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Linear regression is equivalent to ANOVA. Do we need to check for linearity and multicolinearity for ANOVA? I have seen that these assumptions are usually omitted for ANOVA but not for linear regression. Please advise.

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With ANOVA your explanatory/independent/right-hand-side/x-variables are typically categorical, so the linearity assumptions is automaticallly met. Say we have a continuous $x$-varialbe years of education, then we compare the expected $y$ for someone with 0 years of education with 1 year of education, 1 year of education with 2 years of education, 2 years of education with 3 years of education, etc. The linearity assumption says the difference in expected $y$ for each of these steps is the same. With a categorical explanatory variable, e.g. type of diploma, we don't make that assumption: we just compare someone with secondary education with someone with primary education, and someone with tertiary education with someone with primary education.

Absense of multicolinearity is not an assumption, it is something to be aware of because it can influence the stabiltiy of your results and the statistical power of your tests.

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