A Partial Least Squares Algorithm Handling Ordinal Variables

  • Gabriele CantaluppiEmail author
  • Giuseppe Boari
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 173)


The partial least squares (PLS) is a popular path modeling technique commonly used in social sciences. The traditional PLS algorithm deals with variables measured on interval scales while data are often collected on ordinal scales. A reformulation of the algorithm, named Ordinal PLS (OrdPLS), is introduced, which properly deals with ordinal variables. Some simulation results show that the proposed technique seems to perform better than the traditional PLS algorithm applied to ordinal data as they were metric, in particular when the number of categories of the items in the questionnaire is small (4 or 5) which is typical in the most common practical situations.


Partial least squares path modeling (PLS-PM) Robust Methods Ordinal Variables 


  1. Bartolomew, D.: The Statistical Approach to Social Measurement. Academic Press, San Diego (1996)Google Scholar
  2. Bollen, K.: Structural Equations with Latent Variables. John Wiley, New York (1989)CrossRefzbMATHGoogle Scholar
  3. Bollen, K., Maydeu-Olivares, A.: A Polychoric Instrumental Variable (PIV) Estimator for Structural Equation Models with Categorical Variables. Psychometrika 72, 309–326 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Cantaluppi, G.: A Partial Least Squares Algorithm Handling Ordinal Variables also in Presence of a Small Number of Categories. Quaderno di Dipartimento, Università Cattolica del Sacro Cuore, Milano (2012).
  5. Chin, W.: The Partial Least Squares Approach for Structural Equation Modeling. In: Marcoulides, G. (ed.) Modern Methods for Business Research, pp. 295–336. Lawrence Erlbaum Associates, London (1998)Google Scholar
  6. Coenders, G., Satorra, A., Saris, W.: Alternative Approaches to Structural Modeling of Ordinal Data: a Monte Carlo Study. Struct. Equ. Model. 4 (4), 261–282 (1997)CrossRefGoogle Scholar
  7. Drasgow, F.: Polychoric and polyserial correlations. In: Kotz, S., Johnson, N. (eds.) The Encyclopedia of Statistics, vol. 7, pp. 68–74. John Wiley, New York (1986)Google Scholar
  8. Esposito Vinzi, V., Trinchera, L., Amato, S.: PLS Path Modeling: From Foundations to Recent Developments and Open Issues for Model Assessment and Improvement. In: Esposito Vinzi V. et al. (ed.) Handbook of Partial Least Squares, pp. 47–82. Springer-Verlag, Berlin/New York (2010)CrossRefGoogle Scholar
  9. Fornell, C., Cha, J.: Partial Least Squares. In: Bagozzi, R. (ed.) Advanced Methods of Marketing Research, pp. 52–78. Blackwell, Cambridge (1994)Google Scholar
  10. Fox, J.: Polycor: Polychoric and Polyserial Correlations (2010). R package version 0.7-8
  11. Hand, D.J.: Measurement Theory and Practice: The World Through Quantification. John Wiley, New York (2009)zbMATHGoogle Scholar
  12. Jakobowicz, E., Derquenne, C.: A modified PLS path modeling algorithm handling reflective categorical variables and a new model building strategy. Comput. Stat. Data Anal. 51, 3666–3678 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Jöreskog, K.: Structural Equation Modeling with Ordinal Variables using LISREL. Scientific Software Internat. Inc. (2005). Google Scholar
  14. Lohmöller, J.: Latent Variable Path Modeling with Partial Least Squares. Physica-Verlag, Heidelberg (1989)CrossRefzbMATHGoogle Scholar
  15. Nappo, D.: SEM with ordinal manifest variables. An Alternating Least Squares Approach. Ph.D. thesis, Università degli Studi di Napoli Federico II (2009)Google Scholar
  16. Revelle, W.: Psych: Procedures for Psychological, Psychometric, and Personality Research. Northwestern University, Evanston (2012). R package version 1.2.8
  17. Rönkkö, M.: Matrixpls: Matrix-based Partial Least Squares Estimation (2014). R package version 0.3.0
  18. Russolillo, G.: Non-Metric Partial Least Squares. Electron. J. Stat. 6, 1641–1669 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Schneeweiss, H.: Consistency at Large in Models with Latent Variables. In: Haagen et al. K. (ed.) Statistical Modelling and Latent Variables, pp. 299–320. Elsevier, Amsterdam/New York (1993)Google Scholar
  20. Stevens, S.: On the Theory of Scales of Measurement. Science 103, 677–680 (1946)CrossRefzbMATHGoogle Scholar
  21. Tenenhaus, A., Tenenhaus, M.: Regularized Generalized Canonical Correlation Analysis. Psychometrika 76, 257–284 (2011)MathSciNetGoogle Scholar
  22. Tenenhaus, M., Esposito Vinzi, V., Chatelin, Y.M., Lauro, C.: PLS path modeling. Comput. Stat. Data Anal. 48, 159–205 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  23. Thurstone, L.: The Measurement of Values. University of Chicago Press, Chicago (1959)Google Scholar
  24. Wold, H.: Model construction and evaluation when theoretical knowledge is scarce: an example of the use of Partial Least Squares. Cahier 79.06 du Département d’économétrie, Faculté des Sciences Économiques et Sociales, Université de Genève, Genève (1979)Google Scholar
  25. Wold, H.: Soft modeling: the basic design and some extensions. In: Jöreskog, K.G., Wold H. (eds.) Systems Under Indirect Observations, Part II, pp. 1–54. North-Holland, Amsterdam (1982)Google Scholar
  26. Zumbo, B., Gadermann, A., Zeisser, C.: Ordinal Versions of Coefficients Alpha and Theta for Likert Rating Scales. J. Mod. Appl. Stat. Methods 6 (1), 21–29 (2007)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Dipartimento di Scienze statisticheUniversità Cattolica del Sacro CuoreMilanoItaly

Personalised recommendations