General Linear Model

The General Linear Model, Analysis of Covariance, and How ANOVA and Linear Regression Really are the Same Model Wearing Different Clothes

September 17th, 2010 by

Just recently, a client got some feedback from a committee member that the Analysis of Covariance (ANCOVA) model she ran did not meet all the Stage 2assumptions.

Specifically, the assumption in question is that the covariate has to be uncorrelated with the independent variable.

This committee member is, in the strictest sense of how analysis of covariance is used, correct.

And yet, they over-applied that assumption to an inappropriate situation.

ANCOVA for Experimental Data

Analysis of Covariance was developed for experimental situations and some of the assumptions and definitions of ANCOVA apply only to those experimental situations.

The key situation is the independent variables are categorical and manipulated, not observed.

The covariate–continuous and observed–is considered a nuisance variable. There are no research questions about how this covariate itself affects or relates to the dependent variable.

The only hypothesis tests of interest are about the independent variables, controlling for the effects of the nuisance covariate.

A typical example is a study to compare the math scores of students who were enrolled in three different learning programs at the end of the school year.

The key independent variable here is the learning program. Students need to be randomly assigned to one of the three programs.

The only research question is about whether the math scores differed on average among the three programs. It is useful to control for a covariate like IQ scores, but we are not really interested in the relationship between IQ and math scores.

So in this example, in order to conclude that the learning program affected math scores, it is indeed important that IQ scores, the covariate, is unrelated to which learning program the students were assigned to.

You could not make that causal interpretation if it turns out that the IQ scores were generally higher in one learning program than the others.

So this assumption of ANCOVA is very important in this specific type of study in which we are trying to make a specific type of inference.

ANCOVA for Other Data

But that’s really just one application of a linear model with one categorical and one continuous predictor. The research question of interest doesn’t have to be about the causal effect of the categorical predictor, and the covariate doesn’t have to be a nuisance variable.

A regression model with one continuous and one dummy-coded variable is the same model (actually, you’d need two dummy variables to cover the three categories, but that’s another story).

The focus of that model may differ–perhaps the main research question is about the continuous predictor.

But it’s the same mathematical model.

The software will run it the same way. YOU may focus on different parts of the output or select different options, but it’s the same model.

And that’s where the model names can get in the way of understanding the relationships among your variables. The model itself doesn’t care if the categorical variable was manipulated. It doesn’t care if the categorical independent variable and the continuous covariate are mildly correlated.

If those ANCOVA assumptions aren’t met, it does not change the analysis at all. It only affects how parameter estimates are interpreted and the kinds of conclusions you can draw.

In fact, those assumptions really aren’t about the model. They’re about the design. It’s the design that affects the conclusions. It doesn’t matter if a covariate is a nuisance variable or an interesting phenomenon to the model. That’s a design issue.

The General Linear Model

So what do you do instead of labeling models? Just call them a General Linear Model. It’s hard to think of regression and ANOVA as the same model because the equations look so different. But it turns out they aren’t.

Regression and ANOVA model equations

If you look at the two models, first you may notice some similarities.

  • Both are modeling Y, an outcome.
  • Both have a “fixed” portion on the right with some parameters to estimate–this portion estimates the mean values of Y at the different values of X.
  • Both equations have a residual, which is the random part of the model. It is the variation in Y that is not affected by the Xs.

But wait a minute, Karen, are you nuts?–there are no Xs in the ANOVA model!

Actually, there are. They’re just implicit.

Since the Xs are categorical, they have only a few values, to indicate which category a case is in. Those j and k subscripts? They’re really just indicating the values of X.

(And for the record, I think a couple Xs are a lot easier to keep track of than all those subscripts.  Ever have to calculate an ANOVA model by hand?  Just sayin’.)

So instead of trying to come up with the right label for a model, focus instead on understanding (and describing in your paper) the measurement scales of your variables, if and how much they’re related, and how that affects the conclusions.

In my client’s situation, it was not a problem that the continuous and the categorical variables were mildly correlated. The data were not experimental and she was not trying to draw causal conclusions about only the categorical predictor.

So she had to call this ANCOVA model a multiple regression.

3 Situations when it makes sense to Categorize a Continuous Predictor in a Regression Model

July 24th, 2009 by

Stage 2In many research fields, particularly those that mostly use ANOVA, a common practice is to categorize continuous predictor variables so they work in an ANOVA. This is often done with median splits—splitting the sample into two categories—the “high” values above the median and the “low” values below the median. There are many reasons why this isn’t such a good idea:

  • the median varies from sample to sample, making the categories in different samples have different meanings
  • all values on one side of the median are considered equivalent—any variation within the category is ignored, and two values right next to each other on either side of the median are considered different
  • the categorization is completely arbitrary. A ‘High” score isn’t necessarily high. If the scale is skewed, as many are, even a value near the low end can end up in the “high” category.

But it can be very useful and legitimate to be able to choose whether to treat an independent variable as categorical or continuous. Knowing when it is appropriate (more…)

3 Reasons Psychology Researchers should Learn Regression

February 17th, 2009 by

Stage 2Back 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.