analysis of covariance

Every once in a while, I work with a client who is stuck between a particular statistical rock and hard place. It happens when they’re trying to run an analysis of covariance (ANCOVA) model because they have a categorical independent variable and a continuous covariate.

The problem arises when a coauthor, committee member, or reviewer insists that ANCOVA is inappropriate in this situation because one of the following ANCOVA assumptions are not met:

1. The independent variable and the covariate are independent of each other.

2. There is no interaction between independent variable and the covariate.

If you look them up in any design of experiments textbook, which is usually where you’ll find information about ANOVA and ANCOVA, you will indeed find these assumptions.  So the critic has nice references.

However, this is a case where it’s important to stop and think about whether the assumptions apply to your situation, and how dealing with the assumption will affect the analysis and the conclusions you can draw. (more…)

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If you’re like most researchers, your statistical training focused on Regression or ANOVA, but not both. It all depends on whether your field focuses more on experimental data (Biology, Psychology) or observed data (Sociology, Economics). Maybe one class covered a bit of the other, but most people are comfortable in one, but not the other.

This, in my opinion, is a shame. (Okay, I was going to say tragedy, but let’s be real.  Tsunami that kills thousands=tragedy.  Different scale here).

First of all, the distinction between ANOVA and linear regression is arbitrary. They’re really the same model with different outfits on.

Second, regardless of which one you normally use, you’re going to occasionally have to use the other kind of predictor variables–categorical or continuous. And we can come up with nice names for these models–a regression with dummy variables or an Analysis of Covariance.

But real understanding of the relationships among variables comes only when you dispense of the names and can focus on analyzing and interpreting the model using the kinds of variables you have.

There are other examples, but today I’m going to focus on an ANOVA model with a continuous covariate.

A common model is one in which one predictor is categorical (we’ll use 4 categories) and the other is continuous. Here is an example of a scatterplot of just such a model:

There are four groups, each of which received a different training.  The continuous moderator is Age, and the outcome is OverallPost, which is the post-training test score to see how well they learned the material in each training program.

As you can see, the effect of the training program is moderated by age.  Another way to say that is there is a significant interaction between Age and Training Group.  The effect of the training is depending on the trainee’s age.

One way to interpret this significant interaction is to compare the slopes of the four lines, which is easily done with any regression coefficient table.  (Okay, not always easily done, but easily found in…)

But this doesn’t make very much sense when Age is really a moderator–a predictor we want to control for, and see how it affects the relationship between the independent (IV) and dependent variables (DV), but not really the IV we’re interested in.

A better way to do it in this situation is to compare the means among groups at a low value of Age, say 20, and again at a high value of Age, say 50.  You can get p-values, adjusted for multiple comparisons, using either SAS or SPSS GLM.

SAS Proc GLM uses the LSMeans statement and SPSS GLM uses EMMeans.  They do the same thing–calculate the mean of Y for each group, at a specific value of the covariate.

If you use the menus in SPSS, you can only get those EMMeans at the Covariate’s mean, which in this example is about 25, where the vertical black line is.  This isn’t very useful for our purposes.  But we can change the value of the covariate at which to compare the means using syntax.

So it would tell us that at a young age of say 20, the three treatment groups (green, tan, and purple lines) all have means higher than the control (blue).  Young people learned more in all three treatment groups.

But at an older age, say 50, the means of the purple and tan groups were not significantly different from the control group’s (blue), and the green  (EIQ group) did worse!

In SPSS GLM, the syntax would be:

UNIANOVA OverallPost BY group WITH NEWAGE
/METHOD=SSTYPE(3)
/INTERCEPT=INCLUDE
/EMMEANS=TABLES(group) WITH(NEWAGE=MEAN) COMPARE ADJ(SIDAK)
/EMMEANS=TABLES(group) WITH(NEWAGE=45) COMPARE ADJ(SIDAK)
/EMMEANS=TABLES(group) WITH(NEWAGE=20) COMPARE ADJ(SIDAK)
/PRINT=PARAMETER
/CRITERIA=ALPHA(.05)
/DESIGN=NEWAGE group NEWAGE*group.

 

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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 assumptions.

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.

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.

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Here’s a little quiz:

True or False?

1. When you add an interaction to a regression model, you can still evaluate the main effects of the terms that make up the interaction, just like in ANOVA.

2. The intercept is usually meaningless in a regression model. (more…)

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If your graduate statistical training was anything like mine, you learned ANOVA in one class and Linear Regression in another.  My professors would often say things like “ANOVA is just a special case of Regression,” but give vague answers when pressed.

It was not until I started consulting that I realized how closely related ANOVA and regression are.  They’re not only related, they’re the same thing.  Not a quarter and a nickel–different sides of the same coin.

So here is a very simple example that shows why.  When someone showed me this, a light bulb went on, even though I already knew both ANOVA and multiple linear (more…)

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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.

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