ANCOVA and its disturbing assumptions 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?
 A: This is just a nomenclature problem.  ANCOVA in its original incarnation often implied an additive model for which non-parallelism was feared and tested.  If we used the more general name "linear model" we would avoid this connotation (or perhaps the even more general phrase "multivariable regression model").  
Besides rightly worrying about the additivity assumption you should spend a lot of effort examining linearity assumptions (plus constant variance, normality, etc.).  The linearity assumption, in my experience, is the most frequently violated assumption that has high impact, causes such problems as apparently significant interactions that are just stand-ins for omitted main effects or nonlinear terms.
A: +1 to @FrankHarrell.  To be honest, I find a lot of terminology in statistics to be used inconsistently, confusing, or generally unhelpful.  It's best to concentrate on the underlying logical structure of your situation.  For example, an ANOVA isn't fundamentally different from a multiple regression model.  An ANOVA is just a MR where all the explanatory / predictor variables are categorical.  An ANCOVA is just a MR where there are categorical explanatory variables (that are of primary interest), and also some continuous covariates (that are assumed to contribute to the DV, but are regarded as nuisance variables not of substantive interest), but no interactions between the factors and the covariates (hence, the assumption of parallel lines, as you state).  Note that not everyone seems to use the term ANCOVA in this (traditional) way.  Of course, you can also have a MR model with both categorical and continuous variables and interactions between them.  The world does not end when this occurs, you just no longer have an 'ANCOVA'.  
