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Q:

Do the assumptions of linearity, normality, and homoskedasticity apply to categorical variables?

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It is not meaningful to check normality or linearity of a categorical variable. But you can check homoskedacity by plotting the standardized residuals vs. the categorical variable. For each possible value of the categorical variable, you will have a vertical line of points that shows the spread of standardized residuals. You would want these spreads to be roughly equal for all levels of your categorical variable. Note: If you have too many data points, it is hard to read this graph because the data points begin to overlap making it hard to see how things are distributed. In this case, side-by-side box plots can be helpful.
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