1 A bridge between PLS path modeling and ULS-SEM Michel Tenenhaus

2 A SEM tree Svante Wold Harald Martens PLS regression (1983) SIMCA-P The Unscrambler C. Ringle SMART-PLS Chatelin-Esposito Vinzi Fahmy-Jäger-Tenenhaus XLSTAT-PLSPM (2008) W. Chin PLS-Graph Herman Wold NIPALS (1966) PLS approach (1975) J.-B. Lohmöller LVPLS 1.8 (1984) SEM Component-based SEM (Score computation) Covariance-based SEM (CSA) (Model validation) K. Joreskog (LISREL, 1970) R. McDonald(1996) M. Tenenhaus (2001) H. Hwang Y. Takane GSCA (2004) H. Hwang VisualGSCA 1.0 (2007) Generalized Structured Component Analysis (ALS)

3 Covariance-based Structural Equation Modeling Latent Variables : Structural model (Inner model):

4 Structural Equation Modeling Measurement model (outer model) : MV LV MV Endogenous Exogenous

5 Structural Equation Modeling Mixing Structural and measurement models : Residual variances are diagonal matrices

6 Structural Equation Modeling Covariance matrix for manifest variables : Outer model Inner model Exogen. LV Cov. Structural Residual variance Measurement Residual variance

7 Covariance-based SEM ULS algorithm : S = Observed covariance matrix for MV Goodness-of-fit Index (Jöreskog & Sorbum): PCA Generalization

8 Path model describing causes and consequences of Customer Satisfaction: The inner model. value satisfaction Image Perceived Customer Expectation Perceived quality Loyalty Customer Complaints

9 Use of AMOS 6.0 Méthod = ULS This is a computational trick: Residual variances are passed to errors and can always be computed afterwards. First Roderick McDonald’s idea (1996) Measurement residual variances are canceled:

10 Covariance-based SEM ULS algorithm with the McDonald’s constraints: S = Observed covariance matrix for MV Goodness-of-fit Index (Jöreskog & Sorbum):

11 Use of AMOS Méthod = ULS - Measurement residual variances = 0

12 Results Outer LV Estimates: 2 nd McDonald’s idea + Fornell’s idea PLS estimate of LV: - Mode A - LV inner estimate = theoretical LV - LV inner estimate computation is useless. GFI =.869

13 Cross-validation by bootstrap

14 Cross-validation by bootstrap

15 Comparison between the Fornell-ULS and Fornell-PLS standardized weights

16 Comparison between the Fornell-ULS and Fornell-PLS standardized weights and all Cor(LV Fornell-ULS, LV Fornell-PLS ) >.99

17 First particular case : Factor Analysis and Principal Component Analysis

18 First particular case : FA and PCA Factor Analysis Reflective mode Principal component Analysis Formative mode AVE =.69 AVE = a posteriori computation

19 FA vs PCA : Variance reconstruction FA does not care for variance reconstruction PCA cares for variance reconstruction FA (reflective mode) yields to better covariances reconstruction than PCA (formative mode).

20 MIMIC mode (with this new approach) (Multiple effect indicators for multiple causes) Formative mode (multiple cause) Reflective mode (multiple effect)

21 MIMIC mode (usual in CSA) Formative mode (multiple cause) Reflective mode (multiple effect) ? ?

22 MIMIC mode (usual in CSA) Formative mode (multiple cause) Reflective mode (multiple effect) ?

23 MIMIC mode (better) Formative mode (multiple cause) Reflective mode (multiple effect) (Same than before) Proposal: Compute a global score as  PCA oriented vs the dependent block

24 Second particular case : Multi-block data analysis

Sensory analysis of 21 Loire Red Wines (J. Pagès) X 1 = Smell at rest, X 2 = View, X 3 = Smell after shaking, X 4 = Tasting X1X1 X2X2 X3X3 X4X4 3 Appellations4 Soils Illustrative variable 4 blocks of variables

PCA of each block: Correlation loadings

PCA of each block: Correlation loadings GFI =.301

28 Multi-block data analysis = Confirmatory Factor Analysis VIEW SMELL AFTER SHAKING SMELL AT REST SMELL AT REST TASTING GFI =.849

29 First dimension Using MV with significant loadings

30 First global score GFI =.973 2nd order CFA

31 Validation of the first dimension Correlations Rest1 View Shaking1 Tasting1 Score1 Rest1ViewShaking1Tasting1

32 Second dimension

33 2 nd global score GFI =.905

34 Validation of the second dimension Correlations Rest2 Shaking2 Tasting2 Score2 Rest2Shaking2Tasting2

35 Mapping of the correlations with the global scores Score 1 related with quality Score 2 unrelated with quality

36 Correlation with global quality New result. Not obtained with other multi-block data analysis methods, nor with factor analysis of the whole data.

37 Wine visualization in the global score space Wines marked by Appellation

38 Wine visualization in the global score space Wines marked by Soil

DAM = Dampierre-sur-Loire

A soft, warm, blackberry nose. A good core of fruit on the palate with quite well worked tannin and acidity on the finish; Good length and a lot of potential. DECANTER (mai 1997) (DECANTER AWARD ***** : Outstanding quality, a virtually perfect example) Cuvée Lisagathe 1995

42  = Sun exposure  = Sun protection A = Gender (A 1 = Men, A 2 = Women) Data Model  =  0 +  1 A 1 +  2  *A 1 +  3  *A 2 +  (1)  1 < 0?  2 <  3 ? Third particular case : Analysis of covariance between two blocks of binary variables (with C. Guinot et E. Mauger (CERIES))

43   No gender effect   Gender main effect W M  1 = 0,  2 =  3  1  0,  2 =  3   Interaction  *gender W M 2  32  3 Theory: background  =  0 +  1 A 1 +  2  *A 1 +  3  *A 2 +  (1)

44 X 1 X 1 Sun exposure during lifetime (4) X 4 X 4 Sun exposure during hobbies (2) X 5 X 5 Practice of naturism (1) A Gender Y Y Sun protection behavior over the past year (6) X 3 X 3 Sun exposure during nautical sports (2) X 2 X 2 Sun exposure during mountain sports (2) Score for sun protection Theory: background Score for sun exposure

45 Equation (1) is replaced by: Yc =  0 +  1 A 1 +  2 Xw*A 1 +  3 Xw*A 2 +  (2) =  0 +  1 A 1 +  2 (X*A 1 )w +  3 (X*A 2 )w +  (3) Question: How to estimate and test the parameters w, c,  0,  1,  2,  3 ? Theory: background  =  0 +  1 A 1 +  2  *A 1 +  3  *A 2 +  (1)

46 Covariance based SEM with constraints No group effect on the measurement model Y3Y3Y3Y3 Y2Y2Y2Y2 Y1Y1Y1Y1 Y A1A1A1A1 X 1 *A 1 X 2 *A 1 X 3 *A 1 X 1 *A 2 X 2 *A 2 X 3 *A 2 X*A 1 X*A 2 2 2 33 11 w1w1 w1w1 w2w2 w3w3 w2w2 w3w3 c1c1 c2c2 c3c3 Theory: methods

47 We have applied the ULS-SEM to a study on sun- exposure behavior in 8,084 French adults. Development of skin cancers Premature skin ageing. Data came from the SU.VI.MAX study* *Hercberg S. et al. Arch Intern Med. 2004;164: Application: material

48 Use of AMOS method = ULS

Product used besides voluntarily sun exposure periods Nb of days of practice of nautical activites > 400 days Sun exposure during practice of nautical sports Nb of days of lifetime hobbies > 900 days Sun exposure during practice of hobbies Practice of naturism during lifetime Product used for face with SPF > 15 Product used for body with SPF > 15 Several times while sun exposure While sun exposure While sun tanning Nb of days of lifetime hobbies > 900 days Sun exposure during practice of hobbies Nb of days of mountain sport actvities > 200 days Sun exposure during practice of mountain sports Intensity of sun exposure moderate or severe Basking in the sun is important or very important Sun exposure between 11am and 4 pm Sun exposure of body and face. Basking in the sun is important or very important Intensity of sun exposure moderate or severe Nb of days of mountain sport actvities > 200 days Practice of naturism during lifetime... Sun exposure during practice of mountain sports Nb of days of practice of nautical activites > 400 days Sun exposure during practice of nautical sports Sun exposure between 11am and 4 pm Sun exposure of body and face Sun expo. (M) Sun expo. (W) Sun Protection Men d AMOS results [-.20, -.14] [.64,.89] Confidence Interval (Bootstrap) [1.12, 1.38] GFI =.870

50  =  0 +  1 M +  2  *M +  3  *W +  Conclusion 1.Women tend to protect themselves from the sun more than men (  1 < 0). 2.This difference between men and women increases as lifetime sun exposure increases (  3 -  2 > 0) CoefficientsEstimateLowerUpper 22 Sun exposure (Men)  Sun Protection 33 Sun exposure (Women)  Sun Protection 11 Men  Sun Protection Results: AMOS ULS

51 LV estimation using PLS (Mode A, Fornell’s normalisation) Example for Sun Protection:

52 Sun protection over the past year score *SPF: Sun protection factor If sun protection products used while sun tanning If sun protection products used throughout voluntarily sun exposure periods If sun protection products applied several times during sun exposure periods If the sun protection product used for the face has a SPF* over If the sun protection product used for the body has a SPF* over If sun protection products used besides voluntarily sun exposure periods Results

53 Lifetime sun exposure score * Median value of the duration was used as a threshold for dichotomisation 0.14If sun exposure of the body and the face 0.11If sun exposure between 11 a.m. and 4 p.m. 0.07If basking in the sun is declared important or extremely important 0.20If self-assessed intensity of sun exposure is declared moderate or severe 0.10If sun exposure during practice of mountain sports 0.05If the number of days of lifetime mountain sports activities > 200 days* 0.06If sun exposure during practice of nautical sports 0.03If the number of days of lifetime nautical sports activities > 400 days* 0.13If sun exposure during practice of hobbies 0.07If the number of days of lifetime hobby activities > 900 days* 0.03If practice of naturism during lifetime Results

54 Coefficients EstimateLowerUpper 22 Sun exposure (Men)  Sun Protection 33 Sun exposure (Women)  Sun Protection 11 Men  Sun Protection Standard Parameter Estimate Error t Value Pr > |t| Intercept B <.0001 score_x1_protect B <.0001 GENRE Femmes B <.0001 GENRE Hommes B... score_x1_protec*GENRE Femmes B <.0001 score_x1_protec*GENRE Hommes B... Main effect Gender + Interaction Sun exposure*Gender are highly significant. Results: analysis of covariance

55 When mode A is chosen, outer LV estimates using Covariance-based SEM (ULS or ML) or Component based SEM (PLS) are always very close. It is possible to mimic PLS with a covariance-based SEM software (McDonald,1996, Tenenhaus, 2001). Covariance-based SEM authorizes to implement constraints on the model parameters. This is impossible with PLS. Conclusion 1: SEM-ULS > PLS

56 When SEM-ULS does not converge or does not give an admissible solution, PLS is an attractive alternative. PLS offers many optimization criterions for the LV search (but rigorous proofs are still to be found). PLS still works when the number of MV is very high and the number of cases very small. The new software XLSTAT-PLSPM will be available at the beginning of Conclusion 2: PLS > SEM-ULS

57 References - M. Tenenhaus, E. Mauger, C. Guinot : « Use of ULS-SEM and PLS-SEM to measure interaction effect in a regression model relating two blocks of binary variables » in Handbook of Partial Least Squares (PLS): Concepts, Methods and Applications (V. Esposito Vinzi, J. Henseler, W. Chin, H. Wang, Eds), Springer, M. Tenenhaus : « A bridge between PLS path modeling and ULS-SEM » PLS’07, Oslo. - Tenenhaus M., Esposito Vinzi V., Chatelin Y.-M., Lauro C. (2005) : « PLS path modeling » Computational Statistics & Data Analysis, 48,

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