ANCOVA for nonparallel slopes: the Johnson-Neyman technique

The Johnson-Neyman (JN) procedure, as originally formulated ( Stat Res Mem, 1 (1936) 57–93), applies to a situation in which measurements on 1 dependent (response) variable, X, and 2 independent (predictor) variables, Z 1 and Z 2, are available for the members of 2 groups. The expected value of X is assumed to be a linear function of Z 1 and Z 2, but not necessarily the same function for both groups. The JN technique is used to obtain a set of values for the Z variables for which one would reject, at a specified level of significance α (e.g., α = 0.05), the hypothesis that the 2 groups have the same expected X values. This set of values, or ‘region of significance,’ may then be plotted to obtain a convenient description of those values of Z 1 and Z 2 for which the 2 groups differ. The technique can thus be described as a generalization of the analysis of covariance (ANCOVA) which does not make the assumption that the regression coefficients for the regression of X on the covariates, Z 1 and Z 2, are equal in the groups being compared. In this paper we describe, illustrate and make available a menu-driven PC program (TXJN2) implementing the JN procedure.