1. 1 PLS Path Modeling Michel Tenenhaus (tenenhaus@hec.fr)

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7. 7 PLS Methods initiated by Herman Wold, Svante Wold, Harald Martens and Jan-Bernd Lohmöller 1. NIPALS (Nonlinear Iterative Partial Least Squares) 2. PLS Regression (Partial Least Squares Regression) 3. PLS Discriminant Analysis 4. SIMCA (Soft Independent Modeling by Class Analogy) 5. PLS Approach to Structural Equation Modeling 6. N-way PLS 7. PLS Logistic Regression 8. PLS Generalized Linear Model

8. 8 PLS Methods PLS Path Modeling: PLS Approach to Structural Equation Modeling

9. 9 ECSI Path model for a “ Mobile phone provider”

10. 10 Structural Equation Modeling The PLS approach of Herman WOLD Study of a system of linear relationships between latent variables. Each latent variable is described by a set of manifest variables, or summarizes them. Variables can be numerical, ordinal or nominal (no need for normality assumptions). The number of observations can be small compare to the number of variables.

11. 11 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

12. 12 Economic inequality and political instability (Data from Russett, 1964) 47 countries

13. 13 Economic inequality and political instability 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 + + + + - + + + - + + + -

14. 14 Modeling Reflective model (the block is supposed to be uni-dimensionnel) Each manifest variable X jh is written as : X jh =  jh  h +  jh Formative model (the block can be multi-dimensionnel) The latent variable  h is a function of the manifest variables of its block X h : There exists a linear structural relationship between the latent variables: Political instability (  3 ) =  1  Agri. inequality (  1 ) +  2  Ind. development (  2 ) + residual

15. 15 Estimation of latent variables using the PLS approach (1)External (outer) estimation Y h of  h Y h = X h w h (2)Internal (inner) estimation Z h of  h (3)Calculation of w h w hj = cor(Z h, X hj )

16. 16 Estimation of latent variables using the PLS approach (1)External estimation Y h of  h Y h = X h w h Y 1 = w 11 Gini + w 12 Farm + w 13 Rent Y 2 = X 2 w 2 Y 3 = X 3 w 3

17. 17 Estimation of latent variables using the PLS approach (2)Internal estimation Z h of  h (Centroid scheme) Z 1 = sign(cor(  1,  3 )Y 3 = (+1)Y 3 Z 2 = sign(cor(  2,  3 )Y 3 = (-1)Y 3 Z 3 = sign(cor(  3,  1 )Y 1 + sign(cor(  3,  2 )Y 2 = (+1)Y 1 + (-1)Y 2 11 22 33 + -

18. 18 (3)Calculation of the weights w h w hj = cor(X hj, Z h ) w 11 = cor(Gini, Z 1 ) w 12 = cor(Farm, Z 1 ) w 13 = cor(Rent, Z 1 ) And the same way for the other w hj. Estimation of latent variables using the PLS approach

19. 19 Option “1” : All weights are equal to 1. Option “–1” : All weights equal to 1, except the last one put to –1. w 11,initial = 1 w 12,initial = 1 w 13,initial = -1 Weight initialization in PLS-graph This choice allows some sign control: If the variable with the largest weight is put on last position, this weight will have a good chance to be negative.

20. 20 Economic inequality and political instability Estimation of latent variables with PLS Approach (1) External estimation Y 1 = X 1 w 1 Y 2 = X 2 w 2 Y 3 = X 3 w 3 (2) Internal estimation Z 1 = Y 3 Z 2 = -Y 3 Z 3 = Y 1 - Y 2 (3) Calculation of w h w 1j = cor(X 1j, Z 1 ) w 2j = cor(X 2j, Z 2 ) w 3j = cor(X 3j, Z 3 ) Algorithm Begin with arbitrary weights w 1, w 2, w 3. Get new weights w h by using (1) to (3). Iterate until convergence.

21. 21 Use of PLS-Graph (Wynne Chin)

22. 22 Résults (Corrélation) Loading= coeff. de régression de X hj sur Y h, = cor(X hj, Y h ) si les X sont centrées-réduites Outer Model =============================== Variable Weight Loading ------------------------------- Ineg_agri outward gini 0.4567 0.9745 farm 0.5125 0.9857 rent 0.1018 0.5156 ------------------------------- Dev_ind outward gnpr 0.5113 0.9501 labo -0.5384 -0.9551 ------------------------------- Inst_pol outward inst 0.1187 0.3676 ecks 0.2855 0.8241 death 0.2977 0.7910 demostab -0.3271 -0.8635 demoinst 0.0370 0.1037 dictatur 0.2758 0.7227 =================================

23. 23 Results

24. 24 PLS results

25. 25 Economic inequality and political instability GINI FARM RENT GNPR LABO Agricultural inequality (X 1 ) Industrial development (X 2 ) ECKS DEAT D-STB D-UNS INST DICT Political instability (X 3 ) 11 22 33.974.986.516.950 -.955.368.824.791 -.864.104.723.217 -.692 R2 = 0.618

26. 26 Map of countries : Y 1 = agricultural inequality, Y 2 = industrial development

27. 27 Results Inner Model ======================= Block Mult.RSq ----------------------- Inégalit 0.0000 Développ 0.0000 Instabil 0.6180 ======================== Path coefficients ======================================== Ineg_agr Dev_indu Inst_pol ---------------------------------------- Ineg_agr 0.000 0.000 0.000 Dev_indu 0.000 0.000 0.000 Inst_pol 0.217 -0.692 0.000 ======================================== Correlations of latent variables ======================================== Ineg_agr Dev_indu Inst_pol ---------------------------------------- Ineg_agr 1.000 Dev_indu -0.309 1.000 Inst_pol 0.431 -0.759 1.000 ========================================

28. 28 Results Outer Model ======================================================== Variable Weight Loading Communality Redundancy -------------------------------------------------------- Ineg_agri outward gini 0.4567 0.9745 0.9496 0.0000 farm 0.5125 0.9857 0.9716 0.0000 rent 0.1018 0.5156 0.2659 0.0000 -------------------------------------------------------- Dev_indu outward gnpr 0.5113 0.9501 0.9027 0.0000 labo -0.5384 -0.9551 0.9123 0.0000 -------------------------------------------------------- Inst_pol outward inst 0.1187 0.3676 0.1352 0.0835 ecks 0.2855 0.8241 0.6792 0.4197 death 0.2977 0.7910 0.6257 0.3867 demostab -0.3271 -0.8635 0.7457 0.4608 demoinst 0.0370 0.1037 0.0107 0.0066 dictatur 0.2758 0.7227 0.5223 0.3228 ======================================================== Communality = Cor(X hj, Y h ) 2 = Loading 2 For endogenous LV : Redundancy = Cor 2 (X hj, Y h )*R 2 (Y h, LVs explaining Y h ) Average = 0.28

29. 29 Résultats Inner Model =========================================================== Block Mult.RSq AvCommun AvRedund Goodness of Fit ----------------------------------------------------------- Ineg_agri 0.0000 0.7290 0.0000 Dev_indu 0.0000 0.9075 0.0000 Inst_pol 0.6180 0.4531 0.2800 ----------------------------------------------------------- Average 0.6180 0.6110 0.2800.614 =========================================================== = Average Variance of X h explained by Y h = AVE h A latent variable must explain at least 50% of its block variance. Average Communality = (3*AvCommun 1 + 2*AvCommun 2 + 6*AvCommun 3 )/11 Value of the external model Value of the internal model

30. 30 A global index of model fit PLS Goodness of Fit Inner Model =================================================== Block Mult.RSq AvCommun AvRedund GoF --------------------------------------------------- Ineg_agri 0.0000 0.7290 0.0000 Dev_indu 0.0000 0.9075 0.0000 Inst_pol 0.6180 0.4531 0.2800 --------------------------------------------------- Average 0.618 0.6110 0.2800.614 =================================================== Outer model Inner model

31. 31 Discriminant validity A LV explains more its own MVs than the other LVs AVE and square correlations ======================================== Ineg_agr Dev_indu Inst_pol ---------------------------------------- Ineg_agr 0.729 Dev_indu 0.095 0.907 Inst_pol 0.186 0.576 0.453 ======================================== AVE(Y j ) must be larger than the cor 2 (Y j,Y h ) for all h ????

32. 32 Using PLS-Graph (t=1.705) (t=-7.685) t coming from bootstrap re-sampling

33. 33 Bootstrap validation in PLS-Graph Sign control: Individual sign changes / Construct level changes * Outer Model Loadings: ==================================================================== Entire Mean of Standard T-Statistic sample subsamples error estimate Inégalité agricole: gini 0.9745 0.9584 0.0336 28.9616 farm 0.9857 0.9689 0.0329 29.9339 rent 0.5156 0.4204 0.2462 2.0946 Développement industriel: gnpr 0.9501 0.9489 0.0121 78.3692 labo -0.9551 -0.9536 0.0107 -89.1493 Instabilité politique: inst 0.3676 0.3347 0.1756 2.0932 ecks 0.8241 0.8138 0.0699 11.7920 demostab -0.8635 -0.8520 0.0667 -12.9419 demoinst 0.1037 0.0955 0.1611 0.6438 dictatur 0.7227 0.7195 0.0841 8.5915 death 0.7910 0.7977 0.0528 14.9773 ==================================================================== (*) used here

34. 34 Bootstrap validation in PLS-Graph Sign control Individual sign changes Construct level changes (Default) Each bootstrapped sign weight is automatically put equal to the full sample sign weight. For each LV (Construct) the weights are globally inversed if the new loadings (after inversion) are closer to the full sample loadings than the bootstrapped loadings (before inversion).

35. 35 PLS-Graph : Bootsrap Validation Path Coefficients Table (Entire Sample Estimate): ==================================================================== Inég. Agric. Dev. Indust. Instab. Pol. Inég. Agric. 0.0000 0.0000 0.0000 Dev. Indust. 0.0000 0.0000 0.0000 Inst. Pol. 0.2170 -0.6920 0.0000 ==================================================================== Path Coefficients Table (Mean of Subsamples): ==================================================================== Inég. Agric. Dev. Indust. Instab. Pol. Inég. Agric. 0.0000 0.0000 0.0000 Dev. Indust. 0.0000 0.0000 0.0000 Inst. Pol. 0.2328 -0.6743 0.0000 ==================================================================== Path Coefficients Table (Standard Error): ==================================================================== Inég. Agric. Dev. Indust. Instab. Pol. Inég. Agric. 0.0000 0.0000 0.0000 Dev. Indust. 0.0000 0.0000 0.0000 Instabil 0.1272 0.0900 0.0000 ==================================================================== Path Coefficients Table (T-Statistic) ==================================================================== Inég. Agric. Dev. Indust. Instab. Pol. Inég. Agric. 0.0000 0.0000 0.0000 Dev. Indust. 0.0000 0.0000 0.0000 Inst. Pol. 1.7054 -7.6855 0.0000 ====================================================================

36. 36 SPECIAL CASES OF PLS PATH MODELLING Principal component analysis Multiple factor analysis Canonical correlation analysis Redundancy analysis PLS Regression Generalized canonical correlation analysis (Horst) Generalized canonical correlation analysis (Carroll)

37. 37 Options of the PLS algorithm External estimation Y j = X j w j Mode A (for reflective) : w jh = cor(X jh, Z j ) Mode B (for formative) : w j = (X j ´X j ) -1 X j ´Z j Internal estimation Centroid scheme e ji = sign of cor(Y i,Y j ) Factorial scheme e ji = cor(Y i,Y j ) Path weighting scheme e ji = regression coeff. in the regression of Y j on the Y i ’s

38. 38 The general PLS algorithm wjwj Choice of weights e: Centroid, Factorial or Path weighting scheme Initial step Y j =  X j w j Outer Estimation (standardized) Y j2 Y j1 Y jm ZjZj e j1 e j2 e jm Inner estimation Mode A: w j = Mode B: w j = w  cor(X,Z) Look at the loading, not at the w

39. Some modified multi-block methods for SEM PLS : B, Centroid PLS: B, Factorial c jk = 1 if blocks are linked, 0 otherwise PLS : A, Horst NEW APPROACH PLS: B, Horst (New) MAXDIFF B (Hanafi & Kiers, 2006) PLS : A, Factorial NEW APPROACH

40. 40 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.

41. 41 PLS regression (2 components) dim 1 dim 2 - Mode A for X - Mode A for Y - Deflate only X

42. 42 -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

43. 43 Correlation loadings

44. 44 Redundancy analysis of X on Y (2 components) - Mode A for X - Mode B for Y - Deflate only X dim 1 dim 2

45. 45 Inter-battery factor analysis (2 components) dim 1 dim 2 - Mode A for X - Mode A for Y - Deflate both X and Y

46. 46 Canonical correlation analysis (2 components) dim 1 dim 2 - Mode B for X - Mode B for Y - Deflate both X and Y

47. 47 PLS approach : K blocks X1X1 XKXK......    X1X1...... XKXK X Deflation: On the super-block only

48. 48 A new PLS algorithm Arthur & Michel Tenenhaus c ij = 1 if blocks are linked, 0 otherwise

49. 49 CONCLUSION PLS IS TO COVARIANCE-BASED SEM AS PRINCIPAL COMPONENT ANALYSIS IS TO FACTOR ANALYSIS. WHEN INDIVIDUAL DATA ARE AVAILABLE, SIGNIFICANCE TESTS CAN BE CARRIED OUT WITH PLS BY CROSS VALIDATION METHODS.

50. 50 Michel Tenenhaus tenenhaus@hec.fr European Customer Satisfaction Index PLS Path Modelling versus LISREL

51. 51 The European Customer Satisfaction Index (ECSI) ECSI is an economic indicator that measures customer satisfaction. It is an adaptation of the Swedish Customer Satisfaction Barometer and the American Customer Satisfaction Index (ACSI) proposed by Claes Fornell. Fornell’s methodology is presented.

52. 52 Path model describing causes and consequences of Customer Satisfaction. Image Perceived value Customer Expectation Perceived quality Loyalty Customer satisfaction Complaints Full model in red and blue, Reduced model in red

53. 53 Content of the presentation Use of Fornell’s methodology on the full ECSI model Use of Fornell’s methodology on the reduced model Use of SEM-ML on the reduced model (SEM-ML did not work on the full model) Comparison between PLS and SEM-ML results on the reduced model

54. 54 Measurement Instrument for the Mobile Phone Industry : Examples of latent and manifest variables Customer expectationCustomer satisfaction a) Overall satisfaction b) Fulfilment of expectations c) How well do you think “ your mobile phone provider” compares with your ideal mobil phone provider ?

55. 55 Measurement Instrument for the Mobile Phone Industry : Examples of latent and manifest variables Customer loyalty a) If you would need to choose a new mobile phone provider how likely is it that you would choose “your provider” again ? b) Let us now suppose that other mobile phone providers decide to lower fees and prices, but “your mobile phone provider” stays at the same level as today. At which level of difference (in %) would you choose another phone provider ? c) If a friend or colleague asks you for advice, how likely is it that you would recommend “your mobile phone provider” ? And so on for the other latent variables...

56. 56 Each latent variable is estimated as a weighted average of its manifest variables. PLS Path modeling is used to estimate the weights with Mode A and Centroid scheme options. Path coefficients are computed by multiple regression on the estimated latent variables and t-statistics by cross- validation (bootstrap). I. Study of the complete model using the Fornell’s approach Manifest variables V are transformed from a scale “ 1-10 ” to a scale “ 0 -100 ” :

57. 57 Use of PLS-Graph (Wynne Chin)

58. 58 Results : The weights Outer Model ====================== Variable Weight ---------------------- Image outward IMAG1 0.0147 IMAG2 0.0127 IMAG3 0.0137 IMAG4 0.0177 IMAG5 0.0143 ---------------------- Expectat outward CUEX1 0.0232 CUEX2 0.0224 CUEX3 0.0252 ---------------------- Per_Qual outward PERQ1 0.0098 PERQ2 0.0085 PERQ3 0.0118 PERQ4 0.0094 PERQ5 0.0084 PERQ6 0.0095 PERQ7 0.0129 ---------------------- Outer Model ====================== Variable Weight ---------------------- Per_Valu outward PERV1 0.0239 PERV2 0.0247 ---------------------- Satisfac outward CUSA1 0.0158 CUSA2 0.0231 CUSA3 0.0264 ---------------------- Complain outward CUSCO 0.0397 ---------------------- Loyalty outward CUSL1 0.0185 CUSL2 0.0061 CUSL3 0.0225 ======================

59. 59 Fornell’s computation of the latent variables Example : Customer Satisfaction Index

60. 60 Correlations between manifest variables and latent variables Correlations below 0.5 in absolute value are not shown.

61. 61 Image Perceived value Customer Expectation Perceived quality Loyalty Customer satisfaction Complaint.492 (7.67) R 2 =.242.544 (10.71).066 (1.10).037 (1.14).153 (3.07).211 (2.54).541 (6.93).543 (8.62).201 (3.59).468 (5.18).540 (11.08).049 (1.11) R 2 =.296 R 2 =.335 R 2 =.672 R 2 =.432 R 2 =.292 ECSI Path model for a“ Mobile phone provider” Regression on standardized variables and t-statistics provided by PLS-Graph bootstrap, construct level change option

62. 62 II. Study of the reduced model using Fornell’s approach

63. 63 The new PLS weights For each variable the relative weights sum up to 1.

64. 64 III. Study of the reduced model using AMOS: Model 1 (Standardized Results) Chi-Square = 271 DF = 128 Chi-Square /DF = 2.12 (.63) (.83)

65. 65 Reduced model 2 (Standardized results) Chi-Square = 271 DF = 130 Chi-Square /DF = 2.08 RMSEA = 0.066 H 0 : RMSEA  0.05 : p-value = 0.01

66. 66 Reduced model 2 (Unstandardized results)

67. 67 Specific estimation of the latent variables Each latent variable is estimated as a weighted average of its own manifest variables, using the loadings hj. For example is the Customer Satisfaction Index score. Each coefficient 4j is the regression coefficient of  4 in the regression relating the manifest variable X 4j to its latent variable  4 (similar to PLS weight estimation when mode A is used).

68. 68 Loadings and LISREL weights

69. 69 Comparison between the PLS and LISREL weights PLS RELATIVE WEIGHT.6.5.4.3.2.1 LISREL WEIGHT.6.5.4.3.2.1 0.0

70. 70 Correlations between the PLS latent variables and the specific LISREL latent variables All the correlations are above.998

71. 71 First conclusions If COV-BASED SEM works, the PLS results can be derived from the COV-BASED SEM results. If COV-BASED SEM does not work, PLS is still an alternative. If COV-BASED SEM is not adequate (small number of observations and/or large number of variables) PLS can be used for exploratory purposes.

72. 72 Usual estimation of latent variables in LISREL Proc calis covariance modification data =ecsi outstat=a; lineqs CUEX1 = 1 f1 + e1, CUEX2 = Lambda12 f1 + e2, CUEX3 = Lambda13 f1 + e3,. CUSL1 = 1 f5 + e16, CUSL2 = Lambda52 f5 + e17, CUSL3 = Lambda53 f5 + e18, f2 = beta21 f1 + d2, f3 = beta31 f1 + beta32 f2 + d3, f4 = beta41 f1 + beta42 f2 + beta43 f3 + d4, f5 = beta54 f4 + d5; std e1-e18 = vare1-vare18, d2-d5 = vard2-vard5, f1 = varf1; var CUEX1 CUEX2 CUEX3 PERQ1 PERQ2 PERQ3 PERQ4 PERQ5 PERQ6 PERQ7 PERV1 PERV2 CUSA1 CUSA2 CUSA3 CUSL1 CUSL2 CUSL3; run; proc print data=a (where = (_type_="SCORE")); run;

73. 73 Variable weights for the usual estimation of the latent variables in LISREL CUEX1 CUEX2 CUEX3 PERQ1 PERQ2 PERQ3 f1 0.11102 0.074334 0.055776 0.07362 0.024507 0.050785 f2 0.03242 0.021705 0.016287 0.12987 0.043233 0.089590 f3 -0.00083 -0.000558 -0.000418 0.01235 0.004112 0.008522 f4 0.01321 0.008842 0.006634 0.04578 0.015241 0.031583 f5 0.00906 0.006064 0.004550 0.03140 0.010453 0.021661 PERQ4 PERQ5 PERQ6 PERQ7 PERV1 PERV2 CUSA1 0.046274 0.049023 0.046702 0.051524 -0.00071 -0.00440 0.030850 0.081633 0.086483 0.082388 0.090894 0.00466 0.02873 0.047098 0.007765 0.008227 0.007837 0.008646 0.10833 0.66864 0.027074 0.028778 0.030488 0.029044 0.032043 0.00992 0.06122 0.093221 0.019737 0.020910 0.019920 0.021977 0.00680 0.04199 0.063936 CUSA2 CUSA3 CUSL1 CUSL2 CUSL3 0.027752 0.03606 0.00387 0.000423 0.01533 0.042369 0.05506 0.00591 0.000645 0.02340 0.024356 0.03165 0.00340 0.000371 0.01345 0.083861 0.10898 0.01170 0.001277 0.04632 0.057516 0.07474 0.10838 0.011830 0.42895

74. 74 Correlations between the PLS latent variables and the usual estimated LISREL latent variables

75. 75 Final conclusions COV-BASED SEM did not work on the full model. COV-BASED SEM gives better results for the inner model (relating the latent variables between them) because the latent variables are space-free. PLS gives better results for the outer model (relating the manifest variables to their latent variables) because each latent variable is constrained to be in its own manifest variables space. If each COV-BASED SEM latent variable is estimated as a weighted average of its own manifest variables, then COV-BASED SEM and PLS give almost identical latent variable estimates (at least on the examples we have studied).

76. 76 PLS Path Modeling and Multiple Table Analysis Application to the study of the cosmetic habits of women in Ile-de-France Christiane Guinot (CERIES/CHANEL) & Michel Tenenhaus (HEC)

77. 77 Objective of the analysis We have applied the PLS approach to a study of the cosmetic habits of women living in Ile-de-France The aim of the project was to obtain a global score describing the propensity to use cosmetic products in this sample Then, we used behavioural and skin characteristic variables, which are known to account for the variation in use of cosmetic products, to check on the relevance of this score

78. 78 The cosmetic products were divided into four blocks corresponding to different cosmetic practices Sun care Body care Face care make-up and eye make-up removers, tonic lotions, day creams, night creams exfoliation products sun protection products for face and for body after-sun products for face and for body Make -up blushers, mascaras, eye shadows, eye pencils, lipsticks, lip shiners and nail polish Data soap, liquid soap, moisturising body cream, hand creams

79. 79 Construction of a global score Manifest variables Cosmetic practices Latent Variable (A) Global score Partial Scores Latent Variables (B) Body care Body care Face care Face care Make-up Make-up Sun care Sun care  1111 2222 3333 4444 Body care Body care Face care Face care Make-up Sun care Inner model : centroid scheme

80. 80 Propensity to use cosmetic products Global score Results 1111 2222 3333 4444 -.24.47.80.56.44.56.46.55.57.72.58.43.53.47.64.73.71.78 -.12.23.40.28.32.40.33.39.41.39.49.39.29.36.31.39.45.44.49.27.42.43.44  Face care Make -up Sun care soap body liquid soap body cream hand cream Body care make-up rem. tonic lotion eye m.up rem. day cream night cream exfoliation pdt protec face after sun face protec body after sun body blusher mascara eye shadow eye pencil lipstick nail polish Soap body liquid soap body cream hand cream make-up remover tonic lotion eye m.up remover day cream night cream exfoliation pdt blusher mascara eye shadow eye pencil lipstick nail polish sun protec. face after sun face sun protec. body after sun body Correlations Regression coefficients

81. 81 Global score Global score = -3.40 -.11 * soaps and toilet soaps for body care +.20 * liquid soaps for body care +.38 * moisturising body creams and milks +.25 * hand creams and milks +.21 * make-up removers +.26 * tonic lotions +.30 * eye make-up removers +.39 * moisturising day creams +.30 * moisturising night creams +.30 * exfoliation products +.26 * blushers +.41 * mascara +.26 * eye shadows +.20 * eye pencils +.33 * lipsticks and lip shiners +.20 * nail polish +.36 * sun protection products for the face +.31 * moisturising after sun products for the face +.38 * sun protection products for the body +.34 * moisturising after sun products for the body Result : Global score

82. 82 Results: partial scores Score body-care Score facial-care Score make-up Score sun-care

83. 83 S_body-care S_facial-care S_make-up S_sun-care S_facial-care 0.24001 S_make-up 0.13462 0.35035 S_sun-care 0.16500 0.19075 0.14273 S_global 0.50263 0.71846 0.67347 0.62071 Result : global score

84. 84 Relevance of the global score Factors influencing the use of cosmetic products To identify behavioural and skin characteristic variables which best account for the variation in the use of cosmetic products, we can relate the global score to the following variables: 4 Professional activity & Socio-professional category 4 Children 4 Sun exposure habits 4 Practice of sport 4 Importance of physical appearance 4 Type of facial skin & type of body skin 4 Age

85. 85 Relevance of the global score E(Score global) = -1.02 +.21 * professional activity +.07 * housewife or student +.00 * retired +.27 * CSP A (craftsmen, trades people, business managers, managerial staff,academics and professionals) +.09 * CSP B (farmers and intermediary professions) +.05 * CSP C (employees and working class people) +.00 * CSP D (retired and non working people) -.21 * without child +.00 * with child +.40 * habits of deliberate exposure to sunlight +.09 * previous habits of deliberate exposure to sunlight +.00 * no habits of deliberate exposure to sunlight -.17 * no sport practised +.00 * sport practised +1.04 * physical appearance is of extreme importance +.89 * physical appearance is of high importance +.50 * physical appearance is of some importance +.00 * physical appearance is of little importance -.06 * oily facial skin +.16 * combination facial skin -.20 * normal facial skin +.00 * dry facial skin -.32 * oily body skin -.57 * combination body skin -.32 * normal body skin +.00 * dry body skin -.00 * age

86. 86 -1.02 E(Global score)= -1.02 +.21 * professional activity +.07 * housewife or student +.00 * retired +.27 * CSP A (craftsmen, trades people, business managers, managerial staff, academics and professionals) +.09 * CSP B (farmers and intermediary professions) +.05 * CSP C (employees and working class people) +.00 * CSP D (retired and non working people) -.21 * without child +.00 * with child +.40 * habits of deliberate exposure to sunlight +.09 * previous habits of deliberate exposure to sunlight +.00 * no habits of deliberate exposure to sunlight -.17 * no sport practised +.00 * sport practised +1.04 * physical appearance is of extreme importance +.89 * physical appearance is of high importance +.50 * physical appearance is of some importance +.00 * physical appearance is of little importance -.06 * oily facial skin +.16 * combination facial skin -.20 * normal facial skin +.00 * dry facial skin -.32 * oily body skin -.57 * combination body skin -.32 * normal body skin +.00 * dry body skin -.00 * age 1.06 A good profile

87. 87 -1.02 E(Global score)= -1.02 +.21 * professional activity +.07 * housewife or student +.00 * retired +.27 * CSP A (craftsmen, trades people, business managers, managerial staff, academics and professionals) +.09 * CSP B (farmers and intermediary professions) +.05 * CSP C (employees and working class people) +.00 * CSP D (retired and non working people) -.21 * without child +.00 * with child +.40 * habits of deliberate exposure to sunlight +.09 * previous habits of deliberate exposure to sunlight +.00 * no habits of deliberate exposure to sunlight -.17 * no sport practised +.00 * sport practised +1.04 * physical appearance is of extreme importance +.89 * physical appearance is of high importance +.50 * physical appearance is of some importance +.00 * physical appearance is of little importance -.06 * oily facial skin +.16 * combination facial skin -.20 * normal facial skin +.00 * dry facial skin -.32 * oily body skin -.57 * combination body skin -.32 * normal body skin +.00 * dry body skin -.00 * age -2.00 A bad profile

88. 88 Conclusion Using PLS approach, we obtain a score presenting the propensity to use cosmetic products by balancing the different types of cosmetic products better than using principal component analysis.

89. 89 Final conclusion « All the proofs of a pudding are in the eating, not in the cooking ». William Camden (1623)

90. 90 Some references on PLS Path Modeling CHIN W.W. (2001) : PLS-Graph User’s Guide, C.T. Bauer College of Business, University of Houston, Houston. CHIN W.W. (1998) : “The partial least squares approach for structural equation modeling”, in: G.A. Marcoulides (Ed.) Modern Methods for Business Research, Lawrence Erlbaum Associates, pp. 295-336. FORNELL C. (1992) : “A National Customer Satisfaction Barometer: The Swedish Experience”, Journal of Marketing, Vol. 56, 6-21. FORNELL C. & CHA J. (1994) : “Partial Least Squares”, in Advanced Methods of Marketing Research, R.P. Bagozzi (Ed.), Basil Blackwell, Cambridge, MA., pp. 52-78. GUINOT, C., LATREILLE, J. & TENENHAUS M.: “PLS Path Modeling and Analysis of Multiple Tables”, Chemometrics and Intelligent Laboratory Systems, Special issue on PLS methods, 58, 2001 (with C. Guinot and J. Latreille). LOHMÖLLER J.-B. (1987) : LVPLS Program Manual, Version 1.8, Zentralarchiv für Empirische Sozialforschung, Köln.

91. 91 LOHMÖLLER J.-B. (1989) : Latent Variables Path Modeling with Partial Least Squares, Physica-Verlag, Heildelberg. PAGÈS J. & TENENHAUS M. (2001) : "Multiple Factor Analysis and PLS Path Modeling", Chemometrics and Intelligent Laboratory Systems, 58, 261-273. TENENHAUS M. (1998) : La Régression PLS. Éditions Technip, Paris TENENHAUS M. (1999) : “L’approche PLS”, Revue de Statistique Appliquée, vol. 47, n°2, pp. 5-40. TENENHAUS M., ESPOSITO VINZI V., CHATELIN Y.-M., LAURO, C. (2005): "PLS Path Modeling", Computational Statistics and Data Analysis. WOLD H. (1985) : “Partial Least Squares”, in Encyclopedia of Statistical Sciences, vol. 6, Kotz, S & Johnson, N.L. (Eds), John Wiley & Sons, New York, pp. 581-591.