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Using Adjusted Means to Interpret Moderators in Analysis of Covariance

The General Linear Model, Analysis of Covariance, and How ANOVA and

Linear Regression Really are the Same Model Wearing Different Clothes

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

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

This committee member is, in the strictest sense, correct. Analysis of Covariance was developed for experimental situations

in which the independent variables are categorical and usually manipulated, not observed. The covariatecontinuous and

observedis considered a nuisance variable. There are no research questions about how this covariate itself affects 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 would be to compare the math scores of students who were enrolled in three different learning programs

at the end of the school year. The only research question would be about whether the math scores differed on average among

the three programs. It would be useful to control for a covariate like IQ scores, but we are not really interested in therelationship between IQ and math scores.

But thats really just one application of a linear model with one categorical and one continuous predictor. The research

question of interest doesnt have to be about the categorical predictor, and the covariate doesnt have to be a nuisance

variable.

A regression model with one continuous and one dummy variable is the same model (actually, youd need two dummy

variables to cover the three categories, but thats another story).

The focus of that model may differperhaps the main research question is about the continuous predictor. But its the same

model. And your software will run it the same way. YOU may focus on different parts of the output or select different

options, but its the same model.

And thats where the model names can get in the way of understanding the relationships among your variables. The model

itself doesnt care if the categorical variable was manipulated. It doesnt care if the categorical independent variable and her

continuous covariate are mildly correlated.

If those ANCOVA assumptions arent 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 arent about the model. Theyre about the design. Its the design that affects the

conclusions. It doesnt matter if a covariate is a nuisance variable or an interesting phenomenon to the model. Thats a design

issue.

So what do you do instead of labeling models? Just call them a General Linear Model. Its hard to think ofregression and

ANOVA as the same model because the equations look so different. But it turns out they arent.

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 estimatethis portion estimates the mean values of Y at the different values of

X.

og Archive The General Linear Model, Analysis of Covariance, a... http://www.analysisfactor.com/statchat/the-general-linear-model-analys...

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Both equations have a residual, which is the random part of the modelthe variation in Y that is not affected by the Xs.

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

Actually, there are. Theyre 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? Theyre 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 theyre related, and how that affects the conclusions.

Tags: ANOVA, General Linear Model, Linear Regression

This entry was posted on Friday, September 17th, 2010 at 11:24 am and is filed underANOVA, Linear Regression. You can follow any responses to

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