# Centering data in multiple regression

In a multiple regression analysis (with 4 continuous predictors and 2 categorical factors), we mean centered the data (for each continuous variable) due to issues of multicollinearity when the interaction terms are included.

My question is whether I can center the response variable too.

More specifically, the response variable and the 4 continuous predictors are all averaged survey responses (using scales of 1 to 5). I originally thought that if I were centering the explanatory variables, I might as well center the response.. but I realize that most references to centering seem to only apply to the predictor variables.

Any help is much appreciated.

Update: as I mentioned in an earlier comment, I questioned the validity of centering my response variable due to having different ANOVA results using centered versus non-centered response. My interpretation is that the linear model for a non-centered response (using the lm() in R) uses the mean of the 'reference level' of my two factors for computing the intercept. When I centered the response variable, it's being subtracted from the grand mean of this variable, rather than the mean of the reference level. Now I've verified that by 'centering' using the reference level mean and it does indeed yield identical results for p-values as the non-centered response model. I hope I am interpreting this issue correctly. IF someone could further confirm/clarify this I would really appreciate that.

• i don't think centering cures collinearity – Aksakal Mar 2 '14 at 4:07
• It might help with the piurely numerical issues, not the statistical ones! But, with modern numerical linear algebra algorithms, probably not. Centering should be done if it helps interpretation of the model! – kjetil b halvorsen Jun 10 '14 at 8:15
• @Aksakal, centering does affect collinearity. Please see here. – Vivek Subramanian Apr 23 '17 at 20:29
• @VivekSubramanian, there are problems in that answer you refer to. You can't cure collinearity by centering – Aksakal Apr 23 '17 at 21:14

• I am surprised by your argument about categorical responses $Y$. If OLS is being used to fit $Y$, then it is computing $\hat{\beta}=HY$ where $H=(X'X)^{-1}X'$. To fit a centered and/or scaled version $Z=(Y-m)/s$, observe that $HZ$ = $H(Y-m)/s$ = $(HY-Hm)/s$ can be reversed to obtain $\hat{\beta}$ = $sHZ+Hm.$ Nothing at all is lost and nothing is any different when $Y$ is categorical. (However, usually one shouldn't be using OLS for truly categorical responses anyway.) – whuber Jan 29 '14 at 22:09
• I haven't even considered the interpretation, Nick, because this appears to be a purely mathematical issue. All that matters is that (1) $Y$ is a bunch of numbers being fitted with OLS and (2) centering and rescaling are linear transformations. Issues about distributions of residuals, interpretation of $Y$, etc., do not seem to be relevant because one can estimate the model using the centered/scaled responses and then, through a linear back-transformation, recover the correct parameter estimates (and their confidence intervals, p-values, and so on). – whuber Jan 29 '14 at 22:27