Pls path modelling computation of latent variables with the estimation mode b
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PLS PATH MODELLING : Computation of latent variables with the estimation mode B. UNITE DE SENSOMETRIE ET CHIMIOMETRIE Nantes-France. Mohamed Hanafi. References. Herman Wold (1985). Partial Least Squares. Encyclopedia of statistical sciences ,

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Pls path modelling computation of latent variables with the estimation mode b
PLS PATH MODELLING : Computation of latent variables with the estimation mode B

UNITE DE SENSOMETRIE ET CHIMIOMETRIE

Nantes-France

Mohamed Hanafi


References
References the estimation mode B

Herman Wold (1985). Partial Least Squares. Encyclopedia of statistical sciences ,

vol 6 Kotz, S & Johnson, N.L(Eds), John Wiley & Sons, New York, pp 581-591.

Jan-Bernd Lohmöller, 1989. Latent variable path modelling with partial least squares.

Physica-Verlag, Heildelberg


Data sets
Data sets the estimation mode B

  • Several groups of variables

  • Multiple data sets

  • Multiblock data sets

  • Partitioned matrices

p2

p1

pm

n


Path model
Path Model the estimation mode B

p1

n

n

n

n

p2

p3

p4

Path :

  • is specified by the investigator

  • likes to explore a specific point of view from the data

  • directed graph


Pls pm one principle and two models
PLS PM = One principle and two models the estimation mode B

Inner Model(Structural model, Path model)

 relating endogeneous LV to other LVs

 shows the LV as dependent on each other

Principle

All information between blocks of observable is assumed to be conveyed by latent variables (linear combination of variables).

Outer Model ( Factor model, measurement model)

 relating Manifest variables to their LV

shows the manifest variables as depending on the LV


Real application european customer satisfaction model ecsm
Real Application : the estimation mode B European Customer Satisfaction Model (ECSM)

ECSM is based on well-established theories and applicable for a number of different industries

Image

Loyalty

Customer Expectation

Perceived Value

Custumer satisfaction

Complaints

Fornell, C. (1992).Journal of Marketing, 56, 6-21.

Perceived quality


Pls pm for two blocks
PLS PM for two blocks the estimation mode B

  • Applications

    • Ecology

    • Food science

    • Biospectroscopy

    • Ect….

p1

p2

n

n


Pls pm for two blocks models
PLS PM for two blocks : models the estimation mode B

Outer Model ( Factor model, measurement model)

 relating Manifest variables to their LV

shows the manifest variables as depending on the LV

Inner Model(Structural model, Path model)

 relating endogeneous LV to other LVs

 shows the LV as dependent on each other

Inner model


PLS PM for two blocks : Estimation the estimation mode B

Inner and outer models are not estimated simultaneously!!!


Computation of latentes variables two estimation modes
Computation of latentes variables the estimation mode B Two estimation modes

MODE A for X2

MODE B for X2



Link with Power Method the estimation mode B


Link with psychometric methods the estimation mode B

Tucker, L. R. (1958).

Van den Wollenberg. A. L. (1977).

Interbattery method

Redundancy Analysis

Hotelling H. (1936).

Canonical correlation

Redundancy Analysis

Hotelling H. (1936). Biometrika, 28, 321-377.

Tucker, L. R. (1958). Psychometrika, 23, 111-136.

Van den Wollenberg. A. L. (1977). Psychometrika, 42, 2, 207-219


Several blocks
Several blocks the estimation mode B

p1

p2

pm

n

Outer model


Inner model
Inner Model the estimation mode B


Pls pm estimation
PLS PM : Estimation the estimation mode B

parameters


Notations
Notations the estimation mode B


Lohm ller s procedure mode b
Lohmöller’s procedure (mode B) the estimation mode B

Jan-Bernd Lohmöller, 1989. Latent variable path modelling with partial least squares.

Physica-Verlag, Heildelberg Chapter 2. page 29.


Remarks
Remarks the estimation mode B

Lohmöller’s procedure

  • implemented in various softwares :

    • PLS Graph (W. Chin)

    • SPAD

    • SmartPLS (Ringle and al.)


Wold s procedure mode b
Wold’s procedure (Mode B) the estimation mode B

  • Herman Wold (1985). Partial Least Squares. Encyclopedia of statistical sciences ,

  • vol 6 Kotz, S & Johnson, N.L(Eds), John Wiley & Sons, New York, pp 581-591.


Remarks1
Remarks the estimation mode B

  • Wold’s procedure

    • proposed by Wold for

      • six blocks

      • Centroid scheme

    • Extended by Hanafi (2006)

      • arbitrary number of blocks

      • take into account the Factorial scheme

Hanafi, M (2006).Computational Statistics.


Computational overview
Computational Overview the estimation mode B

Two blocks

No problem

More than two Blocks

No problem


Monotony convergence of wold s procedure
Monotony convergence of Wold’s procedure the estimation mode B

.

MODE B + CONTROID SCHEME

MODE B + FACTORIAL SCHEME

Hanafi, M (2006).Computational Statistics


Proof centroid
Proof : Centroid the estimation mode B


Proof factorial
Proof : Factorial the estimation mode B



Path for the exemple
Path for the exemple the estimation mode B


Lohm ller s procedure revisited
Lohmöller’s procedure revisited the estimation mode B

  • Hanafi and al (2005)

    • Update ckk=0 by ckk=1 monotonically convergence of the procedure (Mode B+ centroid scheme)

  • Hanafi and al (2006)

    • Alternative procedure

Hanafi, M and Qannari, EM (2005).Computational Statistics and Data Analysis, 48, 63-67

Hanafi, M and Kiers, H.A.L. (2006).Computational Statistics and Data Analysis.



Value of the Criterion =7.10 the estimation mode B

Value of the Criterion =10.28



Generalized canonical correlation analyses cga
Generalized Canonical Correlation Analyses (CGA) the estimation mode B

Kettering, J.R. (1971), Bimetrika

An overview for five generalizations of canonical correlation analysis

[Kettering (1971)]

[Horst (1965)]


Path model for gca
Path model for GCA the estimation mode B



Conclusions
Conclusions the estimation mode B

  • Two blocks

    • PLS PM = general framewok for psychometric methods

    • The procedures of the computation of the latent variables are equivalent to a power method

  • More than two blocks ( with mode B for all blocks)

    • Monotony property of Wold’s procedure

    • Characterization of the latent variable as a solution (among other) of non linear systems of equations

    • Strong link with generalized canonical correlation analysis

    • PLS PM with the estimation mode B can be seen as an extension of CGA.


Perspectives
Perspectives the estimation mode B

  • To what extend the solutions obtained by wold’s procedure are at least a local maximum?

  • Similar results for mode A and mixed mode ?

  • Optimisation principle for Latent variables ?


Computational overview1
Computational Overview the estimation mode B

Two blocks

No problem

More than two Blocks

No problem



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