# 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 Mar 2, 2014 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! Jun 10, 2014 at 8:15
• @Aksakal, centering does affect collinearity. Please see here. Apr 23, 2017 at 20:29
• @VivekSubramanian, there are problems in that answer you refer to. You can't cure collinearity by centering Apr 23, 2017 at 21:14

With continuous dependent variables, you can center these too if you want. Just don't forget that your predicted values have had the mean subtracted from them; otherwise, you should be able to interpret the results normally. If you're not sure whether you want to center in a case like this, or want to consider other issues, you might find this question useful: When conducting multiple regression, when should you center your predictor variables & when should you standardize them?

With categorical variables, the mean may not be appropriate to use for centering, and the data may not be appropriate for fitting a multiple regression model with ordinary least squares. When averaging a reasonably large number of Likert scale responses (say, across five or more items) with a reasonably wide set of options (five options might be enough), you might be okay in using the mean, but you should probably check whether your response frequencies for each item seem to be approximating a normal distribution (i.e., not a distribution with strong skew, excess kurtosis, a bimodal shape, etc.). When you average them across your set of items, check again to make sure these scores seems roughly normal.

If they're not, you might need to explore other methods for handling ordinal data in regression. Item response theory models like the rating scale model might be more suitable. You could also try fitting a structural equation model that relates the latent factors represented by your Likert rated items to your dependent variables using a polychoric correlation matrix. You might find my answer to a related question useful for this.

• 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, 2014 at 22:09
• I answered this one in a hurry; might've missed some subtlety. I was mostly thinking of mean-centering the averaged survey responses as potentially problematic if those variables don't approximate a normal distribution well enough. If, as you seem to point out, the rule is less strict for categorical dependent variables (or DVs that are averages of Likert ratings, and thus pseudo-continuous), that's news to me, and I should probably edit when I have time! Still, isn't there some potential problem in using the mean of ordinal data that aren't roughly similar to a continuous normal distribution? Jan 29, 2014 at 22:21
• 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, 2014 at 22:27
• I see (I think)! One would definitely want to perform that back-transformation before trying to interpret them in terms of the original scale though, no? Nice to know that's an option anyway, should the need arise. Jan 29, 2014 at 22:42
• Thank you for the help @NickStauner. I did check the distribution of my likert scale predictors and they seem fine. They were indeed average over many items. So in this case, does that mean I should expect the same results for p-values in the models? I started questioning whether this was doable because centering the response variable or not seemed to generate different ANOVA results. Of course, there may be other mistakes that I need to discover, but I would like to know if they should be producing the same results to start. Thanks again!
– Wynn
Jan 29, 2014 at 22:45