A criterionbased approach to PLS path modelling

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A criterion-based approachto PLS path modelling

• Michel Tenenhaus (HEC Paris)
• Arthur Tenenhaus (SUPELEC)

PLS09
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Economic inequality and political instability
Data from Russett (1964), in GIFI

• Economic inequality
• Agricultural inequality
• GINI Inequality of land distributions
• FARM farmers that own half of the land (gt
  50)
• RENT farmers that rent all their land
• Industrial development
• GNPR Gross national product per capita (
  1955)
• LABO of labor force employed in agriculture

• Political instability
• INST Instability of executive (45-61)
• ECKS Nb of violent internal war incidents
  (46-61)
• DEAT Nb of people killed as a result of civic
  group violence (50-62)
• D-STAB Stable democracy
• D-UNST Unstable democracy
• DICT Dictatorship

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Economic inequality and political instability
(Data from Russett, 1964)
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A SEM model
Agricultural inequality (X1)
INST
GINI
ECKS
C13 1
FARM
?1
DEAT
RENT
?3
C12 0
D-STB
GNPR
D-INS
?2
C23 1
LABO
DICT
Industrial development (X2)
Political instability (X3)
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Latent Variable outer estimation
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Some modified multi-block methods for SEM
cjk 1 if blocks are linked, 0 otherwise and cjj
0
GENERALIZED CANONICAL CORRELATION ANALYSIS
GENERALIZED PLS REGRESSION
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Covariance-based criteria for SEM
cjk 1 if blocks are linked, 0 otherwise and cjj
0
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A continuum approach
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Construction of monotone convergent algorithms
for these criteria

• Construct the Lagrangian function related to the
  optimization problem.
• Cancel the derivative of the Lagrangian function
  with respect to each wi.
• Use the Wolds procedure to solve the stationary
  equations (? Gauss-Seidel algorithm or ? MAXDIFF
  algorithm).
• This procedure is monotonically convergent the
  criterion increases at each step of the algorithm.

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The general algorithm
Iterate until convergence of the criterion.
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Specific cases (All ?i 0)
PLS Mode B
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Specific cases (All ?i 1)
With usual PLS regression constraint
PLS New Mode A
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Specific cases (Mixing ?j 0 and ?k 1)
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I. PLS approach 2 blocks
()
() Deflation Working on residuals of the
regression of X on the previous
LVs in order to obtain orthogonal LVs.
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II. Structural Equation Modeling
Agricultural inequality (X1)
INST
GINI
ECKS
FARM
?1
DEAT
RENT
?3
D-STB
GNPR
D-INS
?2
LABO
DICT
Industrial development (X2)
Political instability (X3)
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SABSCOR-PLSPMMode B Centroid scheme
(XLSTAT-PLSPM software)
Y1 X1w1
Y3 X3w3
Y2 X2w2
? One-step hierarchical CCA
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SSQCOR-PLSPMMode B Factorial scheme
(XLSTAT-PLSPM software)
Y1 X1w1
Y3 X3w3
Y2 X2w2
? One-step hierarchical CCA
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Comparison between methods
gt
lt
Practice supports theory
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Usual Mode A Centroid schemeXLSTAT-PLSPM
software
The criterion optimized by the algorithm, if any,
is unknown.
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SABSCOV-PLSPMNew Mode A Centroid
scheme(Arthurs R-program)
weight
0.66
Cor.429
0.17
0.74
Cov1.00
0.44
0.11
0.50
-0.55
Cov-1.69
0.69
0.46
Cor-.764
-0.72
? One-step hierarchical PLS Regression
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SABSCOV-PLSPMNew Mode A Factorial
scheme(Arthurs R-program)
weight
0.66
Cor.4276
0.17
0.74
Cov1.00
0.44
0.10
0.48
-0.56
Cov-1.69
0.69
0.49
Cor-.7664
-0.72
? One-step hierarchical PLS Regression
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Generalized Barker Rayens PLS-DASSQCOV-PLSPMNe
w Mode A for X1 and X2 and Mode B for Y
? One-step hierarchical BR PLS-DA
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Generalized Barker Rayens PLS-DA
Industrial development
Agricultural inequality
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III. Multi-block data analysis
SABSCOR PLS Mode B Centroid scheme
Use of XLSTAT-PLSPM
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Multiblock data analysis
SSQCOR PLS Mode B Factorial scheme
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Practice supports theory

gt

lt
(checked on 50 000 random initial weights)
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IV. Hierarchical model J blocs
. . .


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Conclusion 1

• In the PLS approach of Herman Wold, the
  constraint is
• In the PLS regression of Svante Wold, the
  constraint is
• This presentation unifies both approaches.

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Conclusion 2Comparison between methods

• PLS path modelling (with Horst, centroid or
  factorial schemes) is closer to multi-block data
  analysis than to SEM. PLS uses only links
  between blocks to compute the components.
• Generalized Structured Component Analysis (GSCA)
  is closer to SEM than to multi-block data
  analysis. GSCA uses the structural equations to
  compute the components.
• Consider also using McDonald ULS-SEM to compute
  block components. Very efficient for practical
  applications.
• When all blocks are good (? uni-dimensionnal),
  all methods give practically the same components.

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