Significant quadratic effect but non-sign. linear effect?

[JoyFielding]

Hello all,

I am currently analyzing a model with a hypothesized quadratic effect. However, when running the analysis in SmartPLS I get contradicting results:

(1) Running the model with linear relationships only yields a significant effect (at 5% level) for the analyzed pair of variables
(2) Running the model with the quadratic effect yields a significant effect for the quadratic term, but a non-significant effect for the linear term

Hence, the linear effect changed from being significant to non-significance after inclusion of the quadratic term. How do I interpret this result? So far, I have not seen any comparable results in the literature. Any help would be highly appreciated.

Thanks,
Mara

[jmbecker]

First, I would plot the data and the estimated curves and see whether there is really any non-linear shape in the data visible. With very fuzzy data it can happen that you just extract some chance effects that are not really there, especially with fitting higher-order polynomials in non-linear effects.

Second, the linear effect in a quadratic relationship must not be significant if the quadratic effect is significant.

A good read for non-linear relationships is
Haans, R. F., Pieters, C., & He, Z. L. (2016). Thinking about U: Theorizing and testing U‐and inverted U‐shaped relationships in strategy research. Strategic Management Journal, 37(7), 1177–1195.

In addition, the advanced PLS-SEM book also includes a chapter on quadratic effects: 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. https://www.smartpls.com/documentation/advanced

[JoyFielding]

Thank you Dr Becker for your helpful insights. I already read the paper you mentioned but will have a look at the new PLS book. Also, I already plotted the data which pretty much confirms some form of non-linear relationship. Is there any additional reading that you might be aware of which addresses this special case of a 'mismatch' in significance between the linear and non-linear relationship? So far I only have seen examples where both relationships have been significant.

Thanks a lot for your help.