OptinMon 04 - Interpreting Linear Regression Coefficients

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.

 


Clarifications on Interpreting Interactions in Regression

May 17th, 2010 by

In a previous post, Interpreting Interactions in Regression, I said the following:

In our example, once we add the interaction term, our model looks like:

Height = 35 + 4.2*Bacteria + 9*Sun + 3.2*Bacteria*Sun

Adding the interaction term changed the values of B1 and B2. The effect of Bacteria on Height is now 4.2 + 3.2*Sun. For plants in partial sun, Sun = 0, so the effect of Bacteria is 4.2 + 3.2*0 = 4.2. So for two plants in partial sun, a plant with 1000 more bacteria/ml in the soil would be expected to be 4.2 cm taller than a (more…)


Answers to the Interpreting Regression Coefficients Quiz

January 16th, 2010 by

Yesterday I gave a little quiz about interpreting regression coefficients.  Today I’m giving you the answers.

If you want to try it yourself before you see the answers, go here.  (It’s truly little, but if you’re like me, you just cannot resist testing yourself).

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. (more…)


Interpreting (Even Tricky) Regression Coefficients – A Quiz

January 15th, 2010 by

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…)


Interpreting Regression Coefficients in Models other than Ordinary Linear Regression

January 5th, 2010 by

Someone who registered for my upcoming Interpreting (Even Tricky) Regression Models workshop asked if the content applies to logistic regression as well.

The short answer: Yes

The long-winded detailed explanation of why this is true and the one caveat:

One of the greatest things about regression models is that they all have the same set up: (more…)


To Compare Regression Coefficients, Include an Interaction Term

August 14th, 2009 by

Just yesterday I got a call from a researcher who was reviewing a paper.  She didn’t think the authors had run their model correctly, but wanted to make sure.  The authors had run the same logistic regression model separately for each sex because they expected that the effects of the predictors were different for men and women.

On the surface, there is nothing wrong with this approach.  It’s completely legitimate to consider men and women as two separate populations and to model each one separately.

As often happens, the problem was not in the statistics, but what they were trying to conclude from them.   The authors went on to compare the two models, and specifically compare the coefficients for the same predictors across the two models.

Uh-oh. Can’t do that.

If you’re just describing the values of the coefficients, fine.  But if you want to compare the coefficients AND draw conclusions about their differences, you need a p-value for the difference.

Luckily, this is easy to get.  Simply include an interaction term between Sex (male/female) and any predictor whose coefficient you want to compare.  If you want to compare all of them because you believe that all predictors have different effects for men and women, then include an interaction term between sex and each predictor.  If you have 6 predictors, that means 6 interaction terms.

In such a model, if Sex is a dummy variable (and it should be), two things happen:

1.the coefficient for each predictor becomes the coefficient for that variable ONLY for the reference group.

2. the interaction term between sex and each predictor represents the DIFFERENCE in the coefficients between the reference group and the comparison group.  If you want to know the coefficient for the comparison group, you have to add the coefficients for the predictor alone and that predictor’s interaction with Sex.

The beauty of this approach is that the p-value for each interaction term gives you a significance test for the difference in those coefficients.