ANCOVA has two additional assumptions as compared to two-way factorial ANOVA. They are (1) independence of the covariate and factor (2) homogeneity of slope. Why don't we need to check them for two-way factorial ANOVA?
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2 Answers
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The covariate (i.e., continuous factor) cannot be part of an ANOVA and these assumptions are for the covariate, thus they are only needed for the ANCOVA.
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The first assumption you mentioned is not correct. Controlling for the covariate is typically done because the factor and the covariate could be dependent.
With your second assumption I guess you refer to the idea that the covariate ("age") must have the same effect for each level of the factor. Or maybe more important: the factor should have the same effect no matter what value of age you are looking at. Anyway, this means that there should be NO interaction of the factor and the covariate. In a two-way ANOVA on the other hand, the interaction of the two factors is typically analysed/studied. If it appears to be insignificant, than that is considered an important finding.
Maybe that sometimes you would only like to control for another factor i.e. a categorical independent variable, say gender. In that case, gender would serve more as a covariate, like age above, for which you would only like to control without studying the interaction of gender and your factor of interest. Then, again, you would have to assume that gender does not interact with your factor of interest, meaning that your factor has the same effect for both genders. That would be an assumption then.
