Laura Trinchera, Giorgio Russolillo, Vincenzo Esposito Vinzi

Partial Least Squares Path Modeling (PLS-PM) [1,3,6] is classically regarded as a component-based approach to Structural Equation Models (SEM) and has been more recently revisited as a general framework for multiple table analysis [4,5]. In PLS-PM a key role is played by the computation of the latent variable (LV) scores. In order to obtain such scores, a system of weights for the manifest variables (MVs) is needed. The computation of these weights depends on the type of relation between the MVs and the corresponding LV. We may assume that each block in the model is outwards directed (also called reflective block, or Mode A in PLS-PM literature) or inwards directed (also called formative block, or Mode B in PLS-PM literature). Outwards directed models require blocks to be homogeneous and unidimensional. Inwards directed models, instead, imply considering blocks as full dimensional. However, in most applications blocks are neither unidimensional nor full dimensional. Due to a certain degree of multicollinearity in each block, it is important to consider just a few dimensions, i.e. we need a measurement model estimation capable to yield solutions somewhere between Mode A and Mode B.

Here we will show two new modes for estimating the weights in PLS-PM: the PLScore Mode and the PLScow Mode. Both modes involve integrating PLS Regression [2,7] as an estimation technique within the outer estimation phase of PLS-PM. PLScore Mode is based on correlations between latent variables and on standardized scores. It can be considered as a fine-tuning of the analysis between two extreme cases: Mode A and Mode B. PLScow Mode is based on covariances between latent variables and normalized weights. Links to the New Mode A recently proposed in [4] are also demonstrated.

**References:**

[1] Esposito Vinzi V., Chin W., Henseler J., Wang H.(2010) Handbook of Partial Least Squares: Concepts, Methods and Applications, Computational Statistics Handbook series (Vol. II), Springer-Verlag, Europe.

[2] Tenenhaus M.(1998) La Régression PLS: théorie et pratique, Technip, Paris.

[3] Tenenhaus M., Esposito Vinzi V., Chatelin Y.M., Lauro C.(2005) PLS path modeling, *Computational Statistics and Data Analysis*, 48, pp. 159-205.

[4] Tenenhaus A., Tenenhaus M.(2011) Regularized generalized canonical correlation analysis. *Psychometrika*, in press.

[5] Tenenhaus M., Hanafi (2010) A bridge between PLS path modeling and multi-block data analysis, in: V. Esposito Vinzi et al. (eds.), Handbook of Partial Least Squares: Concepts, Methods and Applications, Computational Statistics Handbook series (Vol. II), Springer-Verlag, Europe.

[6] Wold H.(1975) Modelling in complex situations with soft information, in Third World Congress of Econometric Society, Toronto, Canada.

[7] Wold S., Martens H., Wold H.(1983) The multivariate calibration method in chemistry solved by the PLS method, in: A. Ruhe et al. (eds.), Proc. Conf. Matrix Pencils. Lecture Notes in Mathematics, Springer-Verlag, Heidelberg, pp. 286-293.

**Keywords:** Partial Least Squares; Structural Equation Modeling; Multicollinearity; Shrinkage

**Biography:** Vincenzo Esposito Vinzi, is Professor of Statistics and Chair of the Information Systems and Decision Sciences Department at the ESSEC Business School of Paris. His research interests focus on factorial methods for dimensionality reduction and classification, structural equation modeling, PLS (Partial Least Squares) regression and path modeling. He is associate editor of Computational Statistics and Data Analysis (Elsevier), Computational Statistics (Physica-Verlag), Advances in Data Analysis and Classification (Springer), Statistical Methods and Applications (Springer). He is also serving the international scientific community as Vice President of the International Society for Business and Industrial Statistics (ISBIS), Chairman-Elect of the European Section of the International Association for Statistical Computing (IASC) and elected member of the Council of the International Statistical Institute (ISI). He has acted as the Editor-in-Chief of the “Handbook of Partial Least Squares: Concepts, Methods and Applications” (Springer) and co-authored two top-cited 2005-2010 articles in “Computational Statistics and Data Analysis”.