Standard conceptualizations that underpin cause-effect relationships in PLS path models imply that constructs affect another in a linear fashion. In some instances, however, this assumption does not hold in that relationships may be nonlinear.
In PLS-SEM, relationships between constructs can take various forms. While linear relationships can be represented by straight lines (with positive or negative slopes) when plotting the latent variables’ values in a scatterplot, nonlinear relationships include all associations that are not straight lines but curves.
When the relationship between two constructs is nonlinear, the size of the effect between two constructs not only depends on the magnitude of change in the exogenous construct but also on its value. In analyzing nonlinear effects, researchers have to make an assumption regarding the nature of the effect. While an abundance of different effect types is possible, quadratic effects are most common. The following figure shows the quadratic effect estimation of satisfaction on loyaltiy for the corporate reputation model in SmartPLS.
Hair et al. (2018) describe the analysis of quadratic effects and their use in SmartPLS in more detail.
Moderation Settings in SmartPLS
The selected dependent variable for which a quadratic effect will be estimated.
Field to define the predictor variable for which a quadratic effect will be estimated.
Selects the method of calculating the quadratic construct in PLS path modeling. There are three options:
(1) Product Indicator
This approach uses all possible pair combinations of the indicators of the latent predictor variable. These product terms serve as indicators ("product indicators") of the quadratic effect term in the structural model.
This approach uses the latent variable scores of the latent predictor variable from the main effects model (without the quadratic effect term). These latent variable scores are saved and used to calculate the squared indicator for the second stage analysis that involves the quadratic effect term in addition to the predictor variable.
This approach uses residuals that are calculated by regressing all possible pairwise product terms of the indicators of the latent predictor variable (i.e., product indicators) on all indicators of the latent predictor and the latent moderator variable. These residuals serve as indicators of the quadratic effect term in the structural model.
The residuals will be orthogonal to all indicators of the predictor variable to ensure that the indicators of the quadratic effect term do not share any variance with any of the indicators of the predictor variable.
Product Term Generation
Defines the way how product terms for the quadratic effect will be calculated. There are three options:
Unstandardized data are used for the calculation of the product terms of the quadratic effect.
Mean-centered data are used for the calculation of the product terms of the quadratic effect.
(3) Standardized (default)
Standardized data are used for the calculation of the product terms of the quadratic effect.
Note: If the Two-stage approach is used as Calculation Method, all options should lead to the same results, because the components for the product term calculation (i.e., latent variable scores) are always standardized.
For the Product Indicator and Orthogonalization approach the default option should be standardized.
Quadratic effect Term and Regression Handling
The quadratic effect term (latent variable) is left as it is when entering the final regression of the nonlinear model (i.e., predictor variables as well as quadratic effect term on the selected dependent variable). Hence, it is not standardized which would bias the results
Chin, W. W., Marcolin, B. L., and 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(2): 189-217.
Hair, J. F., Sarstedt, M., Ringle, C. M., and Gudergan S. P. 2018. Advanced Issues in Partial Least Squares Structural Equation Modeling (PLS-SEM), Sage: Thousand Oaks. Link to Book
Henseler, J., and Chin, W. W. 2010. A Comparison of Approaches for the Analysis of Interaction Effects Between Latent Variables Using Partial Least Squares Path Modeling. Structural Equation Modeling: A Multidisciplinary Journal, 17(1): 82-109.
Henseler, J., Fassott, G., Dijkstra, T., and Wilson, B. 2012. Analysing Quadratic Effects of Formative Constructs by Means of Variance-Based Structural Equation Modelling. European Journal of Information Systems, 21(1): 99-112.
Rigdon, E. E., Ringle, C. M., and Sarstedt, M. 2010. Structural Modeling of Heterogeneous Data with Partial Least Squares, in Review of Marketing Research, N. K. Malhotra (ed.), Sharpe: Armonk, 255-296. Link to Article