If you’ve compared two textbooks on linear models, chances are, you’ve seen two different lists of assumptions.
I’ve spent a lot of time trying to get to the bottom of this, and I think it comes down to a few things.
1. There are four assumptions that are explicitly stated along with the model, and some authors stop there.
2. Some authors are writing for introductory classes, and rightfully so, don’t want to confuse students with too many abstract, and sometimes untestable, (more…)
I know you know it–those assumptions in your regression or ANOVA model really are important. If they’re not met adequately, all your p-values are inaccurate, wrong, useless.
But, and this is a big one, linear models are robust to departures from those assumptions. Meaning, they don’t have to fit exactly for p-values to be accurate, right, and useful.
You’ve probably heard both of these contradictory statements in stats classes and a million other places, and they are the kinds of statements that drive you crazy. Right?
I mean, do statisticians make this stuff up just to torture researchers? Or just to keep you feeling stupid?
No, they really don’t. (I promise!) And learning how far you can push those robust assumptions isn’t so hard, with some training and a little practice. Over the years, I’ve found a few mistakes researchers commonly make because of one, or both, of these statements:
1. They worry too much about the assumptions and over-test them. There are some nice statistical tests to determine if your assumptions are met. And it’s so nice having a p-value, right? Then it’s clear what you’re supposed to do, based on that golden rule of p<.05.
The only problem is that many of these tests ignore that robustness. They find that every distribution is non-normal and heteroskedastic. They’re good tools, but these hammers think every data set is a nail. You want to use the hammer when needed, but don’t hammer everything.
2.They assume everything is robust anyway, so they don’t test anything. It’s easy to do. And once again, it probably works out much of the time. Except when it doesn’t.
Yes, the GLM is robust to deviations from some of the assumptions. But not all the way, and not all the assumptions. You do have to check them.
3. They test the wrong assumptions. Look at any two regression books and they’ll give you a different set of assumptions.
This is partially because many of these “assumptions” need to be checked, but they’re not really model assumptions, they’re data issues. And it’s also partially because sometimes the assumptions have been taken to their logical conclusions. That textbook author is trying to make it more logical for you. But sometimes that just leads you to testing the related, but wrong thing. It works out most of the time, but not always.
Should you drop outliers? Outliers are one of those statistical issues that everyone knows about, but most people aren’t sure how to deal with. Most parametric statistics, like means, standard deviations, and correlations, and every statistic based on these, are highly sensitive to outliers.
And since the assumptions of common statistical procedures, like linear regression and ANOVA, are also based on these statistics, outliers can really mess up your analysis.
Despite all this, as much as you’d like to, it is NOT acceptable to drop an observation just because it is an outlier. They can be legitimate observations and are sometimes the most interesting ones. It’s important to investigate the nature of the outlier before deciding.
For example, I once analyzed a data set in which a woman’s weight was recorded as 19 lbs. I knew that was physically impossible. Her true weight was probably 91, 119, or 190 lbs, but since I didn’t know which one, I dropped the outlier.
This also applies to a situation in which you know the datum did not accurately measure what you intended. For example, if you are testing people’s reaction times to an event, but you saw that the participant is not paying attention and randomly hitting the response key, you know it is not an accurate measurement.
Neither the presence nor absence of the outlier in the graph below would change the regression line:
In the following graph, the relationship between X and Y is clearly created by the outlier. Without it, there is no relationship between X and Y, so the regression coefficient does not truly describe the effect of X on Y.
So in those cases where you shouldn’t drop the outlier, what do you do?
One option is to try a transformation. Square root and log transformations both pull in high numbers. This can make assumptions work better if the outlier is a dependent variable and can reduce the impact of a single point if the outlier is an independent variable.
Another option is to try a different model. This should be done with caution, but it may be that a non-linear model fits better. For example, in example 3, perhaps an exponential curve fits the data with the outlier intact.
Whichever approach you take, you need to know your data and your research area well. Try different approaches, and see which make theoretical sense.