The assumptions of normality and constant variance in a linear model (both OLS regression and ANOVA) are quite robust to departures. That means that even if the assumptions aren’t met perfectly, the resulting p-values will still be reasonable estimates.
But you need to check the assumptions anyway, because some departures are so far off that the p-values become inaccurate. And in many cases there are remedial measures you can take to turn non-normal residuals into normal ones.
But sometimes you can’t.
Sometimes it’s because the dependent variable just isn’t appropriate for a linear model. The dependent variable, Y, doesn’t have to be normal for the residuals to be normal (since Y is affected by the X’s).
The errors do.
But the distribution of the errors is related to the distribution of Y. So Y does have to be continuous, unbounded, and measured on an interval or ratio scale.
If you go through the Steps to Statistical Modeling, Step 3 is: Choose the variables for answering your research questions and determine their level of measurement. Part of the reason for doing this is to save yourself from running a linear model on a DV that just isn’t appropriate and will never meet assumptions.
Some of these include DVs that are:
- Discrete counts, bounded at 0, which is often the most common value
- Zero Inflated, where even if the rest of the distribution looks normal, there is a huge spike in the distribution at 0.
- Censored or truncated, including time to event variables
- a Proportion, which is bounded at 0 and 1, or a percentage, which is bounded at 0 and 100.
If you have one of these, Stop. Do not pass Go. Do not run a linear model.
Hopefully you noticed this at Step 3, not when you’re checking assumptions, which is Step 11.
But luckily, there are other types of regression procedures available for all of these variables.