation)
ˆ  LV ( X )  w1 X 1  w2 X 2   w11 X 11  w X

j
j
w1  w2   w11
c1Y1  c2Y2   c6Y6
ˆ  LV (Y ) 
  ck Yk
c1  c2   c6
Example for Sun Protection:
1.00
c1 
 .239
1.00  .82  .90  .62  .40  .45
51
Results
Sun protection over the past year score
+ 0.24 If sun protection products used while sun tanning
+ 0.20 If sun protection products used throughout voluntarily sun exposure periods
+ 0.22 If sun protection products applied several times during sun exposure periods
+ 0.14 If the sun protection product used for the face has a SPF* over 15
+ 0.09 If the sun protection product used for the body has a SPF* over 15
+ 0.11 If sun protection products used besides voluntarily sun exposure periods
*SPF: Sun protection factor
52
Results
Lifetime sun exposure score
0.14 If sun exposure of the body and the face
0.11 If sun exposure between 11 a.m. and 4 p.m.
0.07 If basking in the sun is declared important or extremely important
0.20 If self-assessed intensity of sun exposure is declared moderate or severe
0.10 If sun exposure during practice of mountain sports
0.05 If the number of days of lifetime mountain sports activities > 200 days*
0.06 If sun exposure during practice of nautical sports
0.03 If the number of days of lifetime nautical sports activities > 400 days*
0.13 If sun exposure during practice of hobbies
0.07 If the number of days of lifetime hobby activities > 900 days*
0.03 If practice of naturism during lifetime
* Median value of the duration was used as a threshold for dichotomisation
53
Results: analysis of covariance
ˆ  Protection on ˆ  Sun exposure and Gender
Parameter
Intercept
score_x1_protect
GENRE
GENRE
score_x1_protec*GENRE
score_x1_protec*GENRE
Estimate
Femmes
Hommes
Femmes
Hommes
0.0729460737
0.2473795070
0.1269948620
0.0000000000
0.1613712617
0.0000000000
B
B
B
B
B
B
Standard
Error
t Value
Pr > |t|
0.01213456
0.02574722
0.01557730
.
0.03316612
.
6.01
9.61
8.15
.
4.87
.
<.0001
<.0001
<.0001
.
<.0001
.
Coefficients
Estimate
Lower
Upper
2 Sun exposure (Men)

Sun Protection
.754
.647
.891
3 Sun exposure (Women)

Sun Protection
1.237
1.123
1.385
1 Men

Sun Protection
-.166
-.198
-.139
Main effect Gender + Interaction Sun exposure*Gender
54
are highly significant.
Conclusion 1: SEM-ULS > PLS
• When mode A is chosen, outer LV estimates using
Covariance-based SEM (ULS or ML) or Component
based SEM (PLS) are always very close.
• It is possible to mimic PLS with a covariance-based
SEM software (McDonald,1996, Tenenhaus, 2001).
• Covariance-based SEM authorizes to implement
constraints on the model parameters. This is
impossible with PLS.
55
Conclusion 2: PLS > SEM-ULS
• When SEM-ULS does not converge or does not
give an admissible solution, PLS is an attractive
alternative.
• PLS offers many optimization criterions for the LV
search (but rigorous proofs are still to be found).
• PLS still works when the number of MV is very high
and the number of cases very small.
• The new software XLSTAT-PLSPM will be available
at the beginning of 2008.
56
References
- Tenenhaus M., Esposito Vinzi V., Chatelin Y.-M., Lauro C. (2005) :
« PLS path modeling »
Computational Statistics & Data Analysis, 48, 159-205.
- M. Tenenhaus, E. Mauger, C. Guinot :
« Use of ULS-SEM and PLS-SEM to measure interaction effect
in a regression model relating two blocks of binary variables »
in Handbook of Partial Least Squares (PLS): Concepts, Methods and
Applications (V. Esposito Vinzi, J. Henseler, W. Chin, H. Wang, Eds),
Springer, 2007.
- M. Tenenhaus :
« A bridge between PLS path modeling and ULS-SEM »
PLS’07, Oslo.
57
Final conclusion
« All the proofs of a pudding are in the eating,
not in the cooking ».
William Camden (1623)