1. A criterion-based PLS approach to SEM Michel Tenenhaus (HEC Paris) Arthur Tenenhaus (SUPELEC)

2. 2 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 (> 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

3. 3 Economic inequality and political instability (Data from Russett, 1964)

4. 4 A SEM model GINI FARM RENT GNPR LABO Agricultural inequality (X 1 ) Industrial development (X 2 ) ECKS DEAT D-STB D-INS INST DICT Political instability (X 3 ) 11 22 33 C 13 = 1 C 23 = 1 C 12 = 0

5. 5 Latent Variable outer estimation

6. 6 Some modified multi-block methods for SEM c jk = 1 if blocks are linked, 0 otherwise and c jj = 0 GENERALIZED CANONICAL CORRELATION ANALYSIS GENERALIZED PLS REGRESSION

7. 7 Covariance-based criteria for SEM c jk = 1 if blocks are linked, 0 otherwise and c jj = 0

8. 8 A continuum approach

9. 9 A general procedure to obtain critical points of the criteria Construct the Lagrangian function related to the optimization problem. Cancel the derivative of the Lagrangian function with respect to each w i. Use the Wold’s procedure to solve the stationary equations (  Lohmöller’s procedure). This procedure converges to a critical point of the criterion. The criterion increases at each step of the algorithm.

10. 10 wjwj Initial step Y2Y2 Y1Y1 YJYJ ZjZj e j1 e j2 e jJ Inner estimation Y j = X j w j Outer Estimation The general algorithm Choice of weights e jh : - Horst : e jh = c jh - Centroid : e jh = c jh sign(Cor(Y h, Y j ) - Factorial : e jh = c jh Cov(Y h, Y j ) c jh = 1 if blocks are linked, 0 otherwise and c jj = 0  Iterate until convergence

11. 11 Specific cases PLS Mode B With usual PLS-PM constraints:

12. 12 Specific cases PLS New Mode A With usual PLS regression constraint:

13. 13 I. PLS approach : 2 blocks X1X1   X2X2 (*) (*) Deflation: Working on residuals of the regression of X on the previous LV’s in order to obtain orthogonal LV’s.

14. 14 PLS regression (2 components) dim 1 dim 2 - Mode A for X - Mode A for Y - Deflate only X Use of PLS-GRAPH

15. 15 -4 -3 -2 -1 0 1 2 3 -4-3-2-101234 t[2] t[1] australie belgique canada danemark inde irlande luxembourg pays-bas nouvelle zélande norvège suède suisse royaume-uni états-unis uruguay autriche brésil japon argentine autriche brésil chili colombie costa-rica finlande france grèce italie japon West Germany bolivie cuba rép. dominicaine équateur égypte salvador guatémala honduras irak libye nicaragua panama pèrou philippines pologne sud vietnam espagne taiwan venezuela yougoslavie PLS Regression in SIMCA-P : PLS Scores

16. 16 Correlation loadings

17. 17 Redundancy analysis of X on Y (2 components) - Mode A for X - Mode B for Y - Deflate only X dim 1 dim 2

18. 18 Inter-battery factor analysis (2 components) dim 1 dim 2 - Mode A for X - Mode A for Y - Deflate both X and Y

19. 19 Canonical correlation analysis ( 2 components) dim 1 dim 2 - Mode B for X - Mode B for Y - Deflate both X and Y

20. 20 Barker & Rayens PLS DA - Mode A for X - Mode B for Y - Deflate only on X

21. 21 Barker & Rayens PLS DA Economic inequality vs Political regime Separation between political regimes is improved compared to PLS-DA.

22. 22 II. Hierarchical model : J blocs X1X1 XJXJ......  JJ  X1X1 …….. XJXJ Deflation : On original blocks and/or the super-block Generalized PLS regression Generalized CCA

23. 23 III. Multi-block data analysis SABSCOR : PLS Mode B + Centroid scheme Use of XLSTAT-PLSPM

24. 24 Multiblock data analysis SSQCOR : PLS Mode B + Factorial scheme

25. 25 Practice supports “theory” * * * Criterion optimized by the method (checked on 50 000 random initial weights)

26. 26 IV. Structural Equation Modeling GINI FARM RENT GNPR LABO Agricultural inequality (X 1 ) Industrial development (X 2 ) ECKS DEAT D-STB D-INS INST DICT Political instability (X 3 ) 11 22 33 C 13 =1 C 23 =1 C 12 =0 C ij = 1 if  i and  j are connected = otherwise

27. 27 SABSCOR-PLSPM Mode B + Centroid scheme Y 1 = X 1 w 1 Y 2 = X 2 w 2 Y 3 = X 3 w 3

28. 28 SSQCOR-PLSPM Mode B + Factorial scheme Y 1 = X 1 w 1 Y 2 = X 2 w 2 Y 3 = X 3 w 3

29. 29 Comparison between methods * * * Criterion optimized by the method (checked on 50 000 random initial weights) Practice supports “theory”

30. 30 Mode A + Centroid scheme The criterion optimized by the algorithm, if any, is unknown.

31. 31 SABSCOV-PLSPM New Mode A + Centroid scheme 0.66 0.74 0.11 0.69 -0.72 0.17 0.44 Cov=1.00 0.50 -0.55 0.46 Cov=-1.69 weight  One-step hierarchical PLS Regression Cor=.429 Cor=-.764

32. 32 SABSCOV-PLSPM New Mode A + Factorial scheme 0.66 0.74 0.10 0.69 -0.72 0.17 0.44 Cov=1.00 0.48 -0.56 0.49 Cov=-1.69 weight  One-step hierarchical PLS Regression Cor=.4276 Cor=-.7664

33. Generalized Barker & Rayens PLS-DA SSQCOV-PLSPM New Mode A for X 1 and X 2 and Mode B for Y 0.62 0.75 0.22 0.67 -0.74 0.55 -1.04 -0.72 0.39 weight Cov  One-step hierarchical B&R PLS-DA

34. Generalized Barker & Rayens PLS-DA Agricultural inequality Industrial development

35. 35 Conclusion 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.