It is possible to test for interaction in Analysis of Covariance (ANCOVA) between the independent variable and the co-variate, but isn't “the homogeneity of regression slopes” an assumption that should not be violated in ANCOVA? If this is the case, then if we have an interaction between independent variable and co-variate, does this not indicate non-homogeneity of slopes (and hence violation of the former assumption)?
So if we find a significant interaction in ANCOVA, is it OK to accept the results? Or should we run ANCOVA at a specified level of co-variate?
Furthermore, if ANCOVA is part of the family of regression analyses, why is it inappropriate to have different slopes (different according to the independent variable) when having different slopes would not be a problem in a regression analysis? And so, when interactions exist, should we run a multiple regression analysis instead of an ANCOVA model?