If the qualitative variable represents the presence of three groups, one modified version of Equation 13 can be written as: (17)Y:{group=1Y=A0+B1(X1−C)+Egroup=2Y=A0+B2(X1−C)+Egroup=3Y=A0+B3(X1−C)+E where B1 through B3 are regression slopes on X1 for groups 1 through 3, respectively, and other terms were defined above. Equation 17 contains a single cross-over or convergence point C, so is a restricted re-parameterization of Equation 13. That is, Equation 17 has 5 free parameters, whereas Equation 13 has 6 free parameters. Several alterations could be made to Equation 17 to introduce an additional parameter; for example, one could fit the following model: (18)Y:{group=1Y=A0+B1(X1−C12)+Egroup=2Y=A0+B2(X1−C12)+Egroup=3Y=(A0+B1(C13−C12))+B3(X1−C13)+E where C12 (labeled simply C in Equation 17) and are the points at which regression lines for groups 2 and 3, respectively, cross the line for group 1, and other symbols were defined above. With the additional parameter, Equation 18 has the same number of free parameters and R2 as Equation 13. Thus, a nested-model test of the difference in R2 for Equations 17 and 18 provides a 1 df test of the hypothesis that a single cross-over point holds for groups 1, 2, and 3.
