Nonlinear Effects in PLS-SEM: When a Straight Line Cannot Describe the Relationship

By default, PLS-SEM models relationships between latent variables with straight lines. When an independent variable's effect changes with its own level, for example increasing and then leveling off, a linear model may obscure an important part of the relationship. A quadratic effect is the most common way to introduce this nonlinearity.

PLS-SEM & SMARTPLSPrepared by Lê Minh5 min

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Nonlinear relationships need explicit specification

Theory
Why might the effect change?
Specification
Specify linear and quadratic terms
Plot
Inspect shape within the data range
Uncertainty
Read CIs and assess sensitivity

Conceptual diagram. Do not extrapolate the curve beyond the observed data range.

Source: Lê Minh · Conceptual illustration, not software output.

1. Why PLS-SEM Assumes Linearity by Default

A standard PLS-SEM path model estimates path coefficients assuming a linear relationship between independent and dependent variables: the independent variable's marginal effect is constant across its values. This assumption is simple and adequate for many theoretical models, but it does not always match the mechanism under study.

Many social science phenomena are nonlinear. A factor may have a positive effect at low levels that gradually weakens or even reverses at high levels. Estimating a single linear coefficient ignores that curvature and reflects only an average trend across the range.

2. What Is a Quadratic Effect?

A quadratic effect adds the squared independent variable to the structural equation alongside its original linear term. This is the most common SmartPLS-supported approach to nonlinear relationships, although other shapes are theoretically possible.

Why it resembles self-moderation

SmartPLS documentation describes a quadratic effect as similar to moderation, except that the moderator is the independent variable itself rather than a separate third variable. Quadratic-term estimation therefore follows the logic of interaction-term estimation in moderation analysis.

3. Two-Stage Estimation in SmartPLS

Stage 1: Estimate the main-effects model

Estimate the model with only the main linear effects, without the quadratic term, and save the predictor's latent variable scores.

Stage 2: Add the quadratic term

Square the saved scores to create an indicator, then estimate the final model with this quadratic term and the original linear term.

This two-stage approach separates measurement model estimation from construction of the quadratic term, following familiar interaction-term estimation logic in moderation analysis. Chin, Marcolin, and Newsted (2003) are among the early sources for using latent variable scores to construct interaction terms in PLS. Becker, Ringle, and Sarstedt (2018) compare interaction-term generation and data treatment approaches. Both underpin SmartPLS's construction of quadratic terms.

4. Implementation Steps

Build the model and structural paths

Construct the measurement and structural models as usual in SmartPLS's modeling window.

Select Quadratic effect

Select the quadratic effect option on the toolbar.

Click the path requiring a quadratic term

Click the structural path where nonlinearity is to be tested. SmartPLS automatically constructs the quadratic term and runs two-stage estimation.

5. Interpreting the Quadratic Coefficient

The SmartPLS guide does not explain quadratic coefficient interpretation in detail on its instruction page, instead referring readers to Basco, Hair, Ringle, and Sarstedt (2021) and Hair, Sarstedt, Ringle, and Gudergan (2024). Under general quadratic regression principles, the coefficient's sign indicates curvature: a positive value suggests a U-shaped tendency, with a stronger effect toward the ends of the predictor range; a negative value suggests an inverted-U tendency, with the effect weakening or reversing at higher values.

Concluding statistically significant nonlinearity requires a bootstrap test of the quadratic term itself rather than inspection of its sign alone. If the term is not statistically significant, the simpler linear model remains appropriate for reporting.

6. Limitations to Consider

The nonlinear shape must be assumed in advance

Selecting a quadratic effect assumes a specific nonlinear shape. A quadratic term may not fully capture a more complex underlying relationship. Establish a theoretical reason to expect this form before adding it, rather than adding it merely to see whether it becomes significant.

7. Common Mistakes

Table 1
Weak wording	Why it is inappropriate	Recommended wording
"Add quadratic terms to every path to search for nonlinear effects."	Nonlinearity testing requires theoretical justification for each relationship rather than indiscriminate searching.	State the theory or prior empirical evidence suggesting nonlinearity in the specific relationship.
"The quadratic coefficient is significant, so the relationship is U-shaped."	The curve's shape depends on both linear and quadratic coefficients, beyond the quadratic coefficient's sign.	Interpret the curve using both coefficients and the predictor's observed range.
"The linear-only model does not need reporting for comparison."	Comparing the models helps readers assess the improvement from adding a quadratic term.	Report both the linear model and the model with a quadratic term.

8. Checklist before Reporting

Have you stated the theoretical basis for expecting nonlinearity in the tested path?
Have you described the SmartPLS two-stage estimation procedure?
Have you tested the quadratic term through bootstrapping?
Have you interpreted the curve using both linear and quadratic coefficients?
Have you compared it with a linear-only model to show the improvement?

Conclusion

Quadratic effects allow PLS-SEM to capture relationships that do not follow a simple straight line, using two-stage estimation based on moderation analysis logic. They are useful when a specific nonlinear form has theoretical support, rather than as a search for additional statistical significance.

A quadratic curve is only one possible nonlinear form. A non-significant quadratic term therefore does not prove linearity; it indicates that this particular quadratic form lacks support in the data.

References

Chin, W. W., Marcolin, B. L., & Newsted, P. R. (2003). A partial least squares latent variable modeling approach for measuring interaction effects. Information Systems Research, 14(2), 189-217. https://doi.org/10.1287/isre.14.2.189.16018
Basco, R., Hair, J. F., Ringle, C. M., & Sarstedt, M. (2021). Advancing family business research through modeling nonlinear relationships: Comparing PLS-SEM and multiple regression. Journal of Family Business Strategy, 13(3), Article 100457.
Becker, J.-M., Ringle, C. M., & Sarstedt, M. (2018). Estimating moderating effects in PLS-SEM and PLSc-SEM: Interaction term generation x data treatment. Journal of Applied Structural Equation Modeling, 2(2), 1-21.
Hair, J. F., Sarstedt, M., Ringle, C. M., & Gudergan, S. P. (2024). Advanced issues in partial least squares structural equation modeling (PLS-SEM) (2nd ed.). Sage.

Academic note: This article presents general quadratic-effect principles and procedures following SmartPLS documentation. For detailed coefficient interpretation, consult Basco, Hair, Ringle, and Sarstedt (2021) and Hair, Sarstedt, Ringle, and Gudergan (2024), as cited in the original guide.