# Partial least squares path modeling

The partial least squares path modeling or partial least squares structural equation modeling (PLS-PM, PLS-SEM)[1][2][3] is a method of structural equation modeling which allows estimating complex cause-effect relationship models with latent variables.

## Overview

PLS-PM[4][5] is a component-based estimation approach that differs from the covariance-based structural equation modeling. Unlike covariance-based approaches to structural equation modeling, PLS-PM does not fit a common factor model to the data, it rather fits a composite model.[6][7] In doing so, it maximizes the amount of variance explained (though what this means from a statistical point of view is unclear and PLS-PM users do not agree on how this goal might be achieved).

In addition, by an adjustment PLS-PM is capable of consistently estimating certain parameters of common factor models as well, through an approach called consistent PLS (PLSc).[8] A further related development is factor-based PLS-PM (PLSF), a variation of which employs PLSc as a basis for the estimation of the factors in common factor models; this method significantly increases the number of common factor model parameters that can be estimated, effectively bridging the gap between classic PLS and covariance‐based structural equation modeling.[9] Furthermore, PLS-PM can be used for out-sample prediction purposes,[10] and can be employed as an estimator in confirmatory composite analysis.[11][12]

The PLS structural equation model is composed of two sub-models: the measurement model and structural model. The measurement model represents the relationships between the observed data and the latent variables. The structural model represents the relationships between the latent variables.

An iterative algorithm solves the structural equation model by estimating the latent variables by using the measurement and structural model in alternating steps, hence the procedure's name, partial. The measurement model estimates the latent variables as a weighted sum of its manifest variables. The structural model estimates the latent variables by means of simple or multiple linear regression between the latent variables estimated by the measurement model. This algorithm repeats itself until convergence is achieved.

With the availability of software applications, PLS-PM became particularly popular in social sciences disciplines such as accounting,[13][14] family business,[15] marketing,[16] management information systems,[17][18] operations management,[19] strategic management,[20] and tourism.[21] Recently, areas such as engineering, environmental sciences,[22] medicine,[23] and political sciences more broadly use PLS-PM to estimate complex cause-effect relationship models with latent variables. Thereby, they analyse, explore and test their established and underlying their conceptual models and theory.

PLS is viewed critically by several methodological researchers.[24][25] A major point of contention has been the claim that PLS-PM can always be used with very small sample sizes.[26] A recent study suggests that this claim is generally unjustified, and proposes two methods for minimum sample size estimation in PLS-PM.[27][28] Another point of contention is the ad hoc way in which PLS-PM has been developed and the lack of analytic proofs to support its main feature: the sampling distribution of PLS weights. However, PLS-PM is still considered preferable (over CB-SEM) when it is unknown whether the data's nature is common factor- or composite-based.[29]

## References

1. ^ Hair, J.F.; Hult, G.T.M.; Ringle, C.M.; Sarstedt, M. (2017). A Primer on Partial Least Squares Structural Equation Modeling (PLS-SEM) (2 ed.). Thousand Oaks, CA: Sage. ISBN 9781483377445.
2. ^ Vinzi, V.E.; Trinchera, L.; Amato, S. (2010). Handbook of partial least squares. Springer Berlin Heidelberg.
3. ^ Hair, J.F.; Sarstedt, M.; Ringle, C.M.; Gudergan, S.P. (2018). Advanced Issues in Partial Least Squares Structural Equation Modeling (PLS-SEM). Thousand Oaks, CA: Sage. ISBN 9781483377391.
4. ^ Wold, H. O. A. (1982). "Soft Modeling: The Basic Design and Some Extensions". In Jöreskog, K. G.; Wold, H. O. A. (eds.). Systems Under Indirect Observations: Part II. Amsterdam: North-Holland. pp. 1–54. ISBN 0-444-86301-X.
5. ^ Lohmöller, J.-B. (1989). Latent Variable Path Modeling with Partial Least Squares. Heidelberg: Physica. ISBN 3-7908-0437-1.
6. ^ Henseler, Jörg; Dijkstra, Theo K.; Sarstedt, Marko; Ringle, Christian M.; Diamantopoulos, Adamantios; Straub, Detmar W.; Ketchen, David J.; Hair, Joseph F.; Hult, G. Tomas M. (2014-04-10). "Common Beliefs and Reality About PLS". Organizational Research Methods. 17 (2): 182–209. doi:10.1177/1094428114526928.
7. ^ Rigdon, E. E.; Sarstedt, M.; Ringle, M. (2017). "On Comparing Results from CB-SEM and PLS-SEM: Five Perspectives and Five Recommendations". Marketing ZFP. 39 (3): 4–16. doi:10.15358/0344-1369-2017-3-4.
8. ^ Dijkstra, Theo K.; Henseler, Jörg (2015-01-01). "Consistent and asymptotically normal PLS-PM estimators for linear structural equations". Computational Statistics & Data Analysis. 81: 10–23. doi:10.1016/j.csda.2014.07.008.
9. ^ Kock, N. (2019). From composites to factors: Bridging the gap between PLS and covariance‐based structural equation modeling. Information Systems Journal, 29(3), 674-706.
10. ^ Shmueli, Galit; Ray, Soumya; Velasquez Estrada, Juan Manuel; Chatla, Suneel Babu (2016-10-01). "The elephant in the room: Predictive performance of PLS models". Journal of Business Research. 69 (10): 4552–4564. doi:10.1016/j.jbusres.2016.03.049.
11. ^ Hair, J. F.; Howard, M. C.; Nitzl, C. (2020). "Assessing Measurement Model Quality in PLS-SEM Using Confirmatory Composite Analysis". Journal of Business Research. 109: 101–110. doi:10.1016/j.jbusres.2019.11.069.
12. ^ Schuberth, Florian; Henseler, Jörg; Dijkstra, Theo K. (2018). "Confirmatory Composite Analysis". Frontiers in Psychology. 9: 2541. doi:10.3389/fpsyg.2018.02541. PMC 6300521. PMID 30618962.
13. ^ Nitzl, C. (2016). "The Use of Partial Least Squares Structural Equation Modelling (PLS-SEM) in Management Accounting Research: Directions for Future Theory Development". Journal of Accounting Literature. 37: 19–35. doi:10.1016/j.acclit.2016.09.003.
14. ^ Nitzl, C.; Chin, W. W. (2017). "The Case of Partial Least Squares (PLS) Path Modeling in Managerial Accounting". Journal of Management Control. 28: 137–156. doi:10.1007/s00187-017-0249-6. S2CID 113867355.
15. ^ Sarstedt, M.; Ringle, C.M.; Smith, D.; Reams, R.; Hair, J.F. (2014). "Partial Least Squares Structural Equation Modeling (PLS-SEM): A Useful Tool for Family Business Researchers". Journal of Family Business Strategy. 5 (1): 105–115. doi:10.1016/j.jfbs.2014.01.002.
16. ^ Sarstedt, M.; Ringle, C.M.; Hair, J.F.; Mena, J.A. (2012). "An Assessment of the Use of Partial Least Squares Structural Equation Modeling in Marketing Research". Journal of the Academy of Marketing Science. 40 (3): 414–433. doi:10.1007/s11747-011-0261-6. S2CID 167672022.
17. ^ Schmitz, K. W., Teng, J. T., & Webb, K. J. (2016). Capturing the complexity of malleable IT use: Adaptive structuration theory for individuals. Management Information Systems Quarterly, 40(3), 663-686.
18. ^ Ringle, C.M.; Sarstedt, M.; Straub, D.W. (2012). "A Critical Look at the Use of PLS-SEM in MIS Quarterly" (PDF). MIS Quarterly. 36 (1): iii-xiv. doi:10.2307/41410402. JSTOR 41410402. Archived from the original (PDF) on 2018-04-03. Retrieved 2015-08-02.
19. ^ Peng, D.X.; Lai, F. (2012). "Using Partial Least Squares in Operations Management Research: A Practical Guideline and Summary of Past Research". Journal of Operations Management. 30 (6): 467–480. doi:10.1016/j.jom.2012.06.002.
20. ^ Hair, J.F.; Sarstedt, M.; Pieper, T.; Ringle, C.M. (2012). "The Use of Partial Least Squares Structural Equation Modeling in Strategic Management Research: A Review of Past Practices and Recommendations for Future Applications". Long Range Planning. 45 (5–6): 320–340. doi:10.1016/j.lrp.2012.09.008.
21. ^ Rasoolimanesh, S.M., Jaafar, M., Kock, N. and Ahmad, A. G. (2017). The effects of community factors on residents’ perceptions toward World Heritage Site inscription and sustainable tourism development. Journal of Sustainable Tourism, 25(2), 198-216.
22. ^ Brewer, T.D., Cinner, J.E., Fisher, R., Green, A., & Wilson, S.K. (2012). Market access, population density, and socioeconomic development explain diversity and functional group biomass of coral reef fish assemblages. Global Environmental Change, 22(2), 399-406.
23. ^ Berglund, E., Lytsy, P., & Westerling, R. (2012). Adherence to and beliefs in lipid-lowering medical treatments: A structural equation modeling approach including the necessity-concern framework. Patient Education and Counseling, 91(1), 105-112.
24. ^ Rönkkö, M.; McIntosh, C.N.; Antonakis, J.; Edwards, J.R. (2016). "Partial least squares path modeling: Time for some serious second thoughts". Journal of Operations Management. 47–48: 9–27. doi:10.1016/j.jom.2016.05.002.
25. ^ Goodhue, D. L., Lewis, W., & Thompson, R. (2012). Does PLS have advantages for small sample size or non-normal data? MIS Quarterly, 981-1001.
26. ^ Kock, N., & Hadaya, P. (2018). Minimum sample size estimation in PLS-SEM: The inverse square root and gamma-exponential methods. Information Systems Journal, 28(1), 227–261.
27. ^ Kock, N., & Hadaya, P. (2018). Minimum sample size estimation in PLS-SEM: The inverse square root and gamma-exponential methods. Information Systems Journal, 28(1), 227–261.
28. ^ Sarstedt, Marko; Cheah, Jun-Hwa (2019-06-27). "Partial least squares structural equation modeling using SmartPLS: a software review". Journal of Marketing Analytics. 7 (3): 196–202. doi:10.1057/s41270-019-00058-3. ISSN 2050-3318.
29. ^ Sarstedt, M.; Hair, J.F.; Ringle, C.M.; Thiele, K.O.; Gudergan, S.P. (2016). "Estimation issues with PLS and CBSEM: Where the bias lies!". Journal of Business Research. 69 (10): 3998–4010. doi:10.1016/j.jbusres.2016.06.007.