A more directly informative understanding of a linear X qualitative interaction is obtained using the re-parameterized equation: (15)Y=A0+B1(X1−C)+B3((X1−C)·D2)+E where all symbols were defined above. The following equation is an equivalent formulation: (16)Y:{group=1Y=A0+B1(X1−C)+Egroup=2Y=A0+B2(X1−C)+E where B1 and B2 are slopes on X1 for groups 1 and 2, respectively, and other symbols were defined above. Equations 15 and 16 lead to exactly the same R2 as Equations 11 and 12. Thus, Equations 11, 12, 15, and 16 are equivalent regression models, with the same number of free parameters and the same R2. But, Equations 15 and 16 have a unique advantage over Equations 11 or 12: the direct estimate for the cross-over point C and its SE. The difference between Equations 15 and 16 is the way in which the slope on X1 for group 2 is represented. In Equation 15, B3 is the difference between slopes on X1 for groups 1 and 2, so the slope for group 2 must be calculated as B1 + B3; in Equation 16, B2 is a direct estimate of the slope on X1 for group 2.
