# Regression: centered vs. uncentered predictors [duplicate]

I'm trying to understand how/why centering predictors in a 2 predictor regression model would change the coefficients

Lets say I have 2 centered predictors and an interaction term, predicting a continuous Y variable that is not centered, and I have gone through the process of calculating all the regression coefficients (b0-3).

Now, the question is, if I were to uncenter the 2 predictors, how would the coefficients change (if at all)?

From my understanding, the distribution of data points would not change, nor would the regression plane. However, by centering/uncentering we change the (X1, X2, Y) value of each datapoint. I don't believe this would change the b3 interaction coefficient because the relationship between the simple slopes would not be affected. The coefficients for the intercept and the 2 predictors, however, should change because we've essentially changed the scale for each of the predictors.

Can anyone shed some light on why/how this happens? and how do you determine the magnitude/direction of these changes?

## marked as duplicate by Scortchi♦Oct 8 '15 at 15:49

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• When there is no interaction term centering just change the intercept not the other coefficients. When there is an interaction term then the coefficient will change after you centering. I think this can be proved mathematically, intuitively, you can think when you centering (add or decrease some value) the effect of one variable on the other variable is not linear then the coefficient before the centering and after centering will change. – Deep North Oct 8 '15 at 1:29
• – Scortchi Oct 8 '15 at 15:50

## 1 Answer

The literature related to models with curvilinear, second-order or higher terms is among the more contradictory and controversial areas in statistics. Modern development of these topics goes back to the 60s with Jacob Cohen's work on the relationships between correlation, ANOVA and multiple regression. More recently, Aiken and West's (A&W) Multiple Regression goes into much greater methodological detail about these issues but important contributions have also been made by political scientists such as Harvard's Gary King who argues for substantive interpretation of the output, as opposed to the simple reporting of p-values and significance.

Main effects only models are most typically defined as the constant effect of one variable across values for the other variables in the model. With higher-order term (interaction) models, this is not the case. The coefficients of higher-order models represent the effects of the predictors at the mean of the other predictors and requires centering to make the *mean” be the same – at zero – for all predictors. A&W take pains to note that, e.g., in the social sciences, many predictors are rated on an interval scale (e.g., 1 to 7) where zero has no meaning. If the data isn't centered in these instances, then the evaluation of a higher-order model will occur at a value that isn't even on the scale.

A&W compare models with raw, uncentered inputs vs appropriately mean-centered predictors and find that, while the coefficients for main effects only models will not change, the main effects coefficients in higher order models do change, and potentially change dramatically.