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 from the assumptions that the p-value 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 GLM. The dependent variable, Y, doesn’t have to be normal for the residuals to be normal (since Y is affected by the X’s).
But 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:
- Categorical
- Ordinal
- 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.
But luckily, there are other types of regression procedures available for all of these variables.
If you’d like to learn more about planning a data analysis based on the types of variables involved, check out our webinar recording: The First 3 Steps to Statistical Modeling: How to Clarify the Research Question, the Design, and the Variables.
And if you want to learn all the ins and outs of dealing with logistic regression, check out our 8-hour live workshop Binary, Ordinal, and Multinomial Logistic Regression.
Send to Kindle
{ 2 comments… read them below or add one }
Nice post.
“Unbounded” is interesting. if the bounds are very far from the mean (in standardized terms) it can be OK. Take, for example, weight of human adults. This has a lower bound. It certainly can’t be less than 0! Yet that’s fine, because that is so far from the mean.
Thanks, Peter.
I agree. I think it’s not even that the bound is so far from the mean, but even that it’s so far from any data points. The practical problem is when you get ceiling and floor effects–when a lot of observations are butted up against the bound.
It’s similar to the idea of using a linear regression instead of logistic, when all the probabilities are in the middle (say between .2 and .8). Because the sigmoidal logistic regression function is linear in the middle, you’ll get pretty much the same results. It’s close to 1 and 0 (the bounds) where logistic regression can accommodate the fact that the relationship isn’t linear.
Karen
{ 1 trackback }