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…)

When NOT to Center a Predictor Variable in Regression

February 9th, 2009 by

There are two reasons to center predictor variables in any type of regression analysis–linear, logistic, multilevel, etc.

1. To lessen the correlation between a multiplicative term (interaction or polynomial term) and its component variables (the ones that were multiplied).

2. To make interpretation of parameter estimates easier.

I was recently asked when is centering NOT a good idea? (more…)

Centering for Multicollinearity Between Main effects and Quadratic terms

December 10th, 2008 by

One of the most common causes of multicollinearity is when predictor variables are multiplied to create an interaction term or a quadratic or higher order terms (X squared, X cubed, etc.).

Why does this happen?  When all the X values are positive, higher values produce high products and lower values produce low products.  So the product variable is highly correlated with the component variable.  I will do a very simple example to clarify.  (Actually, if they are all on a negative scale, the same thing would happen, but the correlation would be negative).

In a small sample, say you have the following values of a predictor variable X, sorted in ascending order:

2, 4, 4, 5, 6, 7, 7, 8, 8, 8

It is clear to you that the relationship between X and Y is not linear, but curved, so you add a quadratic term, X squared (X2), to the model.  The values of X squared are:

4, 16, 16, 25, 49, 49, 64, 64, 64

The correlation between X and X2 is .987–almost perfect.

Plot of X vs. X squared
Plot of X vs. X squared

To remedy this, you simply center X at its mean.  The mean of X is 5.9.  So to center X, I simply create a new variable XCen=X-5.9.

These are the values of XCen:

-3.90, -1.90, -1.90, -.90, .10, 1.10, 1.10, 2.10, 2.10, 2.10

Now, the values of XCen squared are:

15.21, 3.61, 3.61, .81, .01, 1.21, 1.21, 4.41, 4.41, 4.41

The correlation between XCen and XCen2 is -.54–still not 0, but much more managable.  Definitely low enough to not cause severe multicollinearity.  This works because the low end of the scale now has large absolute values, so its square becomes large.

The scatterplot between XCen and XCen2 is:

Plot of Centered X vs. Centered X squared
Plot of Centered X vs. Centered X squared

If the values of X had been less skewed, this would be a perfectly balanced parabola, and the correlation would be 0.

Tonight is my free teletraining on Multicollinearity, where we will talk more about it.  Register to join me tonight or to get the recording after the call.


Centering and Standardizing Predictors

December 5th, 2008 by

I was recently asked about whether centering (subtracting the mean) a predictor variable in a regression model has the same effect as standardizing (converting it to a Z score).  My response:

They are similar but not the same.

In centering, you are changing the values but not the scale.  So a predictor that is centered at the mean has new values–the entire scale has shifted so that the mean now has a value of 0, but one unit is still one unit.  The intercept will change, but the regression coefficient for that variable will not.  Since the regression coefficient is interpreted as the effect on the mean of Y for each one unit difference in X, it doesn’t change when X is centered.

And incidentally, despite the name, you don’t have to center at the mean.  It is often convenient, but there can be advantages of choosing a more meaningful value that is also toward the center of the scale.

But a Z-score also changes the scale.  A one-unit difference now means a one-standard deviation difference.  You will interpret the coefficient differently.  This is usually done so you can compare coefficients for predictors that were measured on different scales.  I can’t think of an advantage for doing this for an interaction.