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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 , - PowerPoint PPT Presentation
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Mohamed HanafiMohamed Hanafi
PLS PATH MODELLING : Computation of latent PLS PATH MODELLING : Computation of latent variables with the estimation mode B variables with the estimation mode B
UNITE DE SENSOMETRIE ET CHIMIOMETRIENantes-France
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References References
Jan-Bernd Lohmöller, 1989. Latent variable path modelling with partial least squares.Physica-Verlag, Heildelberg
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.
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Data sets Data sets
1X2X mX
Several groups of variables Multiple data sets Multiblock data sets Partitioned matrices
n
p1p2 pm
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Path Model Path Model
1Xn
p1
4X
p4
n
2X
p2
n
3X
p3
n
Path : • is specified by the investigator• likes to explore a specific point of view from the data• directed graph
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PLS PM = One principle and two models PLS PM = One principle and two models
All information between blocks of observable is assumed to be All information between blocks of observable is assumed to be conveyed by conveyed by latent variables (linear combination of variables)latent variables (linear combination of variables)..
PrinciplePrinciple
Outer Model ( Factor model, measurement model) relating Manifest variables to their LVshows 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
1z
3z 4z
2z
z
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Real Application : Real Application : European Customer Satisfaction Model (ECSM)European Customer Satisfaction Model (ECSM)
1z
2z
3z 5z
Perceived quality
Customer Expectation
Perceived Value
Custumer satisfaction
4z
Image
6z
Loyalty
7z
Complaints
ECSM is based on well-established theories and applicable for a number of different industries
Fornell, C. (1992).Journal of Marketing, 56, 6-21.
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PLS PM for two blocksPLS PM for two blocks
1X2Xn n
Applications Applications
Ecology Food science Biospectroscopy Ect….
p1 p2
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PLS PM for two blocks : models PLS PM for two blocks : models
2z1z
,,...,2,1 ,0 kkjk
kj
kkj pj ezx
uzz 1102 bb
Inner model
Outer Model ( Factor model, measurement model) relating Manifest variables to their LVshows the manifest variables as depending on the LV
z
Inner Model(Structural model, Path model) relating endogeneous LV to other LVs shows the LV as dependent on each other
1z 2z
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PLS PM for two blocks : Estimation PLS PM for two blocks : Estimation
Estimated parametersEstimated parameters ComputationComputation
Latent variablesLatent variables
IterativeIterative
Outer model Outer model
Inner modelInner model
OLSOLS
1'' kkkk wXXw
kkk wXz
kj
k ,0
2,1k
10 ,bb
Inner and outer models are not estimated simultaneously!!!
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Computation of latentes variables Computation of latentes variables Two estimation modesTwo estimation modes
1,2 normalize 4
3
A mode
A mode 2
1,2 , 1
1
11
1'2
12
2'1
11
k
k
sk
skk
sk
ss
ss
skk
sk
z
wXz
zXw
zXw
wXz
s1z
ss1
'2
12 zXw
ss1
'2
12
'2
12 zXXXw
MODE A for X2
MODE B for X2
s1z
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Compact description of the algorithm Compact description of the algorithm
XX11MODE AMODE A MODE BMODE B
XX22
MODE AMODE A
MODE BMODE B
ss
ss
1'2
12
2'1
11
zXw
zXw
ss
ss
1'2
1
2'2
12
2'1
1
1'1
11
zXXXw
zXXXw
ss
ss
1'2
1
2'2
12
2'1
11
zXXXw
zXw
ss
ss
1'2
12
2'1
1
1'1
11
zXw
zXXXw
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Link with Power Method Link with Power Method
XX11MODE AMODE A MODE BMODE B
XX22
MODE AMODE A
MODE BMODE B
s
ss
2
212
Az
Azz
'11
'22 XXXX
'11
'22 PPPP
'11
'22 PPXX
'11
'22 XXPP
2
1'
kkkk XXXP
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Link with psychometric methodsLink with psychometric methods
XX11MODE AMODE A MODE BMODE B
XX22
MODE AMODE A
MODE BMODE B
212 ,covmax zz
2
212
var
,covmax
z
zz 212 ,max zzr
1
212
var
,covmax
z
zz
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
Canonical correlation
Interbattery method
Redundancy Analysis
Redundancy AnalysisTucker, L. R. (1958). Van den Wollenberg. A. L. (1977).
Hotelling H. (1936).
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Several blocksSeveral blocks
kjk
kjj
kj ezx 0
kpj 1
mk 1
Outer model
1X2X mXn
p1 p2 pm
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Inner ModelInner Model
1z
2z
3z3z
4
431,40,44 uzz bb
322,311,30,33 uzzz bbb
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PLS PM : EstimationPLS PM : Estimation
Estimated parametersEstimated parameters ComputationComputation
Latent variablesLatent variables
IterativeIterative
Outer modeOuter mode
Inner modelInner model
OLSOLS
1'' kkkk wXXw
kkk wXt mk ,....,2,1
parametersparameters
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Notations Notations
1z
3z 4z
2z
0100
1011
0100
0100
klcC
4321 zzzz
4
3
2
1
z
z
z
z
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Lohmöller’s procedure (mode B)Lohmöller’s procedure (mode B)
1
11
1'1
1
normalize 5
4
1 3
1 , 2
1 , 1
sk
skk
sk
skkkk
sk
sl
skl
k
lkl
sk
sk
mk
mkc
mk
z
wXz
ZXXXw
zZ
z
Jan-Bernd Lohmöller, 1989. Latent variable path modelling with partial least squares.Physica-Verlag, Heildelberg Chapter 2. page 29.
Factorial SchemeFactorial Scheme Centroid SchemeCentroid Scheme
0,1
0,1sl
sk
sl
sk
kl r
r
zz
zzv lkkl r zz ,
Mode AMode A Mode BMode B
skk
sk
sk
sk ZX
ZZw '
'1 1
skkk
sk 1
'1'1 ZXXXw
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1z
3z 4z
2z
3311 , zzzZ r 3322 , zzzZ r 2231133 ,, zzzzzzZ rr 4344 , zzzZ r
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RemarksRemarks
Lohmöller’s procedureLohmöller’s procedure
implemented in various softwares :• PLS Graph (W. Chin)• SPAD • SmartPLS (Ringle and al.)
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Wold’s procedure (Mode B)Wold’s procedure (Mode B)
1
11
1'1
11
normalize 5
4
3
,)),(()),(( 2
, 1
000
00000
0
0
00
0
00
0
sk
skk
sk
skkkk
sk
sl
sk
sl
kllk
sl
sk
sl
kllk
sk
sk
rsigncrsignc
z
wXz
ZXXXw
zzzzzzZ
z
(1) 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.
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RemarksRemarks
Wold’s procedureWold’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.Hanafi, M (2006).Computational Statistics.
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Computational OverviewComputational Overview
algorithmalgorithm ConvergenceConvergence
Latent variablesLatent variables Iterative Iterative YESYES
Outer modelsOuter models
Inner modelsInner models OLSOLS YESYES
Two blocks
algorithmalgorithm ConvergenceConvergence
Latent variablesLatent variables Iterative Iterative ??
Outer modelsOuter models
Inner modelsInner models
OLSOLS YESYES
No problem
More than two Blocks
No problem
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Monotony convergence of Wold’s procedure Monotony convergence of Wold’s procedure
m
lklkklm rcf
1,21 ,,...,, zzzzz
112
1121 ,...,,,...,, s
msss
mss ff zzzzzz
m
lklkklm rch
1,
221 ,,...,, zzzzz
112
1121 ,...,,,...,, s
msss
mss hh zzzzzz
Hanafi, M (2006).Computational StatisticsHanafi, M (2006).Computational Statistics
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Proof : CentroidProof : Centroid
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Proof : FactorialProof : Factorial
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Not the case for Lohmöller’s procedureNot the case for Lohmöller’s procedure
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Path for the exemplePath for the exemple
1z
3z
2z
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CentroidCentroid FactorialFactorial
Wold’s procedureWold’s procedure 79 79 iterationsiterations
73 iterations73 iterations
Lohmöller’ s Lohmöller’ s procedureprocedure
159 159 iterationsiterations
128 iterations128 iterations
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Lohmöller’s procedure revisitedLohmöller’s procedure revisited
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, 63-67Hanafi, 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. Hanafi, M and Kiers, H.A.L. (2006).Computational Statistics and Data Analysis.
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Wold’s procedure depends on starting vectorsWold’s procedure depends on starting vectors
1z
3z
2z
3z
1z
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Value of the Criterion =7.10
Value of the Criterion =10.28
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Characterization of latent variablesCharacterization of latent variables
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Generalized Canonical Correlation Analyses Generalized Canonical Correlation Analyses (CGA)(CGA)
m
kllklkrSUMCOR
,1,
,Max zz
m
kllklkrS
,1,
2 ,Max SQCOR zz
Kettering, J.R. (1971), Bimetrika
[Horst (1965)]
[Kettering (1971)]
An overview for five generalizations of canonical correlation analysis
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Path model for GCAPath model for GCA
1z
3z
2z
3z
1z
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PLS PM and Generalized canonical correlation PLS PM and Generalized canonical correlation
SSQCORFactorialBMODE
SUMCORCentroidBMODE
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ConclusionsConclusions
Two blocksTwo blocks
PLS PM = general framewok for psychometric methodsThe procedures of the computation of the latent variables are equivalent to a power method
More than two blocks ( with mode B for all blocks) 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.
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Perspectives Perspectives
To what extend the solutions obtained by wold’s To what extend the solutions obtained by wold’s procedure are at least a local maximum? procedure are at least a local maximum?
Similar results for mode A and mixed mode ? Similar results for mode A and mixed mode ?
Optimisation principle for Latent variables ?Optimisation principle for Latent variables ?
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Computational OverviewComputational Overview
algorithmalgorithm ConvergenceConvergence OptimalitOptimality y
Latent variablesLatent variables Iterative Iterative YESYES YESYES
Outer modelsOuter models
Inner modelsInner models OLSOLS YESYES
YesYes
No problem
Two blocks
algorithmalgorithm ConvergenceConvergence OptimalitOptimality y
Latent variablesLatent variables Iterative Iterative ?? ??
Outer modelsOuter models OLSOLS YESYES YesYes
Inner modelsInner models OLSOLS YESYES YesYes
No problem
More than two Blocks
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Characterization of latent variablesCharacterization of latent variables
1' kkuuKk ,,2,1
KKKKK
K
K
u
u
u
u
u
u
AA
AA
AA
22
11
2
1
21
221
112
0
0
0
lkklklc PPu '
2
1'
,
kkk
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Kk 1