I just came across this great article by Frank Harrell: Problems Caused by Categorizing Continuous Variables
It’s from the Vanderbilt University biostatistics department, so the examples are all medical, but the points hold for any field.
It goes right along with my recent post, Continuous and Categorical Variables: The Trouble with Median Splits.
Back when I was doing psychology research, I knew ANOVA pretty well. I’d taken a number of courses on it and could run it backward and forward. I kept hearing about ANCOVA, but in every ANOVA class that was the last topic on the syllabus, and we always ran out of time.
The other thing that drove me crazy was those stats professors kept saying “ANOVA is just a special case of Regression.” I could not for the life of me figure out why or how.
It was only when I switched over to statistics that I finally took a regression class and figured out what ANOVA was all about. And only when I started consulting, and seeing hundreds of different ANOVA and regression models, that I finally made the connection.
But if you don’t have the driving curiosity about ANOVA and regression, why should you, as a researcher in Psychology, Education, or Agriculture, who is trained in ANOVA, want to learn regression? There are 3 main reasons.
1. There a many, many continuous independent variables and covariates that need to be included in models. Without the tools to analyze them as continuous, you are left forcing them into ANOVA using an arbitrary technique like median splits. At best, you’re losing power. At worst, you’re not publishing your article because you’re missing real effects.
2. Having a solid understanding of the General Linear Model in its various forms equips you to really understand your variables and their relationships. It allows you to try a model different ways–not for data fishing, but for discovering the true nature of the relationships. Having the capacity to add an interaction term or a squared term allows you to listen to your data and makes you a better researcher.
3. The multiple linear regression model is the basis for many other statistical techniques–logistic regression, multilevel and mixed models, Poisson regression, Survival Analysis, and so on. Each of these is a step (or small leap) beyond multiple regression. If you’re still struggling with what it means to center variables or interpret interactions, learning one of these other techniques becomes arduous, if not painful.
Having guided thousands of researchers through their statistical analysis over the past 10 years, I am convinced that having a strong, intuitive understanding of the general linear model in its variety of forms is the key to being an effective and confident statistical analyst. You are then free to learn and explore other methodologies as needed.
A Median Split is one method for turning a continuous variable into a categorical one. Essentially, the idea is to find the median of the continuous variable. Any value below the median is put it the category “Low” and every value above it is labeled “High.”
This is a very common practice in many social science fields in which researchers are trained in ANOVA but not Regression. At least that was true when I was in grad school in psychology. And yes, oh so many years ago, I used all these techniques I’m going to tell you not to.
There are problems with median splits. The first is purely logical. When a continuum is categorized, every value above the median, for example, is considered equal. Does it really make sense that a value just above the median is considered the same as values way at the end? And different than values just below the median? Not so much.
So one solution is to split the sample into three groups, not two, then drop the middle group. This at least creates some separation between the two groups. The obvious problem, here though, is you’re losing a third of your sample.
The second problem with categorizing a continuous predictor, regardless of how you do it, is loss of power (Aiken & West, 1991). It’s simply harder to find effects that are really there.
So why is it common practice? Because categorizing continuous variables is the only way to stuff them into an ANOVA, which is the only statistics method researchers in many fields are trained to do.
Rather than force a method that isn’t quite appropriate, it would behoove researchers, and the quality of their research, to learn the general linear model and how ANOVA fits into it. It’s really only a short leap from ANOVA to regression but a necessary one. GLMs can include interactions among continuous and categorical predictors just as ANOVA does.
If left continuous, the GLM would fit a regression line to the effect of that continuous predictor. Categorized, the model will compare the means. It often happens that while the difference in means isn’t significant, the slope is.
Reference: Aiken & West (1991). Multiple Regression: Testing and interpreting interactions.
I just discovered something in SPSS GLM that I never knew.
When you have an interaction in the model, the order you put terms into the Model statement affects which parameters SPSS gives you.
The default in SPSS is to automatically create interaction terms among all the categorical predictors. But if you want fewer than all those interactions, or if you want to put in an interaction involving a continuous variable, you need to choose Model–>Custom Model.
In the specific example of an interaction between a categorical and continuous variable, to interpret this interaction you need to output Regression Coefficients. Do this by choosing Options–>Regression Parameter Estimates.
If you put the main effects into the model first, followed by interactions, you will find the usual output–the regression coefficients (column B) for the continuous variable is the slope for the reference group. The coefficients for the interactions in the other categories tell you the difference between the slope for that category and the slope for the reference group. The coefficient for the reference group here in the interaction is 0.
What I was surprised to find is that if the interactions are put into the model first, you don’t get that.
Instead, the coefficients for the interaction of each category is the actual slope for that group, NOT the difference.
This is actually quite useful–it can save a bit of calculating and now you have a p-value for whether each slope is different from 0. However, it also means you have to be cautious and make sure you realize what each parameter estimate is actually estimating.
I first encountered the Great Likert Data Debate in 1992 in my first statistics class in my psychology graduate program.
My stats professor was a brilliant mathematical psychologist and taught the class unlike any psychology grad class I’ve ever seen since. Rather than learn ANOVA in SPSS, we derived the Method of Moments using Matlab. While I didn’t understand half of what was going on, this class roused my curiosity and led me to take more theoretical statistics classes. The rest is history.
A large section of the class was dedicated to the fact that Likert data was not interval and therefore not appropriate for statistics that assume normality such as ANOVA and regression. This was news to me. Meanwhile, most of the rest of the field either ignored or debated this assertion.
16 years later, the debate continues. A nice discussion of the debate is found on the Research Methodology blog by Hisham bin Md-Basir. It’s a nice blog with thoughtful entries that summarize methodological articles in the social and design sciences.
To be fair, though, this blog entry summarizes an article on the “Likert scales are not interval” side of the debate. For a balanced listing of references, see Can Likert Scale Data Ever Be Continuous?
I just wanted to follow up on my last post about Regression without Intercepts.
Regression through the Origin means that you purposely drop the intercept from the model. When X=0, Y must = 0.
The thing to be careful about in choosing any regression model is that it fit the data well. Pretty much the only time that a regression through the origin will fit better than a model with an intercept is if the point X=0, Y=0 is required by the data.
Yes, leaving out the intercept will increase your df by 1, since you’re not estimating one parameter. But unless your sample size is really, really small, it won’t matter. So it really has no advantages.