Structural Equation Modelling for Small Samples

8/18/2019 Structural Equation Modelling for Small Samples

1/52

1

Structural Equation Modelling for small samples

Michel Tenenhaus HEC School of Management (GRECHEC),

1 rue de la Libération, Jouy-en-Josas, France [[email protected]]

Abstract

Two complementary schools have come to the fore in the field of Structural Equation Modelling

(SEM): covariance-based SEM and component-based SEM.

The first approach developed around Karl Jöreskog. It can be considered as a generalisation of both

principal component analysis and factor analysis to the case of several data tables connected by

causal links.

The second approach developed around Herman Wold under the name "PLS" (Partial Least

Squares). More recently Hwang and Takane (2004) have proposed a new method namedGeneralized Structural Component Analysis. This second approach is a generalisation of principal

component analysis (PCA) to the case of several data tables connected by causal links.

Covariance-based SEM is usually used with an objective of model validation and needs a large

sample (what is large varies from an author to another: more than 100 subjects and preferably more

than 200 subjects are often mentioned). Component-based SEM is mainly used for score

computation and can be carried out on very small samples. A research based on 6 subjects has been

published by Tenenhaus, Pagès, Ambroisine & Guinot (2005) and will be used in this paper.

In 1996, Roderick McDonald published a paper in which he showed how to carry out a PCA using

the ULS (Unweighted Least Squares) criterion in the covariance-based SEM approach. He

concluded from this that he could in fact use the covariance-based SEM approach to obtain resultssimilar to those of the PLS approach, but with a precise optimisation criterion in place of an

algorithm with not well known properties.

In this research, we will explore the use of ULS-SEM and PLS on small samples. First experiences

have already shown that score computation and bootstrap validation are very insensitive to the

choice of the method. We will also study the very important contribution of these methods to multi-

block analysis.

Key words: Multi-block analysis, PLS path modelling, Structural Equation Modelling, Unweighted

Least Squares

Introduction

Compare to covariance-based SEM, PLS suffers from several handicaps: (1) the diffusion of path

modelling softwares is much more confidential than that of covariance-based SEM softwares, (2) the

PLS algorithm is more an heuristic than an algorithm with well known properties and (3) the

possibility of imposing value or equality constraints on path coefficients is easily managed in

covariance-based SEM and does not exist in PLS. Of course, PLS has also some advantages on

covariance-based SEM (that’s why PLS exists) and we can list some of them: systematic

convergence of the algorithm due to its simplicity, possibility of managing data with a small number

of individuals and a large number of variables, practical meaning of the latent variable estimates,

general framework for multi-block analysis.


2/52

2

It is often mentioned that PLS is to covariance-based SEM as PCA is to factor analysis. But the

situation has seriously changed when Roderick McDonald showed in his 1996 seminal paper that he

could easily carry out a PCA with a covariance-based SEM software by using the ULS ( Unweighted

Least Squares) criterion and cancelling the measurement error variances. Furthermore, the

estimation of the latent variables proposed by McDonald is similar to using the PLS mode A and the

SEM scheme (i.e. using the “theoretical” latent variables as inner LV estimates). Thus, it became possible to use a covariance-based SEM software to mimic PLS.

In the first section of this paper, it is reminded how to use the ULS criterion for covariance-based

SEM and the PLS way of estimating latent variables for mimicking PLS path modelling. Then, the

second section is devoted to show how to carry out a PCA with a covariance-based SEM software

and to comment the interest of this approach for taking into account parameter constraints and for

bootstrapping. Multi-block analysis is presented in the third section as a confirmatory factor

analysis.

We have used AMOS 6.0 (Arbuckle, 2005) and XLSTAT-PLSPM, a module of the XLSTAT

software (XLSTAT, 2007), on practical examples to illustrate the paper. Listing the pluses and

minuses of ULS-SEM and PLS finally concludes the paper.

I. Using ULS and PLS estimation methods for structural equation modelling

We describe in this section the use of the ULS estimation method applied to the SEM parameter

estimates and that of the PLS estimation method for computing the LV values.

In a first part we remind the structural equation model following Bollen (1989). A structural

equation model consists of two models: the latent variable model and the measurement model.

The latent variable model

Letη

be a column vector consisting of m endogenous (dependent) centred latent variables, and ξ a

column vector consisting of k exogenous (independent) centred latent variables. The structural

model connecting the vectorη

to the vectorsη and ξ is written as

(1) η = Bη + Γξ + ζ

where B is a zero-diagonal m m× matrix of regression coefficients, a m k × matrix of regressioncoefficients and

ζ

a centred random vector of dimension m.

The measurement model

Each latent (unobservable) variable is described by a set of manifest (observable) variables. Thecolumn vector y j of the centred manifest variables linked to the dependent latent variable η j can be

written as a function of η j through a simple regression with usual hypotheses.

(2) y j j j j

η y = λ + ε

The column vector y, obtained by concatenation of the y j’s, is written as

(3) yy = Λ η + ε

where1

m y

j j=

= ⊕yΛ λ is the direct sum of 1 ,...,

y y

mλ λ and ε is a column vector obtained by concatenation

of theε j’s. It may be reminded that the direct sum of a set of matrices A1, A2,…, Am is a block

diagonal matrix in which the blocks of the diagonal are formed by matrices A1, A2,…, Am.


3/52

3

Similarly, the column vector x of the centred manifest variables linked to the latent independent

variables is written as a function of ξ:

(4) xx = Λ ξ + δ

Adding the usual hypothesis that the matrix I-B is non-singular, equation (1) can also be written as:

(5) 1( ) ( )−η = I - B Γξ + ζ

and consequently (3) becomes

(6) 1[( ) ( )]−y

y = Λ I - B Γξ + ζ + ε

Factorisation of the manifest variable covariance matrix

Let = Cov(ξ) = E (ξξ’), ψ = Cov(ζ) = E (ζζ’), Θε = Cov(ε) = E (εε’) and Θδ = Cov(δ) = E (δδ’).Suppose that the random vectorsξ

,ζ

,ε

andδ

are independent of each other and that the covariance

matrices Ψ, Θε

, Θδ

of the error terms are diagonal. Then, we get:

' xx = x x δΣ Λ ΦΛ + Θ ,

1 1[( ) ( )][( ) '] ' yy

− −y y ε

Σ = Λ I - B ΓΦΓ' + Ψ I - B Λ + Θ ,

[ ]1

' ( ) ' ' xy

−=

x yΣ Λ ΦΓ I - B Λ

From which we finally obtain:

(7)

[ ]11 1

1

[( ) ( )][( ) '] ' ( ) '

' ( ) ' ' '

yy yx

xy xx

−− −

−

⎡ ⎤⎡ ⎤⎢ ⎥= =⎢ ⎥

⎡ ⎤⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

y y ε y x

x y x x δ

Λ I - B ΓΦΓ'+ Ψ I - B Λ + Θ Λ I - B ΓΦΛΣ ΣΣ

Σ Σ Λ ΦΓ I - B Λ Λ ΦΛ + Θ

Let { }θ , , , , ,= x y ε δΛ Λ B Γ,Φ, Ψ Θ Θ be the set of parameters of the model and (θ)Σ the matrix (7).

Model estimation using the ULS method

Let S be the empirical covariance matrix of the MV’s. The object is to seek the set of parametersˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆθ { , , , , } x y= ε δΛ Λ B Γ,Φ, Ψ Θ ,Θ minimizing the criterion

(8)2

ˆ(θ)−S Σ

The aim is therefore to seek a factorisation of the empirical covariance matrix S as a function of the

parameters of the structural model. In SEM softwares, the covariance matrix estimations

ˆ ( )Cov=εΘ ε and ˆ ( )Cov=δΘ δ of the residual terms are computed in such a way that the diagonal of

the reconstruction error matrix ˆ(θ)−E = S Σ is null, even when it yields to negative variance(Heywood case).


4/52

4

Let’s denote by îi

σ the i-th term of the diagonal of ˆ ˆ ˆˆ ˆ ˆ( , , , , )x yΣ Λ Λ B Γ,Φ, Ψ 0,0 and byîi

θ the i-th term

of the diagonal of ˆ ˆ⊕ε δΘ Θ . From the formula:

(9) ˆîi ii ii

s σ θ = +

we may conclude that îiσ is the part of the variance sii of the i-th MV explained by its LV (except in

a Heywood case) and îi

θ is the estimate of the variance of the measurement error relative to this

MV. As all the error terms ˆˆ( )ii ii ii iie s σ θ = − + are null, this method is not oriented towards the

research of parameters explaining the MV variances. It is in fact oriented towards the reconstruction

of the covariances between the MV’s, variances excluded.

The McDonald approach for parameter estimation

In his 96 paper, McDonald proposes to estimate the model parameters subject to the constraints that

all the îiθ are null. The object is to seek the parameters ˆ ˆ ˆˆ ˆ ˆ, , , x yΛ Λ B Γ,Φ, Ψ minimizing the criterion

(10)2

ˆ ˆ ˆˆ ˆ ˆ( , , , , , )−x y

S Σ Λ Λ B Γ,Φ, Ψ 0 0

The estimations of the variances of the residual termsε

andδ

are integrated in the diagonal terms of

the reconstruction error matrix ˆ ˆ ˆˆ ˆ ˆ( , , , , , )= −x y

E S Σ Λ Λ B Γ,Φ, Ψ 0 0 . This method is therefore oriented

towards the reconstruction of the full MV covariance matrix, variances included. On a second step,

final estimation ˆ ˆ andε δ

Θ Θ of the variances of the residual termsε

andδ

are obtained by using again

formula (9).

Goodness of Fit

The quality of the fit can be measured by the GFI (Goodness of Fit Index) criterion of Jöreskog &

Sorbum, defined by the formula

(11)

2

2

ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ( , , , , , )1GFI

−= −

x y ε δS Σ Λ Λ B Γ,Φ, Ψ Θ Θ

S

i.e. the proportion of2

S explained by the model. By convention, the model under study is

acceptable when the GFI is greater than 0.90.

The quantity2

ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ( , , , , , )−x y ε δ

S Σ Λ Λ B Γ,Φ, Ψ Θ Θ can be deduced from the CMIN criterion given in

AMOS:

(12)21 ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ( , , , , , )

2

N CMIN

−= × − x y ε δS Σ Λ Λ B Γ,Φ, Ψ Θ Θ

where N is the number of cases.


5/52

5

In practical applications of the McDonald approach, the difference between the GFI given by AMOS

and the exact GFI computed with formula (11) will be small:

(13)

2

2

22

2

ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ( , , , , , )1

ˆ ˆ ˆ ˆˆ ˆ ˆ( , , , , )

1ii

i

GFI

θ

−= −

− −= −

∑

x y ε δ

x y

S Σ Λ Λ B Γ,Φ, Ψ Θ Θ

S

S Σ Λ Λ B Γ,Φ, Ψ 0,0

S

Using the McDonald approach, the GFI given by AMOS is equal to

(14)

2

2

ˆ ˆ ˆˆ ˆ ˆ( , , , , )1GFI

−= −

x yS Σ Λ Λ B Γ,Φ, Ψ 0,0

S

and

22

ˆ /iii

θ ∑ S is usually small. Furthermore, the exact GFI will always be larger than the GFI given by AMOS.

Evaluation of the latent variables

After having estimated the parameters of the model, we now present the problem of evaluating the

latent variables. Three approaches can be distinguished: the traditional SEM approach, the

"McDonald" approach, and the "Fornell" approach. As it is usual in the PLS approach, we now

designate one manifest variable with the letter x and one latent variable with the letter ξ , regardless

of whether they are of the dependent or independent type. The total number of latent variables is

n = k + m and the number of manifest variables related to the latent variable ξ j is p j.

The traditional SEM approach

To construct an estimation ˆ j

ξ of j

ξ , one proceeds by multiple regression of j

ξ on the whole set of

the centred manifest variables 11 11,..., n nnp np x x x x− − . In other words, if one denotes asˆ

xxΣ the

implied (i.e. predicted by the structural model) covariance matrix between the manifest variables,

and as ˆ j xξ

Σ the vector of the implied covariances between the manifest variables x and the latent

variable ξ j, one obtains an expression of ˆ jξ as a function of the whole set of manifest variables:

(15) 1ˆ ˆ ˆ j j xx x

X ξ ξ −= Σ Σ

where 11 11,..., n nnp np X x x x x⎡ ⎤= − −⎣ ⎦

. This method is not really usable, as it is more natural to

estimate a latent variable solely as a function of its own manifest variables.


6/52


7/52


8/52

8

summarize the manifest variables of the block. This relationship may also be reflective: each

manifest variable is then a reflection of a latent variable existing a priori, a theoretical concept one

would try to outline with measures. The formative mode does not require the blocks to be one-

dimensional, while that is compulsory for the reflective mode. Here, we are more in a formative

mode for the physico-chemical and sensorial blocks and reflective mode by construction for the

hedonic block. The two modes are indicated by the direction of the arrows in Figure 1.

With regard to the PLS algorithm, it is recommended that the method of calculating outer estimates

of the latent variables is selected depending on the type of relationship between the manifest

variables and their latent variables: Mode A for the reflective type and Mode B for the formative

type (Wold, 1985). The low number of products has obliged us to use Mode A to calculate the outer

estimates of the latent variables (although the mode of relationship between the manifest and latent

variables is formative for the physico-chemical and sensorial blocks).

The ULS-SEM approach presented here is clearly oriented towards the reflective mode. Therefore

this orange juice example will be analyzed with ULS-SEM and PLS approaches using the reflective

mode for the three blocks. Concerning the physico-chemical and sensorial blocks, the direction of

the arrows connecting the MV’s to their LV’s shown in Figure 1 should thus be inversed.

Figure 1: Theoretical model of relationships between the hedonic, physico-chemical and sensorial

data

Glucose

Fructose

Saccharose

Sweetening power

pH before processing

pH after centrifugation

Titer

Citric acid

VitaminC

Smell intensity

Odor typicity

Pulp

Taste intensity

Acidity

Bitterness

Sweetness

ξ

1

ξ

2

ξ

3

Judge 2Judge 3

Judge 96

1. Use of ULS-SEM

We now use the ULS-SEM approach on the orange juice data. Following McDonald, the

measurement error variances are put to 0. The results are given in Figure 2 and in Table 2.

All manifest variables have been standardized. The value 1 has been given to the path coefficients

related to the manifest variables pH before centrifugation, Sweetness and Judge2.


9/52


10/52

10

Table 2: Outputs of AMOS 6.0

Table 2.1: Regression Weights (non significant weights in bold):

Parameter Estimate Lower (90%) Upper (90%) P

SENSORIAL


11/52

11

Tableau 2.2: Variances

Parameter Estimate Lower Upper P

PHYSICO-CHEMICAL .921 .429 1.120 .020

d1 .298 .028 .364 .010

d2 .034 .000 .044 .177

Tableau 2.3: Squared Multiple Correlations

Parameter Estimate Lower Upper P

SENSORIAL .655 .529 .951 .010

HEDONIC .946 .919 1.000 .020

Tableau 2.4: Model Fit Summary

Model NPAR CMIN

Default model 42 105.613

Model RMR GFI AGFI PGFI

Default model .175 .904 .898 .855

Comment: The GFI is equal to .904 and suggests that the model is acceptable.

Figure 3: Loading plot for the PCA of judges


12/52

12

The main objective of component-based SEM is the construction of scores. Following the

McDonald approach, we use the path coefficients given in Figure 2 and in Table 2.1. We obtain the

following constructs:

For the Physico-chemical block

Scor e( Physi co- Chemi cal ) ∝ - . 765*Gl ucose - . 764*Fr uct ose +. 890*Saccharose+. 219*( Sweet eni ng power) + 1*( pH bef ore cent r i f ugat i on) + . 998*( pH af t ercent r i f ugat i on) - . 869*Ti t er - . 877*( Ci t r i c aci d) - . 064*( Vi t ami n C)

where all the variables (score and manifest variables) are standardized.

For the Sensorial block

Scor e( Sensor i al ) ∝ . 244*( Smel l i nt ensi t y) + . 935*( Odor t ypi ci t y) +. 657*Pul p - . 565*( Tast e i nt ensi t y) - . 946*Aci di t y - . 974*Bi t t er ness +

1*Sweetness

with the same standardization than for the previous block.

For the Hedonic block

Scor e( Hedoni c) = 1*J udge2 + . 956*J udge3 + … + . 821*J udge96

with the same standardization than for the previous blocks.

The latent variable scores are given in Table 2.5 and their correlations in Table 2.6. The correlations

between these scores and the manifest variables are given in Table 2.7.

Tableau 2.5: ULS-SEM latent variable scores

Physico-chemical Sensorial Hedonic

Pampryl r.t. -0.72 -1.26 -1.10

Tropicana r.t. 1.05 0.43 0.66

Fruivita refr. 0.81 0.87 1.17

Joker r.t. -1.54 -0.77 -0.84

Tropicana refr. 0.56 1.27 0.85

Pampryl refr. -0.16 -0.53 -0.74

Tableau 2.6: ULS-SEM latent variable score correlation matrix


Physico-chemical 1.000 .810 .867

Sensorial .810 1.000 .961

Hedonic .867 .961 1.000


13/52

13

Tableau 2.7: Correlations between the ULS-SEM LV scores and the MV’s


Glucose -0.898 -0.585 -0.673

Fructose -0.898 -0.575 -0.673

Saccharose 0.926 0.755 0.817Sweetening power 0.078 0.288 0.242

pH before centrifugation 0.950 0.896 0.947

pH after centrifugation 0.939 0.904 0.946

Titer -0.973 -0.735 -0.765

Citric acid -0.977 -0.740 -0.774

Vitamin C -0.195 -0.040 -0.001

Smell intensity 0.229 0.410 0.174

Odor typicity 0.806 0.976 0.893

Pulp 0.558 0.704 0.625

Taste intensity -0.401 -0.646 -0.552

Acidity -0.745 -0.927 -0.950

Bitterness -0.775 -0.951 -0.976

Sweetness 0.871 0.967 0.979

Judge2 0.640 0.928 0.887Judge3 0.647 0.756 0.877Judge6 0.656 0.662 0.794

Judge11 0.872 0.785 0.919Judge12 0.718 0.929 0.823


Judge35 0.771 0.936 0.926

Judge48 0.460 0.837 0.834Judge52 0.791 0.840 0.944

Judge55 0.504 0.878 0.863Judge59 0.534 0.592 0.458

Judge60 0.870 0.854 0.924


Judge79 0.718 0.929 0.823Judge84 0.953 0.934 0.941

Judge86 0.453 0.685 0.762Judge91 0.827 0.845 0.927

Judge92 0.724 0.419 0.595Judge96 0.554 0.679 0.744

The estimate of the hedonic score, shown in Table 2.5, enables us to classify the products by order of

preference:

Fruivita refr. > Tropicana refr. > Tropicana r.t. > Pampryl refr. > Joker r.t. > Pampryl r.t.

Using the significant regression weights of Table 2.1 and the correlations given in Table 2.7, we may

conclude that the physico-chemical score is correlated negatively with the fructose, glucose, titer and

citric acid characteristics and positively with the saccharose, pH before and after centrifugation


14/52

14

characteristics. The sensorial score is correlated positively with odor typicity and sweetness and

negatively with acidity and bitterness.

The hedonic score related to the homogenous group of judges is correlated positively with the

physico-chemical (.867) and sensorial scores (.961). Consequently, this group of judges likes

products with odor typicity and sweetness (Fruivita refr., Tropicana r.t., Tropicana refr.) and rejects

products with an acidic and bitter nature (Joker r.t., Pampryl refr., Pampryl r.t.). This result is

verified in Table 3.

Table 3: Sensorial characteristics of the products ranked according to the hedonic score

odor hedonic

Product sweeteness typicity acidity bitterness score

_____________________________________________________________________

Fruivita refr. 3.4 2.88 2.42 1.76 1.17

Tropicana refr. 3.3 3.02 2.33 1.97 0.85

Tropicana r.t. 3.3 2.82 2.55 2.08 0.66

---------------------------------------------------------------------

Pampryl refr. 2.9 2.73 3.31 2.63 -0.74Joker r.t. 2.8 2.59 3.05 2.56 -0.84

Pampryl r.t. 2.6 2.53 3.15 2.97 -1.10

2. Use of PLS Path modeling

For estimating the parameters of the model, we have used the module XLSTAT-PLSPM of the

XLSTAT software (XLSTAT, 2007). The variables have all been standardized. To calculate the

inner estimates of the latent variables, we have used the centroid scheme recommended by Herman

Wold (1985).

Table 4 contains the output of this modelling of the orange juice data with comments. Figure 4includes the regression coefficients between the latent variables of the model shown in Figure 1 and

the correlation coefficients between the manifest and latent variables.

Coefficient validation

Although it gives robust and stable results with the various methods used on these orange juice data

(the same items appear to be significant in Tenenhaus, Pagès, Ambroisine, Guinot (2005) and in the

present paper), we may think that bootstrap validation carried out on only 6 cases cannot be very

reliable. One reason is the following: In this example, data structure comes from the opposition

between the two groups {Fruivita refr., Tropicana r.t., Tropicana refr.} on one side and {Pamprylrefr., Joker r.t., Pampryl r.t.} on the other side. If one of these groups of products is not selected in

the bootstrap sampling selection, then the correlations between the latent variables disappear.

Maybe, non representative samples should be eliminated.

Bootstrap has been based on 200 samples and 90% confidence intervals have been asked for.

Results of bootstrap validation for the inner model are shown in Table 3.7. The confidence intervals

indicate the regression coefficients which are significant. We can also look to the usual Student t

related to the regression coefficients. By convention, a coefficient is significant if the absolute value

of t is larger than 2. In this specific example, both methods give the same results. The relationship

between the hedonic data and the physico-chemical data is not significant (t = 1.522), while that

between the hedonic data and the sensorial data is (t = 3.546). There is also a significant connection

between the physico-chemical and the sensorial data (t = 2.864).


15/52

15

Figure 4: XLSTAT-PLSPM software output for the orange juice data


16/52

16

However, the strong correlation between the hedonic data and the physico-chemical data suggests

that a PLS regression of the hedonic score should be carried out on the physico-chemical and

sensorial scores. This PLS regression (with one component) leads to the following equation:

Hedonic score = 0.49*(Physico-Chemical score) + 0.53*(Sensorial score)

with an R 2

= 0.948 to be compared with R 2

= 0.960 in the model shown in Figure 4.Bootstrap validation for PLS regression yields to the same significant regression coefficients as for

OLS regression (see Table 4). If the PLS regression is validated by Jack-knife on the observed latent

variables, both coefficients are now significant (Table 4 and Figure 5).

Figure 5: XLSTAT-PLSPM software output: Validation of the PLS regression of the hedonic score

on the physico-chemical and sensorial scores

Hedonic / Standardized coefficients

(95% conf. inter val)

Physico-chemical

Sensorial

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Variable

S t a n d a r d i z e d c o e f f i c i e n t s

Table 3: XLSTAT-PLSPM outputs for the orange juice example

Table 3.1: Block dimensionality

Latent variable Dimensions Critical value Eigenvalues

Physico-chemical 9 1.800 6.213

1.410

1.046

0.317

0.013

Sensorial 7 1.400 4.7441.333

0.820

0.084

0.019

Hedonic 23 4.600 14.655

3.663

2.199

1.837

0.646

Comment: The critical value is equal to the average eigenvalue. In this example the number of

eigenvalues is equal to 5 as the number of observations (6) is smaller than the number of variables

and because the variables are centered. Each block can be considered as unidimensional.


17/52

17

Table 3.2: Checking block dimensionality (larger correlation per row is in bold )

Variables/Factors correlations (Physico-chemical):

F1 F2 F3 F4 F5

Glucose 0.914 0.388 -0.057 0.109 -0.013

Fructose 0.913 0.378 -0.083 0.121 -0.050

Saccharose -0.912 0.261 0.286 -0.127 0.049

Sweetening power -0.035 0.947 0.319 -0.020 0.017

pH before centrifugation -0.945 0.019 -0.026 0.325 0.006

pH after centrifugation -0.933 0.071 -0.069 0.346 -0.006

Titer 0.974 -0.144 0.070 0.150 0.062

Citric acid. 0.978 -0.136 0.049 0.144 0.052

Vitamin C 0.212 -0.328 0.916 0.080 -0.035

Variables/Factors correlations (Sensorial):

F1 F2 F3 F4 F5Smell intensity 0.460 0.754 -0.468 0.008 0.004

Odor typicity 0.985 0.134 -0.058 0.077 0.041

Pulp 0.722 0.617 0.298 -0.096 -0.031

Taste intensity -0.650 0.429 0.626 0.005 0.048

Acidity -0.913 0.348 -0.021 0.205 -0.057

Bitterness -0.935 0.188 -0.285 -0.028 0.093

Sweetness 0.955 -0.159 0.187 0.161 0.048

Variables/Factors correlations (Hedonic):

F1 F2 F3 F4 F5

judge2 0.894 -0.218 0.307 -0.203 0.132 judge3 0.890 -0.318 -0.310 -0.018 0.104

judge6 0.798 -0.039 -0.166 0.522 -0.247

judge11 0.919 0.051 -0.177 0.278 0.212

judge12 0.814 0.221 0.213 -0.429 -0.243

judge25 0.849 0.422 -0.177 -0.166 0.205

judge30 0.625 0.399 -0.631 -0.058 -0.221

judge31 0.733 -0.565 0.321 0.179 0.090

judge35 0.925 0.035 0.329 0.179 -0.060

judge48 0.852 -0.474 0.207 -0.039 -0.074

judge52 0.948 -0.049 -0.286 0.068 -0.115

judge55 0.878 -0.398 0.131 -0.164 -0.166

judge59 0.438 0.475 0.740 0.176 -0.062

judge60 0.922 0.051 -0.149 -0.154 0.317

judge63 0.733 -0.565 0.321 0.179 0.090

judge68 0.638 0.763 -0.063 -0.072 -0.041

judge77 0.363 0.862 0.235 -0.169 0.203

judge79 0.814 0.221 0.213 -0.429 -0.243

judge84 0.928 0.352 0.115 0.032 0.012

judge86 0.778 -0.410 -0.372 -0.229 -0.187

judge91 0.927 0.043 0.069 0.364 0.043

judge92 0.585 0.348 -0.282 0.676 0.000

judge96 0.755 -0.285 -0.331 -0.430 0.231


18/52

18

Table 3.3: Model validation

Goodness of fit index:

GoF GoF (Bootstrap) Standard error Critical ratio (CR)

Absolute 0.731 0.732 0.049 14.943Relative 0.823 0.801 0.048 17.146

Outer model 0.911 0.852 0.039 23.286

Inner model 0.903 0.940 0.048 18.815

Lower bound(90%)

Upper bound(90%) Minimum

1stQuartile Median

3rdQuartile Maximum

Absolute 0.645 0.799 0.522 0.711 0.738 0.762 0.821

Relative 0.707 0.847 0.525 0.790 0.811 0.824 0.855

Outer model 0.784 0.893 0.707 0.816 0.863 0.865 0.911

Inner model 0.885 0.999 0.669 0.921 0.941 0.966 1.000

Comment: Number of bootstrap samples = 200. Level of the confidence intervals: 90%

Absolute Goodness-of-Fit

2 2

Endogenous VL

1 1( , ) ( ; explaining )

Nb of endogenous LV jh j j i j

j h

GoF Cor x R J

= ξ × ξ ξ ξ∑∑ ∑

where j

j

J p= ∑

Relative Goodness-of-Fit

22

1

2endogenous LV j

Outer model Inner model

( , )( ; explaining )1 1

Nb of endogenous LV

j p

jh j j i jh

j j

Cor x R

J

ξ ξ ξ ξ

λ ρ

= ×∑

∑ ∑

where:- λ j is the first eigenvalue computed from the PCA of block j

- j

ρ is the first canonical correlation between the dependent block j and the

concatenation of all the blocks i explaining the dependent block j.


19/52

19

Table 3.4: Latent Variable validation

Cross-loadings (Monofactorial manifest variables):


Glucose -0.889 -0.584 -0.689Fructose -0.889 -0.574 -0.689

Saccharose 0.931 0.758 0.832

Sweetening power 0.099 0.294 0.242



Titer -0.972 -0.738 -0.789

Citric acid. -0.977 -0.743 -0.798

Vitamin C -0.194 -0.045 -0.023

Smell intensity 0.236 0.411 0.199

Odor typicity 0.814 0.977 0.904

Pulp 0.574 0.709 0.637

Taste intensity -0.397 -0.639 -0.549

Acidity -0.751 -0.925 -0.942

Bitterness -0.784 -0.952 -0.972

Sweetness 0.877 0.968 0.982

judge2 0.646 0.925 0.880

judge3 0.654 0.755 0.860

judge6 0.665 0.667 0.787

judge11 0.873 0.785 0.916

judge12 0.729 0.930 0.834

judge25 0.972 0.817 0.879

judge30 0.750 0.524 0.648

judge31 0.349 0.690 0.689 judge35 0.777 0.936 0.926

judge48 0.470 0.835 0.815

judge52 0.801 0.843 0.938

judge55 0.517 0.876 0.847

judge59 0.533 0.593 0.479

judge60 0.872 0.853 0.924

judge63 0.349 0.690 0.689

judge68 0.910 0.673 0.695

judge77 0.727 0.474 0.432

judge79 0.729 0.930 0.834

judge84 0.957 0.935 0.953

judge86 0.467 0.685 0.742

judge91 0.831 0.846 0.925

judge92 0.724 0.424 0.602

judge96 0.559 0.677 0.731

Comment:- Sweetening power and Vitamin C are not correlated to their block.- Smell intensity is not correlated to its own block- Judges 59 and 77 are weakly correlated to their block.


20/52

20

Table 3.5: Latent Variable weights (non significant weights are in bold )

Latent variable Manifest variablesOuterweight

Outerweight

(Bootstrap)Standard

errorCritical

ratio (CR)

Lowerbound(90%)

Upperbound(90%)

Glucose -0.124 -0.113 0.050 -2.491 -0.147 -0.057

Fructose -0.123 -0.111 0.049 -2.498 -0.144 -0.057

Saccharose 0.154 0.140 0.041 3.720 0.096 0.180

Sweetening power 0.052 0.038 0.093 0.560 -0.115 0.151

pH before centrifugation 0.180 0.159 0.021 8.460 0.121 0.184

pH after centrifugation 0.180 0.159 0.020 9.073 0.121 0.185

Titer -0.148 -0.137 0.017 -8.808 -0.169 -0.111

Citric acid. -0.150 -0.139 0.016 -9.127 -0.171 -0.113

Physico-chemical

Vitamin C -0.007 -0.010 0.085 -0.077 -0.126 0.130

Smell intens ity 0.052 0.033 0.099 0.527 -0.136 0.166

Odor typicity 0.206 0.190 0.039 5.255 0.143 0.241

Pulp 0.145 0.114 0.082 1.759 -0.065 0.207

Taste intensity -0.113 -0.107 0.080 -1.406 -0.196 0.069

Acidity -0.203 -0.190 0.045 -4.472 -0.227 -0.143

Bitterness -0.210 -0.197 0.034 -6.136 -0.240 -0.143

Sensorial

Sweetness 0.223 0.208 0.036 6.148 0.143 0.267

judge2 0.059 0.056 0.012 4.740 0.043 0.064

judge3 0.053 0.051 0.015 3.435 0.034 0.065

judge6 0.050 0.047 0.018 2.749 0.023 0.071

judge11 0.062 0.059 0.019 3.268 0.048 0.080

judge12 0.062 0.058 0.013 4.605 0.037 0.083

judge25 0.067 0.063 0.013 5.247 0.051 0.078

judge30 0.048 0.040 0.020 2.364 0.000 0.065

judge31 0.039 0.036 0.023 1.695 0.000 0.066 judge35 0.064 0.062 0.012 5.530 0.050 0.075

judge48 0.049 0.047 0.016 3.047 0.020 0.064

judge52 0.062 0.061 0.012 5.320 0.049 0.082

judge55 0.052 0.049 0.014 3.717 0.029 0.061

judge59 0.042 0.036 0.026 1.642 -0.021 0.066

judge60 0.065 0.060 0.016 4.065 0.047 0.074

judge63 0.039 0.036 0.023 1.695 0.000 0.066

judge68 0.059 0.051 0.018 3.305 0.000 0.072

judge77 0.045 0.037 0.027 1.649 -0.021 0.065

judge79 0.062 0.058 0.013 4.605 0.037 0.083

judge84 0.071 0.066 0.013 5.420 0.055 0.084 judge86 0.043 0.039 0.020 2.168 -0.007 0.057

judge91 0.063 0.059 0.018 3.499 0.050 0.074

judge92 0.043 0.038 0.028 1.554 -0.007 0.081

Hedonic

judge96 0.046 0.044 0.020 2.345 0.013 0.066

Comment:- Sweetening power and Vitamin C are not significant in block physico-chemical- Smell intensity, Pulp and Taste intensity are not significant in block sensorial- Judges 59, 77, 86 and 92 are not significant in block hedonic.


21/52

21

Table 3.6: Correlations between MV and LV

Correlations:

Latent variable Manifest variables

Standardized

loadings Communalities Redundancies

Standardizedloadings

(Bootstrap)

Standard

errorGlucose -0.889 0.790 -0.850 0.226

Fructose -0.889 0.790 -0.847 0.227

Saccharose 0.931 0.867 0.876 0.268

Sweetening power 0.099 0.010 0.101 0.591pH beforecentrifugation 0.952 0.906 0.968 0.078

pH after centrifugation 0.942 0.887 0.964 0.073

Titer -0.972 0.946 -0.950 0.066

Citric acid. -0.977 0.954 -0.956 0.063

Physico-chemical

Vitamin C -0.194 0.038 -0.203 0.435

Smell intensity 0.411 0.169 0.113 0.285 0.497

Odor typicity 0.977 0.954 0.641 0.940 0.110

Pulp 0.709 0.503 0.338 0.612 0.382

Taste intensity -0.639 0.408 0.274 -0.589 0.404

Acidity -0.925 0.856 0.575 -0.915 0.206

Bitterness -0.952 0.907 0.609 -0.949 0.087

Sensorial

Sweetness 0.968 0.936 0.629 0.967 0.069

judge2 0.880 0.774 0.743 0.859 0.182

judge3 0.860 0.740 0.710 0.828 0.236

judge6 0.787 0.619 0.594 0.736 0.254

judge11 0.916 0.840 0.806 0.885 0.230

judge12 0.834 0.695 0.667 0.852 0.115

judge25 0.879 0.773 0.742 0.896 0.159

judge30 0.648 0.419 0.403 0.570 0.284

judge31 0.689 0.475 0.456 0.619 0.340

judge35 0.926 0.858 0.824 0.923 0.147

judge48 0.815 0.664 0.638 0.785 0.233

judge52 0.938 0.879 0.844 0.936 0.130

judge55 0.847 0.717 0.688 0.809 0.215

judge59 0.479 0.230 0.221 0.447 0.406

judge60 0.924 0.853 0.820 0.893 0.214

judge63 0.689 0.475 0.456 0.619 0.340

judge68 0.695 0.483 0.464 0.685 0.258

judge77 0.432 0.186 0.179 0.426 0.404 judge79 0.834 0.695 0.667 0.852 0.115

judge84 0.953 0.909 0.873 0.943 0.137

judge86 0.742 0.551 0.529 0.670 0.308

judge91 0.925 0.856 0.822 0.895 0.215

judge92 0.602 0.363 0.348 0.531 0.411

Hedonic

judge96 0.731 0.535 0.514 0.709 0.281

Comments:

- Standardized loading = correlation- Communality = squared correlation

- Redundancy = Communality*R2(Dep. LV; Explanatory related LVs)


22/52

22

Table 3.6: Correlations between MV and LV (continued )

Correlations:

Latent variable Manifest variablesCritical ratio

(CR)Lower bound

(90%)Upper bound

(90%)

Glucose -3.938 -0.995 -0.569

Fructose -3.913 -0.998 -0.589

Saccharose 3.476 0.729 0.996

Sweetening power 0.167 -0.860 0.950



Titer -14.729 -1.000 -0.860

Citric acid. -15.413 -1.000 -0.872

Physico-chemical

Vitamin C -0.445 -0.970 0.656

Smell intensity 0.827 -0.684 0.930

Odor typicity 8.912 0.763 0.999

Pulp 1.856 -0.174 0.998

Taste intensity -1.580 -0.998 0.203

Acidity -4.495 -1.000 -0.752

Bitterness -10.967 -1.000 -0.897

Sensorial

Sweetness 13.953 0.940 1.000

judge2 4.832 0.639 0.981

judge3 3.642 0.469 0.999

judge6 3.102 0.347 0.982

judge11 3.993 0.773 0.994

judge12 7.258 0.648 0.999

judge25 5.541 0.742 0.997

judge30 2.277 0.000 0.936 judge31 2.030 0.000 0.991

judge35 6.293 0.825 0.997

judge48 3.493 0.408 0.997

judge52 7.234 0.858 0.998

judge55 3.942 0.425 0.996

judge59 1.179 -0.426 0.948

judge60 4.315 0.716 0.997

judge63 2.030 0.000 0.991

judge68 2.694 0.000 0.997

judge77 1.069 -0.426 0.896

judge79 7.258 0.648 0.999 judge84 6.934 0.911 0.998

judge86 2.411 -0.093 0.992

judge91 4.301 0.783 0.986

judge92 1.465 -0.192 0.982

Hedonic

judge96 2.602 0.173 0.997

Comment: (identical with those for weights) - Sweetening power and Vitamin C are not significant in block physico-chemical- Smell intensity, Pulp and Taste intensity are not significant in block sensorial- Judges 59, 77, 86 and 92 are not significant in block hedonic.


23/52

23

Table 3.7: Inner model

R² (Sensor ial):

R² R²(Bootstrap)Standard

error Critical ratio (CR)Lower bound

(90%) Upper bound (90%)

0.672 0.791 0.157 4.276 0.588 0.999

Path coefficients (Sensorial):

Latent variable Value Standard error t Pr > |t| Value(Bootstrap)

Physico-chimique 0.820 0.286 2.864 0.046 0.835

Latent variableStandard

error(Bootstrap)Critical ratio

(CR)Lower bound

(90%) Upper bound (90%)

Physico-chemical 0.308 2.660 0.757 0.994

Comment:

The usual Student t test and the bootstrap approach give here the same results.

R² (Hedonic):

R² R²(Bootstrap)Standard

error Critical ratio (CR) Lower bound (90%) Upper bound (90%)

0.960 0.986 0.017 58.017 0.947 1.000

Path coefficients (Hedonic):

Latent variable Value Standard error t Pr > |t|


Sensorial 0.713 0.201 3.546 0.038


Latent variable Value(Bootstrap)Standard

error(Bootstrap)Critical ratio

(CR)Lower bound

(90%)Upper bound

(90%)

Physico-chemical 0.331 0.698 0.438 -0.642 1.000

Sensorial 0.651 0.674 1.058 0.000 1.397

Comment:

The usual Student t test and the bootstrap approach give here the same results. But the non-

significance of the physico-chemical can also be due to a multicolinearity problem. PLS regression

for estimating the structural regression equations can be used and is presented in Table 4.


24/52

24

Table 3.8: Impact and contribution of the variables to Hedonic

Impact and contribution of the variables to Hedonic:

Sensorial Physico-chemical

Correlation 0.964 0.891Path coefficient 0.713 0.306

Correlation * path coefficient 0.688 0.273

Contribution to R² (%) 71.612 28.388

Cumulative % 71.612 100.000

Impact and contr ibution of the variables to Hedonic

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Senso rial P hysico -chemical

Latent variable

P a t h c o e f f i c i e n t s

0

20

40

60

80

100

C o n t r i b u t i o n t o R ² ( % )

Path coeff icient Cumulative %

Comment:

-2

1ˆ( ; ,..., ) ( , )*

k j j R Y X X Cor Y X β = ∑

- When all the terms ˆ( , )* j j

Cor Y X β are positive, it makes sense to compute the

relative contribution of each explanatory variable X j to the R square.


25/52

25

Table 3.9: Model assessment

Latent variable Type Mean R² Adjusted

R²

Weighted MeanCommunalities

(AVE) Mean Redundancies

Physico-chemical Exogenous 0.000 0.687

Sensorial Endogenous 0.000 0.672 0.672 0.676 0.454

Hedonic Endogenous 0.000 0.960 0.950 0.634 0.609

Mean 0.816 0.654 0.532

Comments:

- The weighted mean takes into account the number of MVs in each block.- (Absolute GoF)2 = (Mean R2)*(Weighted mean communalities)

Table 3.10: Correlation between the latent variables

Correlations (Latent variable):


Physico-chemical 1.000 0.820 0.891

Sensorial 0.820 1.000 0.964

Hedonic 0.891 0.964 1.000

Table 3.11: Direct, indirect and total effects

Direct effects (Latent variable):


Physico-chemical

Sensorial 0.820

Hedonic 0.306 0.713

Comment :

- Sensorial = .820*Physico-chemical

- Hedonic = .306*Physico-chemical + .713*Sensorial


26/52

26

Table 3.11: Direct, indirect and total effects (continued )

Indirect effects (Latent variable):


Physico-chemicalSensorial 0.000

Hedonic 0.585 0.000

Comment :

- Hedonic = .306*Physico-chemical + .713*.820*Physico-chemical- Indirect effect of Physico-chemical on Hedonic = .713*.820 = 0.585

Total effects (Latent variable):


Physico-chemical

Sensorial 0.820

Hedonic 0.891 0.713

Comment :

Hedonic = .306*Physico-chemical + .713*.820*Physico-chemical

= .891*Physico-chemical

Table 3.12: Discriminant validity

Discriminant validity (Squared correlations < AVE) :


Physico-chemical 1 0.672 0.793

Sensorial 0.672 1 0.930

Hedonic 0.793 0.930 1

Mean Communalities (AVE) 0.687 0.676 0.634

Comment: Due to non significant MV’s, the AVE criterion is too small for the three LV’s.


27/52

27

Table 3.13: Latent variable score

Summary statistics / Latent variable scores:

Variable Observations Minimum Maximum MeanStd.

Deviation

Physico-chemical 6 -1.680 1.120 0.000 1.000

Sensorial 6 -1.381 1.378 0.000 1.000

Hedonic 6 -1.203 1.253 0.000 1.000

Latent variable scores :


pampryl r. t. -0.810 -1.381 -1.203

tropicana r. t. 1.120 0.462 0.742

fruvita refr. 0.917 0.964 1.253

joker r. t. -1.680 -0.852 -0.991

tropicana refr. 0.630 1.378 0.946

pampryl refr. -0.176 -0.570 -0.747

Table 4: PLS regression of Hedonic score on Physico-chemical and sensorial scores

Goodness of fit statistics (Variable Hedonic):

R² 0.948

Bootstrap validation


Latent variable ValueValue

(Bootstrap)Standard error

(Bootstrap)Critical ratio

(CR)Lower bound

(90%)Upper bound

(90%)

Physico-chemical 0.490 0.267 0.408 1.201 -0.422 0.893

Sensorial 0.531 0.744 0.402 1.320 0.103 1.397

Jack-knife validation on the observed latent variables

Standardized coefficients (Variable Hedonic):

Variable CoefficientStd.

deviationLower bound

(95%)Upper bound

(95%)


Sensorial 0.531 0.022 0.488 0.573


28/52

28

3. Comparison between PLS, ULS-SEM and PCA

Comparison between weights

When we compare the weight confidence intervals computed with PLS (Table 3.5) with those

coming from ULS-SEM (Table 2.1), we find that both methods yield to the same non significantweights with only one exception for Judge 86 (non significant for PLS and significant for ULS-

SEM). These weights are compared in Figure 6.

Figure 6: Comparison between the PLS and ULS-SEM weights


29/52

29

Comparison between PLS and ULS-SEM scores

The scores coming from PLS and ULS-SEM are compared in Figure 7. They are highly correlated.

This confirms our previous findings and a general remark of Noonan and Wold (1982) on the fact

that the final outer LV estimates depend very little on the selected scheme of calculation of the inner

LV estimates.

Figure 7: Comparison between the PLS and ULS-SEM scores


30/52

30

Comparison between the PLS and ULS-SEM scores and the block principal components

The correlations between the PLS and USL-SEM scores with the block principal components are

given in Table 5.

Table 5: Correlation between the PLS and ULS-SEM scores and the block principal components

ULS-SEM scores PLS scores

Physico-chemical 1st PC .999 .997

Sensorial 1st PC .998 .998

Hedonic 1st PC .999 .997

We may conclude that ULS-SEM, PLS and principal component analysis are giving practically the

same scores on this orange juice example.

II. Exploratory factor analysis, ULS-SEM and PCA

If the structural model is limited to one standardized latent variable (or common factor) ξ described by a vector x composed of p centred manifest variables, one gets the decomposition

(20) ξ = +xx λ δ

It is usual to add the following hypotheses:

(21)

( ) 0,

( ) ( ') is diagonal( , ) 0

E

Cov E Cov ξ

=

= ==

δ

δ

Θ δ δδδ

Under these hypotheses, the covariance matrixΣ

of the random vector x is written as

(22) '( ') E = = +x x δΣ xx λ λ Θ

The parametersλx and Θδ in model (22) can now be estimated using the ULS method. This means

searching for the parameters ˆ xλ andˆ

δΘ , minimizing the criterion

(23)2

'ˆ ˆ ˆ( )− +x xx δS λ λ Θ

where S is the matrix of empirical covariances. To remove the indetermination on the global sign of

the vector ˆxλ (if

ˆxλ is a solution, then -

ˆxλ is also a solution), the solution can be chosen to make the

sum of the coordinates positive. This is the option chosen in AMOS 6.0.

The advantage of the ULS method over the other more frequently used GLS (Generalized Least

Squares) or ML (Maximum Likelihood) methods lies in its ability to function with a singular

covariance matrix S, particularly in situations where the number of observations is less than the

number of variables.


31/52

31

The quality of the fit is measured by the GFI written here as

(24)

2'

2

ˆ ˆ ˆ( )1GFI

− += −

x xx δS λ λ Θ

S

Principal component analysis (PCA) is found again if one imposes the additional condition

(25) ˆ δΘ = 0

In this case, one seeks to minimise the criterion

(26)2

'ˆ ˆ− x xS λ λ

The vector ˆ xλ is now equal to 1 1λ u , where u1 is the normed eigenvector of the covariance matrix

S associated with the largest eigenvalue λ1.

For each MV x j, the explained variance (or communality) is therefore2

1 1ˆ

jj juσ λ = . The residual

variance (or specificity) θ j is then estimated by2

1 1ˆ j jj j

s uθ λ = − .

The quality of the fit can still be measured by the GFI :

(27)

( )2 2

2

ˆ ˆ ˆ

1

jj jj

j

s

GFI

σ − −

= −∑'x xS - λ λ

S

The square of the norm of S is equal to the sum of the squared eigenvalues λh of S. In PCA,2

ˆ ˆ 'x x

S - λ λ is equal to the sum of the squares of the p-1 last eigenvalues of S. Consequently, in PCA

one obtains

(28)

( )2

2 2

1 1 1

2

1

jj j

j

p

h

h

s u

GFI

λ λ

λ

=

+ −

=∑

∑

Moreover, SEM softwares allow the computation of confidence intervals for parameters by

bootstrapping. They also allow criterion (26) to be minimised by imposing value constraints or

equality constraints on the coordinates of the vector ˆxλ . We can continue to use criterion (27) to

measure the quality of the model.


32/52

32

Link between ULS-SEM, Factor Analysis, PLS and Principal Component Analysis

A central point in PLS path modelling concerns the relation between the MV’s related to one LV and

this LV.

Reflective mode

The reflective mode is common to PLS and SEM. In this mode, each MV is related to its LV by a

simple regression:

(29) j j j x λ ξ δ = +

This model corresponds to the usual one-dimension factor analysis (FA) model. Minimization of

criterion (23) allows the estimation of the parameters of this model. As the diagonal terms of the

residual matrix 'ˆ ˆ ˆ( )− +x xx δ

S λ λ Θ are automatically null, the path coefficient λ j are computed with the

objective of reconstruction of the covariance matrix terms outside the diagonal. The averagevariance extracted ( AVE ), defined by ˆ /

jj jj

j j

sσ ∑ ∑ , measures the summary power of the LV. It is

not the first objective in this approach. It is an a posteriori value of the model.

In a one block of variables situation, it is natural to estimate the LV ξ using the first principal

component of the MV’s. The minimization of criterion (26) yields to this solution. Furthermore, the

diagonal terms of the residual 'ˆ ˆ−x xS λ λ are now taken into account in the minimization. The path

coefficients λ j are now computed with the objective of reconstruction of the whole covariance matrix

terms, diagonal included. The AVE still measures the summary power of the LV. But it is now a

part of the objective in this approach. Consequently, in the ULS-SEM context, PCA can be obtained

by considering the FA model (22) and then by cancelling in a first step the residual measurementvariances.

In PLS path modelling softwares, the one block situation has been implemented. In this situation,

the outer estimate of the block LV is also taken as the inner estimate. Therefore, Mode A yields to

the following equation:

(30) ˆ ˆ( , ) j j

j

Cov x xξ ξ ∝ ∑

The PLS algorithm will converge to the first principal component of the block of MV’s, solution of

equation (30).

Formative mode

The formative mode is easy to implement in PLS. In this mode, each LV is related to its MV’s by a

multiple regression:

(31) j j

j

xξ β δ = ∑ +

But in a one block situation, it is an indeterminate problem.


33/52

33

Conclusion

The residual sum of squares ( RESS ), defined by ( )2

ˆij ij

i j

s σ <

−∑ , is smaller for FA than for PCA. On

the other hand the AVE is larger for PCA than for FA.

Example 2

We use data on the cubic capacity, power output, speed, weight, width and length of 24 car models

in production in 2004 given in Tenenhaus (2007). We compare on these data FA and PCA with

respect to the RESS and AVE criterions. The analyses are carried out on standardized variables. The

correlation matrix is given in Table 6.

Table 6: Car example: Correlation matrix

Capacity Power Speed Weight Width Length

Capacity 1 0.954 0.885 0.692 0.706 0.664Power 1 0.934 0.529 0.730 0.527

Speed 1 0.466 0.619 0.578

Weight 1 0.477 0.795

Width 1 0.591

Length 1

The path models for one-dimension FA and PCA are given in Figure 8. The common factor is

denominated as F1. The implied covariance matrices and the residual matrices produced by AMOS

are given in Table 7.

FA

1.00

F1

Capacity

.02

e1

.99

1

Power

.14

e2

.93

1

Speed

.25

e3.87

1

Weight

.53

e4

.68 1

Width

.45

e5

.74

1

Length

.47

e6

.73

1

PCA

1.00

F1

Capacity

.00

e1

.96

1

Power

.00

e2

.92

1

Speed

.00

e3.89

1

Weight

.00

e4

.76 1

Width

.00

e5

.80

1

Length

.00

e6

.80

1

Figure 8 : Path model for FA and PCA


34/52

34

Table 7: Car example: Implied covariance matrices and residuals produced by AMOS


Capacity 1 .918 .860 .678 .737 .722

Power 1 .804 .633 .689 .674

Speed 1 .593 .645 .632

Weight 1 .508 .498

Width 1 .541

Implied

correlations

Length 1


Capacity 0 0.036 0.025 0.014 -0.031 -0.058

Power 0 0.130 -0.104 0.041 -0.147

Speed 0 -0.127 -0.026 -0.054

Weight 0 -0.031 0.297

Width 0 0.050

FA

Residuals

Length 0

Capacity Power Speed Weight Width LengthCapacity .926 .889 .853 .738 .771 .765

Power .853 .818 .699 .740 .734

Speed .785 .671 .710 .705

Weight .573 .606 .602

Width .642 .637

Implied

correlations

Length .632


Capacity 0.074 0.065 0.032 -0.046 -0.065 -0.101

Power 0 0.147 0.116 -0.170 -0.010 -0.207

Speed 0 0 0.215 -0.205 -0.091 -0.127

Weight 0 0 0 0.427 -0.129 0.193

Width 0 0 0 0 0.358 -0.046

PCA

Residuals

Length 0 0 0 0 0 0.368

The comparison between FA and PCA results is shown in Table 8.

Table 8: Comparison between FA and PCA approaches

RESS AVE GFI

FA .169 .690 .983

PCA .230 .735 .978

For PCA, the GFI produced by AMOS has to be modified according to formula (27). The usual

PCA of standardized data results in the following eigenvalues: 4.4113, .8534, .4357, .2359, .0514

and .0124. The quality of the approximation of S by '1 1 1ˆλ + δ

u u Θ is therefore measured by the

following value of the GFI :

(32)

( )2

2 2

1 1 1

2

1

19.459 .519.978

20.436

jj j

j

q

h

h

s u

GFI

λ λ

λ =

+ −+

= = =∑

∑


35/52

35

We then used AMOS 6.0 to carry out a first-order PCA of these standardized data under the

hypothesis of equality of weights for the engine variables "cubic capacity, power, speed" and

similarly equality of weight for the passenger compartment variables "weight, width, length".

Figure 9 shows the results of this estimation and Table 9 the 90% bootstrap confidence intervals.

The bootstrap intervals contain values greater than 1 because the bootstrap samples no longer consist

of standardized variables.

1.00

F1

Capacity

.00

e1

.92

1

Power

.00

e2

.92

1

Speed

.00

e3.92

1

Weight

.00

e4.78 1

Width

.00

e5

.78

1

Length

.00

e6

.78

1

Figure 9 : PCA under constraints on the "Auto 2004" data ( AMOS 6.0 output )

Table 9: PCA under constraints for the "Auto 2004" data ( AMOS 6.0 output )

Estimation and bootstrap confidence interval for the coordinates of x

Parameter Estimate Inf (90%) Sup (90%)

Capacity - F1 .924 .542 1.195

Power - F1 .924 .542 1.195

Speed - F1 .924 .542 1.195

Weight - F1 .784 .555 1.003

Width - F1 .784 .555 1.003Length - F1 .784 .555 1.003

The GFI for the model with constraints has the following value provided by AMOS:

(33)

2

*

2

ˆ ˆ

1 .9505GFI = − ='

x xS - λ λ

S


36/52

36

Using the modified formula yields to:

(34)

( )2

2

*

2

ˆ.509

.9505 .97520.436

j jj x

j

s

GFI GFI

λ −

= + = + =∑

S

The very slight reduction of the GFI (.975 vs .978) means that one can accept the model withconstraints.

In this example, we obtain the component ξ̂ as the "McDonald" estimation of the factor ξ ,

calculated as follows:

ξ̂ ∝ .924(capacity* + power * + speed*) + .784(weight* + length* + width*)

where the asterisk means that the variable is standardized.

III. Confirmatory factor analysis, ULS-SEM and analysis of multi-block data

We assume now that the random column vector x breaks down into J blocks of random vectors

1( ,..., ) ' j j j jp x x=x . A specific model with one standardized latent variable (and usual hypotheses) is

constructed for each block x j:

(35) , 1,..., j j j j

j J ξ = + =x λ δ

This model is similar to model (4) with1

J

j j=

= ⊕xΛ λ . For each block j we have

(36) ' j j j j

= +x δΣ λ λ Θ

and for two blocks j and k we get

(37) ' j k jk j k

ϕ =x xΣ λ λ

where ( , ) jk j k Cor ϕ ξ ξ = .

Decomposition (7) thus becomes

(38) ' j j

⎡ ⎤ ⎡ ⎤= ⊕ ⊕ +⎣ ⎦ ⎣ ⎦ δ

Σ λ Φ λ Θ

The parametersλ1,…, λ J , and Θδ in model (38) can now be estimated by using the ULS method.

This means seeking the parameters 1ˆ ˆ,...,

J λ λ , Φ̂ and ˆ

δΘ minimizing the criterion

(39)2

ˆ ˆ ˆˆ( ) ( ) ' j j

⎡ ⎤− ⊕ ⊕ +⎣ ⎦δS λ Φ λ Θ

Adding constraint (25) gives a new criterion to be minimized:

(40)2

ˆ ˆˆ( ) ( ) ' j j− ⊕ ⊕S λ Φ λ


37/52

37

This results in a new factorisation of the covariance matrix allowing an estimation to be made of

both the loadings and also the correlations between the factors. The quality of the fit is still

measured by the GFI criterion.

Example 3

In this example we are going to study data about wine tasting described in detail in Pagès, Asselin,

Morlat & Robichet (1987).

Description of the data

A collected of 21 red wines of Bourgueil, Chinon and Saumur appellations is described by a set of

27 taste variables divided into 4 blocks:

X 1 = Smell at rest

Rest1 = smell intensity at rest, Rest2 = aromatic quality at rest, Rest3 = fruity note at rest, Rest4 =floral note at rest, Rest5 = spicy note at rest

X 2 = View

View1 = visual intensity, View2 = shading (from orange to purple), View3 = surface impression

X 3 = Smell after shaking

Shaking1 = smell intensity, Shaking2 = smell quality, Shaking3 = fruity note, Shaking4 = floral

note, Shaking5 = spicy note, Shaking6 = vegetable note, Shaking7 = phenolic note, Shaking8 =

aromatic intensity in mouth, Shaking9 = aromatic persistence in mouth, Shaking10 = aromatic

quality in mouth

X 4 = Tasting

Tasting1 = intensity of attack, Tasting2 = acidity, Tasting3 = astringency, Tasting4 = alcohol,

Tasting5 = balance (acidity, astringency, alcohol), Tasting6 = mellowness, Tasting7 = bitterness,

Tasting8 = ending intensity in mouth, Tasting9 = harmony

These data have already been analysed using PLS in Tenenhaus & Esposito Vinzi (2005) and in

Tenenhaus & Hanafi (2007). We present here the ULS-SEM solution on the standardized variables

with cancellation of the residual measurement variances. First of all, we present the PCA for each

separate block in Table 10.


38/52

38

Table 10: Principal component analysis of each block for the "Wine" data

Smell at rest

.741 .551

.915 -.144

.854 -.191

.345 -.537

.077 .933

Smell intensity at rest Aromatic quality at rest

Fruity note at rest

Floral note at rest

Spicy note at rest

1 2

Component

View

.986 -.146

.983 -.163

.947 .320

Visual intensity

Shading (from orange to purple)

Surface impression

1 2

Component

Smell after shaking

.472 .743

.881 -.180

.819 -.176

.328 -.500

.089 .746

-.635 .593.370 .633

.895 .277

.888 .307

.882 -.372

Smell intensity

Smell quality

Fruity note

Floral note

Spicy note

Vegetable notePhelonic note

Aromatic intensity in mouth

Aromatic persistence in mouth

Aromatic quality in mouth

1 2

Component

Tasting

.937 .082

-.257 .691

.775 .427

.774 .378

.844 -.423

.901 -.380

.377 .760

.967 .117

.958 -.233

Intensity of attack

Acidity

Astringency

Alcohol

Balance (acidity, astringency, alcohol)

Mellowness

Bitterness

Ending intensiry in mouth

Harmony

1 2

Component


39/52

39

Use of ULS-SEM for the analysis of multi-block data

All the variables are standardized: S = R . The correlation matrix R is now approximated using

criterion (40), with the aid of the following factorisation formula:

'12 13 141 1

'21 23 242 2

1 4 '31 32 343 3

'41 42 434 4

' ' ' '

1 1 12 1 2 13 1 3 14 1 3

' '

21 2 1 2 2 2

10 0 0 0 0 0

10 0 0 0 0 0( ,..., , )

10 0 0 0 0 0

10 0 0 0 0 0

ϕ ϕ ϕ

ϕ ϕ ϕ

ϕ ϕ ϕ

ϕ ϕ ϕ

ϕ ϕ ϕ

ϕ ϕ

⎡ ⎤⎡ ⎤⎡ ⎤ ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥Φ =⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

=

λ λ

λ λ R λ λ

λ λ

λ λ

λ λ λ λ λ λ λ λ

λ λ λ λ ' '3 2 3 24 2 4' ' ' '

31 3 1 32 3 2 3 3 34 3 4

' ' ' '41 4 1 42 4 2 43 4 3 4 4

ϕ

ϕ ϕ ϕ

ϕ ϕ ϕ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

λ λ λ λ



In this way, confirmatory factor analysis (or perhaps rather confirmatory PCA), in the context

described here, allows the best first order reconstruction of the intra- and inter-block correlations.

First analysis

Using AMOS 6.0, we obtained Table 11 and the diagram shown in Figure 10. Confirmatory factor

analysis of the four blocks echoes in essence the results of the first principal components of the

separate PCA’s for each block. We have put the significant loadings in bold in Table 11. They well

correspond to the strongest variable×PC1 correlations given in Table 10. There are two exceptions:

Smell intensity at rest in block 1 and Astringency in block 4. It should be noted that these variables

have fairly high correlations with the second principal components. The GFI is less than 0.9. This is

due to the existence of the second dimensions for blocks 1, 3 and 4.

Second analysis

In order to better identify the first dimension of the phenomenon under study, it is usual in

confirmatory factor analysis to "purify" the scales: the analysis is repeated, omitting the non-

significant variables. This is where Table 12 and Figure 11 come from. All the correlations between

the manifest and latent variables of the corresponding block, and the correlations between the latent

variables, are now strongly positive. All the correlations are significant. The first dimension of the phenomenon under study has therefore been perfectly identified. The GFI of 0.983 is excellent and

well shows the unidimensionality of the selected variables.


40/52

40

Table 11: Confirmatory factor analysis of the "Wine" data ( AMOS 6.0 output )

(Significant coefficient in bold, non-significant in italic)

Parameter Estimate Inf (95%) Sup (95%) P

Smell intensity at rest


41/52


42/52

42

Table 12: Confirmatory factor analysis of the "Wine" data on

the significant variables ( p-value < .05) of Table 11 ( AMOS 6.0 output )


Aromatic quality at rest


43/52

43

1.00

Rest 1

rest3

.00

e2

.89

rest2

.00

e3

1

1.00

View

view3

.00

e5

1

view2

.00

e6

1

view1

.00

e7

.94

1

1.00

Shaking 1shaking3

.00

e11 .82

1

shaking2

.00

e12.88

1.00

Tasting 1

tasting6

.00

e14tasting5

.00

e15

tasting4

.00

e16

1

tasting1

.00

e19

1

shaking8

.00

e221

shaking9

.00

e231

shaking10

.00

e24

tasting8 .00

e26

tasting9

.00

e27

1

.96

.93

.89.79

1

1

.99

.92

.92

.85

.89

1

1

.97

.92.65

.87

.69

.84

.90

1.04

1

1

.79

Figure 11: Confirmatory factor analysis of the "Wine" data on

the significant variables of Table 9 ( AMOS 6.0 output )


44/52

44

Third analysis

As the four LV’s appearing in Figure 11 are highly correlated, it is natural to summarize these LV’s

through a second order confirmatory factor analysis. This yields to Figure 12. The regression

coefficient of one MV of each block has been put to 1. The second order LV “Score 1” is similar to

the standardized first principal component of the first order LV’s as the error variances have been put to zero. The first order LV’s are evaluated by using the McDonald approach. For example,

using the path coefficients shown in Figure 12, we get:

* *Score(Rest 1) 1 rest2 .88 rest3∝ × + ×

Rest 1

rest3

.00

e2

.88

rest2

.00

e3

1

View

view3

.00

e5

1

view2

.00

e6

1

view1

.00

e7

1.00

1

Shaking 1

shaking3

.00

e11 .961

shaking2

.00

e121.04

Tasting 1

tasting6

.00

e14tasting5

.00

e15

tasting4

.00

e16

1

tasting1

.00

e19

1

shaking8

.00

e221

shaking9

.00

e231

shaking10

.00

e24

tasting8 .00

e26

tasting9

.00

e27

1

.99

.95

.91.82

1

1

1.00

1.09

1.08

1.00

.91

1

1

1.00

.981.12

1

1

1.00

Score 1

.84.83

.82

.94

.00

d2

.00

d1

.00

d3

.00

d4

11

1

1

Figure 12: Second order confirmatory factor analysis of the "Wine" data on

the significant variables of Table 9 ( AMOS 6.0 output )


45/52

45

In the same way, the second order LV (“Score 1”) can be computed as a weighted sum of all the

MV’s. The regression coefficient of “Score 1” in the regression of an MV on “Score 1” is equal to

the product of the path coefficients related to the link between this MV and its LV and to the link

between the LV and “Score 1”. For example

12 12 12 12

(rest2,Score 1)

(Rest 1,Score 1) ( Rest 1 ,Score 1) (Rest 1,Score 1)

(Score 1)

Cov

CovCov Cov

Var λ ε λ λ = + = = ×

as the latent variable “Score 1” is standardized. This leads to:

( ) ( )* * * *Score 1

.83 1 rest2 .88 rest3 .94 .91 tasting1 1 tasting9∝ × × + × + + × × + + ×

But this formula has a severe drawback: it gives more weight to a block containing many variables

than to a block with few variables. From a pragmatic point of view, we prefer to compute aweighted sum of the first order standardized LV estimates, using the path coefficients relating the

first order LV’s to the second order LV. In fact, these weights are reflecting the quality of the

approximation of the second order LV by the first order LV’s. This leads to what is called here

Global score (1):

Global score (1)

.83 Score (Rest 1) .94 Score (Tasting 1)∝ × + + ×

The correlation table between these scores is given in Table 13. All the computed first order LV’s

are well positively correlated and very well summarized by the computed second order LV.

Table 13: Correlation between scores related to the first dimensions of the wine data

Correlations

1 .671 .687 .546 .802

.671 1 .794 .838 .921

.687 .794 1 .897 .942

.546 .838 .897 1 .920

.802 .921 .942 .920 1

Rest 1

View 1

Shaking 1

Tasting 1

Global score 1

Rest 1 View 1 Shaking 1 Tasting 1Globalscore 1

Fourth analysis

To identify the second dimension of the phenomenon under study, we will construct a new

confirmatory PCA model for the manifest variables not taken into account in the second analysis.The non-significant variables were eliminated iteratively as before. This is where Figure 13 and

Table 14 come from. All the correlations between the manifest and latent variables of the

corresponding block, and the correlations between the latent variables, are now strongly positive.All the correlations are significant. The second dimension of the phenomenon studied has therefore

been identified. The value of the GFI is 0.919; this means that this second dimension can beaccepted.


46/52

46

1.00

Rest 2

rest1

.00

e4

1.00

Shaking 2shaking5

.00

e9.78

shaking1

.00

e131

1.00

Tasting 2

tasting3

.00

e17

1

rest5

.00

e20

shaking7

.00

e21

tasting7

.00

e25

.91

1

.94

1

.85

.60

1 .79

.77

.75

.96

.78

1

1

Figure 13: Confirmatory factor analysis of the "Wine" dataon the variables of Table 8 ( AMOS 6.0 output )

Table 14: Confirmatory factor analysis of the "Wine" data on the non-significant variables of Table

9 ( AMOS 6.0 output ). Results after iterative elimination of non-significant variables.


Smell intensity at rest


47/52

47

Fifth analysis

The three LV’s appearing in Figure 13 being highly correlated, they are summarized as above

through a second order confirmatory factor analysis. This yields to Figure 14. Scores related to thesecond dimension are computed in the same way as those related to the first dimension. The

correlation table related to these scores is given in Table 15. Comments are the same as for Table13.

Rest 2

rest1

.00

e4

1

Shaking 2shaking5

.00

e91.34

shaking1

.00

e131

Tasting 2

tasting3

.00

e17

1

rest5

.00

e20

1

shaking7

.00

e21

tasting7

.00

e25

1.08

1

1.62

1.25

1

1.00

1.00

1.00

1

1.00

Score 2

.70

.54

.78

.00

d1

.00

d3

.00

d4

1

1

1

Figure 14: Second order confirmatory factor analysis of the "Wine" data on

the variables of Table 12 ( AMOS 6.0 output )

Table 15: Correlation between scores related to the second dimensions of the wine data

Correlations

1 .758 .776 .908

.758 1 .793 .933

.776 .793 1 .925

.908 .933 .925 1

Rest 2

Shaking 2

Tasting 2

Global score 2

Rest 2 Shaking 2 Tasting 2Globalscore 2


48/52

48

Remarks:

1. The first dimension consists of variables all positively correlated with the global quality grade

(available elsewhere). These correlations are given in Table 16. The second dimension, on the otherhand, is relative to variables not correlated with the global quality grade.

2. It may be wished to obtain orthogonal components in each block. Then, it would be necessary touse the deflation process, i.e. to construct a new analysis on the residuals of the regression of each

original block X j on its first computed latent variable LV j.

Table 16: Correlation between the variables related to the two dimensions

and the global quality grade

Variables related to dimension 1 Global quality

Aromatic quality at rest 0.62

Fruity note at rest 0.50

Visual intensity 0.54 Shading (from orange to purple) 0.51

Surface impression 0.67

Smell quality 0.76

Aromatic intensity in mouth 0.61

Aromatic persistence in mouth 0.68

Aromatic quality in mouth 0.85

Intensity of attack 0.77

Alcohol 0.52

Balance (acidity, astringency, alcohol) 0.95

Mellowness 0.92

Ending intensity in mouth 0.80 Harmony 0.88

Global score 1 0.73

Variables related to di mension 2 Global quality

Smell intensity at rest 0.04

Spicy note at rest -0.31

Smell intensity after shaking 0.17

Spicy note after shaking -0.08

Phelonic note 0.09

Astringency 0.41

Bitterness 0.05 Global score 2 0.08

Graphical displays

Using Global scores (1) and (2), we obtain three graphical displays. The variables are describedwith their correlations with Global scores (1) and (2). The individuals are visualized with these two

global scores using appellation and soil markers. These graphical displays are given in Figures 15,

16 and 17. Figures 16 and 17 show clearly that soil is a much better predictor of wine quality thanappellation. All the wines produced on a reference soil are positive on Score 1. The reader

interested in wine can even detect that the two Saumur 1DAM and 2DAM are the best wines from


49/52

49

this sample. I can testify that I drank outstanding Saumur-Champigny produced at Dampierre-sur-

Loire.

Figure 15: Graphical display of the variables

Figure 16: Graphical display of the wine with appellation markers


50/52

50

Figure 17: Graphical display of the wine with soil markers

IV. Comparison between the ULS-SEM and PLS approaches.

The die is not cast and the ULS-SEM approach is not uniformly more powerful than the PLSapproach. We have set out the "pluses" and "minuses" of each approach in Table 16.

V. Conclusion

Roderick McDonald has thrown a bridge between the SEM and PLS approaches by making use ofthree ideas: (1) using the ULS method, (2) setting the variances of the residual terms of the

measurement model to 0, and (3) estimating the latent variables by using the loadings of the MV’son their LV’s. The McDonald approach has some very promising implications. Using a SEM

software such as AMOS 6.0 makes it possible to get back to PCA, to the analysis of multi-block data

and to a "data analysis" approach for SEM completely similar to the PLS approach. We haveillustrated this process with three examples, corresponding to these different themes. We have listed

the advantages and disadvantages of the two approaches. We end this paper with a wish: that thisULS-SEM approach be included in a PLS-SEM software. The user would then have access to a very

comprehensive toolbox for a "data analysis" approach to structural equation modelling.


51/52

51

Table 16: Comparison between the ULS-SEM and PLS approaches.

ULS-SEM PLS

The

"pluses"

- Global criterion well identified- Use of SEM softwares

- Parameters can be subject toconstraints

- Use of bootstrapping on all themodel parameters

- Better measurement of thequality of the theoretical model

- Non-recursive model allowed

- No identification problem- Systematic convergence of the

PLS algorithm- General framework for multi-

block data analysis- Robust method for small-size

samples

- Possibility of several LV’s per block exists in PLS-Graph

software- Explicit calculation of LV’s

integrated in PLS softwares

- Easy handling of missing data

The

"minuses"

- Possible difficulty in modelidentification

- Possible non-convergence of thealgorithm

- Explicit calculation of LV’s isoutside the SEM software

- Missing data are not permitted

- Algorithm is often closer to anheuristic than to the optimisation

of a global criterion

- It is impossible to imposeconstraints on the parameters

- Measurement of the quality of theinner model is underestimated

- Measurement of the quality of theouter model is overestimated

- Non-recursive model prohibited

References

Arbuckle, J.L. (2005): AMOS 6.0. AMOS Development Corporation, Spring House, PA.

Bollen, K. A. (1989): Structural Equations with Latent Variables, John Wiley & Sons.

Chin W.W. (2001): “PLS-Graph User’s Guide”, C.T. Bauer College of Business, University of Houston,

USA.

Hwang, H. & Takane Y. (2004) : Generalized structured component analysis, Psychometrika, 69, 1, 81-99.

McDonald, R.P. (1996): Path analysis with composite variables, Multivariate Behavioral Research,

31 (2), 239-270.

Noonan, R. & Wold, H. (1982): PLS path modeling with indirectly observed variables: a comparison

of alternative estimates for the latent variable. In: Jöreskog, K.G., Wold, H. (Eds.), Systems under Indirect Observation. North-Holland, Amsterdam, pp. 75–94.

Pagès J., Asselin C., Morlat R., Robichet J. (1987): Analyse factorielle multiple dans le traitement de

données sensorielles : Application à des vins rouges de la vallée de la Loire, Sciences des aliments,

7, 549-571)

Tenenhaus, M. (2007): Statistique : Méthodes pour décrire, expliquer et prévoir , Dunod, Paris.

Tenenhaus M., Esposito Vinzi V., Chatelin Y.-M., Lauro C. (2005): PLS path modeling.Computational Statistics & Data Analysis, 48, 159-205.


52/52

Tenenhaus, M. & Esposito Vinzi, V. (2005): PLS regression, PLS path modeling and generalized

Procustean analysis: a combined approach for multiblock analysis, Journal of Chemometrics, 19,145-153.

Tenenhaus, M. & Hanafi M. (2007): A bridge between PLS path modelling and multi-block data

analysis », in Handbook of Partial Least Squares (PLS): Concepts, Methods and Applications (V.

Esposito Vinzi, W. Chin, J. Henseler, H. Wang, Eds), Volume II in the series of the Handbooks ofComputational Statistics, Springer, in press.

Tenenhaus, M., Pagès, J., Ambroisine, L. & Guinot, C. (2005): PLS methodology to study

relationships between hedonic judgements and product characteristics, Food Quality and Preference, vol. 16, n° 4, pp. 315-325.

XLSTAT (2007): XLSTAT-PLSPM module, XLSTAT software, Addinsoft, Paris.

Date post:	07-Jul-2018
Category:	Documents
Upload:	franckiko2
View:	222 times
Download:	1 times

Structural Equation Modelling for Small Samples

Documents