Here’s a little reminder for those of you checking assumptions in regression and ANOVA:
The assumptions of normality and homogeneity of variance for linear models are not about Y, the dependent variable. (If you think I’m either stupid, crazy, or just plain nit-picking, read on. This distinction really is important).
The distributional assumptions for linear regression and ANOVA are for the distribution of Y|X — that’s Y given X. You have to take out the effects of all the Xs before you look at the distribution of Y. As it turns out, the distribution of Y|X is, by definition, the same as the distribution of the residuals. So the easiest way to check the distribution of Y|X is to save your residuals and check their distribution.
I’ve seen too many researchers drive themselves crazy trying to transform skewed Y distributions before they’ve even run the model. The distribution of the dependent variable can tell you what the distribution of the residuals is not—you just can’t get normal residuals from a binary dependent variable.
But it cannot always tell what the distribution of the residuals is.
If a categorical independent variable had a big effect, the dependent variable would have a continuous, bimodal distribution. But the residuals (or the distribution within each category of the independent variable) would be normally distributed.
And what are those distributional assumptions of Y|X?
3. Constant Variance
You can check all three with a few residual plots–a Q-Q plot of the residuals for normality, and a scatter plot of Residuals on X or Predicted values of Y to check 1 and 3.