JohnsonNeyman

Johnson-Neyman, also written Johnson–Neyman, is a statistical procedure used in moderation analysis to identify the values of a moderator variable for which the effect of a predictor on an outcome is statistically significant. It is applied in linear regression models that include an interaction term between a focal predictor X and a moderator Z. The core idea is to determine regions of Z where the conditional effect of X on Y, expressed as the simple slope ∂Y/∂X = b1 + b3Z, is significantly different from zero.

The method works by considering how the standard error of the conditional effect depends on Z. The Johnson–Neyman procedure solves for the values of Z at which the t-statistic for the simple slope equals the chosen critical value. This yields boundary points ZL and ZU; between these boundaries the effect of X on Y is statistically significant at the preset alpha level, while outside them it is not (or vice versa, depending on the sign of the slope).

Implementation and use: The Johnson–Neyman approach is widely used in psychology, social sciences, and other fields that employ regression-based moderation analyses. It can be computed with statistical software and is implemented in R packages such as those for probing interactions (for example, the interactions and rockchalk packages), as well as in SPSS and SAS macros that support moderation analysis.

Limitations: The procedure relies on standard regression assumptions (linearity, normality of errors, homoscedasticity) and model specification. Its interpretation should be contextual and consider sample size, measurement reliability, and potential model misspecification.