Extensions of the Johnson-Neyman Technique to Linear Models With Curvilinear Effects: Derivations and Analytical Tools

The past decade has witnessed renewed interest in the use of the Johnson-Neyman (J-N) technique for calculating the regions of significance for the simple slope of a focal predictor on an outcome variable across the range of a second, continuous independent variable. Although tools have been developed to apply this technique to probe 2- and 3-way interactions in several types of linear models, this method has not been extended to include quadratic terms or more complicated models involving quadratic terms and interactions. Curvilinear relations of this type are incorporated in several theories in the social sciences. This article extends the J-N method to such linear models along with presenting freely available online tools that implement this technique as well as the traditional pick-a-point approach. Algebraic and graphical representations of the proposed J-N extension are provided. An example is presented to illustrate the use of these tools and the interpretation of findings. Issues of reliability as well as "spurious moderator" effects are discussed along with recommendations for future research.