Linear Regression Analysis - Centering For Multicollinearity Between Main Effects and Quadratic Term

In Regression Analysis, 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

Linear Regression Analysis - Centering For Multicollinearity Between Main Effects and Quadratic Term

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.

To remedy this, 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 manageable. 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.

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

Linear Regression Analysis - Centering For Multicollinearity Between Main Effects and Quadratic Term
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And now I would like to invite you to learn all about what multicollinearity is, how to diagnose it, and what to do about it in one of my FREE monthly Analysis Factor Teleseminars: "Correlated Predictors in Linear Regression: How to Detect and What to Do about Multicollinearity." Visit Teletraining 1 to get started today.

© 2008 Karen Grace-Martin -- Statistical Consultant and founder of The Analysis Factor

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