Consistent Partial Least Squares for Nonlinear Structural Equation Models
- 749 Downloads
Partial Least Squares as applied to models with latent variables, measured indirectly by indicators, is well-known to be inconsistent. The linear compounds of indicators that PLS substitutes for the latent variables do not obey the equations that the latter satisfy. We propose simple, non-iterative corrections leading to consistent and asymptotically normal (CAN)-estimators for the loadings and for the correlations between the latent variables. Moreover, we show how to obtain CAN-estimators for the parameters of structural recursive systems of equations, containing linear and interaction terms, without the need to specify a particular joint distribution. If quadratic and higher order terms are included, the approach will produce CAN-estimators as well when predictor variables and error terms are jointly normal. We compare the adjusted PLS, denoted by PLSc, with Latent Moderated Structural Equations (LMS), using Monte Carlo studies and an empirical application.
Key wordsconsistent partial least squares latent moderated structural equations nonlinear structural equation model interaction effect quadratic effect
The authors wish to acknowledge the constructive comments and suggestions by the editor, an associate editor, and three reviewers, which led to substantial improvements in contents and readability.
- Bollen, K.A., & Paxton, P. (1998). Two-stage least squares estimation of interaction effects. In R.E. Schumacker & G.A. Marcoulides (Eds.), Interaction and nonlinear effects in structural equation modeling, Mahwah: Erlbaum. Google Scholar
- Chin, W.W., Marcolin, B.L., & Newsted, P.R. (2003). A partial least squares latent variable modeling approach for measuring interaction effects. Results from a Monte Carlo simulation study and an electronic-mail emotion/adoption study. Information Systems Research, 14, 189–217. CrossRefGoogle Scholar
- Cramér, H. (1946). Mathematical methods of statistics. Princeton: Princeton University Press. Google Scholar
- DasGupta, A. (2008). Asymptotic theory of statistics and probability. New York: Springer. Google Scholar
- Dijkstra, T.K. (1981, 1985). Latent variables in linear stochastic models (PhD thesis 1981, 2nd ed. 1985). Amsterdam, The Netherlands: Sociometric Research Foundation. Google Scholar
- Dijkstra, T.K. (2011). Consistent Partial Least Squares estimators for linear and polynomial factor models. Unpublished manuscript, University of Groningen, The Netherlands. Available from http://www.rug.nl/staff/t.k.dijkstra/research.
- Dijkstra, T.K. (2013). The simplest possible factor model estimator, and successful suggestions how to complicate it again. Unpublished manuscript, University of Groningen, The Netherlands. Available from http://www.rug.nl/staff/t.k.dijkstra/research.
- Dijkstra, T.K., & Henseler, J. (2012). Consistent and asymptotically normal PLS-estimators for linear structural equations. In preparation. Google Scholar
- Gray, H.L., & Schucany, W.R. (1972). The generalized statistic. New York: Dekker Google Scholar
- Headrick, T.C. (2010). Statistical simulation: power method polynomials and other transformations. Boca Raton: Chapman & Hall/CRC. Google Scholar
- Henseler, J., & Fassott, G. (2010). Testing moderating effects in PLS path models: an illustration of available procedures. In V. Esposito Vinzi, W.W. Chin, J. Henseler, & H. Wang (Eds.), Handbook of Partial Least Squares: concepts, methods and applications (pp. 713–735). Berlin: Springer. CrossRefGoogle Scholar
- Jöreskog, K.G., & Yang, F. (1996). Non-linear structural equation models: the Kenny–Judd model with interaction effects. In G.A. Marcoulides & R.E. Schumacker (Eds.), Advanced structural equation modeling (pp. 57–87). Mahwah: Erlbaum. Google Scholar
- Kharab, A., & Guenther, R.B. (2012). An introduction to numerical methods: a MATLAB ® approach. Boca Raton: CRC Press. Google Scholar
- Moosbrugger, H., Schermelleh-Engel, K., Kelava, A., & Klein, A.G. (2009). Testing multiple nonlinear effects in structural equation modeling: a comparison of alternative estimation approaches. In T. Teo & M.S. Khine (Eds.), Structural equation modeling in educational research: concepts and applications (pp. 103–136). Rotterdam: Sense Publishers. Google Scholar
- Schermelleh-Engel, K., Klein, A., & Moosbrugger, H. (1998). Estimating nonlinear effects using a latent moderated structural equations approach. In R.E. Schumacker & G.A. Marcoulides (Eds.), Interaction and nonlinear effects in structural equation modeling (pp. 203–238). Mahwah: Erlbaum. Google Scholar
- Wold, H.O.A. (1966). Nonlinear estimation by iterative least squares procedures. In F.N. David (Ed.), Research papers in statistics: festschrift for J. Neyman (pp. 411–444). New York: Wiley. Google Scholar
- Wold, H.O.A. (1982). Soft modelling: the basic design and some extensions. In K.G. Jöreskog & H.O.A. Wold (Eds.), Systems under indirect observation, Part II (pp. 1–55). Amsterdam: North-Holland. Google Scholar