Johnson–Neyman Type Technique in Hierarchical Linear Models

Abstract

In hierarchical linear models we often find that group indicator variables at the cluster level are significant predictors for the regression slopes. When this is the case, the average relationship between the outcome and a key independent variable are different from group to group. In these settings, a question such as “what range of the independent variable is the difference in the outcome variable statistically significant among groups?” naturally arises. The Johnson–Neyman (J-N) technique answers this kind of question in the analysis of covariance (ANCOVA) settings. In the hierarchical modeling context, the F test, which is widely used in ANCOVA, cannot be applied because the assumption of homogeneity of variance within cluster units is violated. Instead, the approximate Wald test can be used to determine the region of significance. To illustrate the application of the J-N technique in the context of hierarchical linear modeling, an example from research in education is provided.