This means that the coefficients are unstable. Please look at your model again before making any decisions.

Hyungkun Park Are you are discussing the problem of multicollinearity? Both correlation coefficient and VIF are used to verify multicollinearity. It must be noted that multicollinearity disables interpretation of separately considered (ceteris paribus) experimental variables.

Pradeep Paraman I do understand the concept of multicollinearity. But the thing is that, while the correlation is low, the VIFs are really high. If this is the case, should I just disregard the result from the mean-centred variables, and use the former results for the hypothesis test?

This means that the coefficients are unstable. Please look at your model again before making any decisions.

Maybe these articles will provide some clarity:

Iacobucci, Schneider, Popovich, & Bakamitsos (2016):

https://link.springer.com/content/pdf/10.3758/s13428-015-0624-x.pdf

Iacobucci, Schneider, Popovich, & Bakamitsos (2017):

https://link.springer.com/content/pdf/10.3758/s13428-016-0827-9.pdf

As you already noticed, the regression coefficient of your interaction term does not change, neither does the result of your significance test for the interaction. Thus, your interpretation for the moderation remains the same. Centering reduced multicollinearity in your case. Therefore, at least in my opinion, it makes sense to use the results based on the centered IV and MV. Ioacobucci et al. (2017) write that " doing so [mean centering] can help (per Irwin and McClelland, 2001) and it will not hurt (per Echambadi and Hess, 2007)" (p. 404).

However, please consider that the interpretation of the regression coefficients for the two predictor variables is different after mean centering.

Excellent answer by Julia. What you are maybe missing Hyungkun Park : it is not only about the correlation between the MV and IV, but also between the zero order predictors and the interaction term. Since the interaction is built out of the predictots (MV and IV), it is expected that collinearity increases. Centering the variables is typically done, but this does not affect the inferential statistics for the interaction term, but only for the zero oder predictors, as Julia mentioned.

I would (nearly) always use centered predictors, since it makes interpretation easier. For multicollinearity, it is only especially useful if you are interested in the effect of the zero order predictots, which is very often not the case or at least not so much, if you include the interaction.

But I would slightly contradict Julia's statement that the interpretation changes. The interpretation remains the same, i.e. the conditional effect of the predictors, when the other predictors have the value "zero", but the meaning of the "zero-value" of the predictors changes, since they represent the mean value of that variable, now.

Just to try and explain Rainer Duesing 's point.

A regression coefficient is the change in Y associated with a 1 unit change in X when all other predictors in the model are kept constant (and specifically with the constant value of zero).

In a model with no interactions the effect of each predictor is independent of the other predictors - it has a fixed effect representing a change in Y. Because the effects are independent a transformation of any X that adds or subtracts a constant won't change the regression coefficient. (It changes the intercept though).

This all changes in the interaction model. By definition the effect of X now depends on M - so a change in the value of X or M changes the regression coefficient. So evaluating the coefficient when X or M = 0 will produce a different result from evaluating the coefficient when X or M is at some other value such as the mean. For some purposes this doesn't matter but it does matter for interpreting what the regression coefficients are. In many cases when X=0 or M=0 is a terrible point to interpret the effects of the model. For example if I'm looking at income as a function of height and age and height*age of workers then I'm looking at a coefficients that are evaluated for someone with zero height and zero age. Centering makes zero the mean and so it all makes.(a bit more) sense. However, the coefficients (including the intercept) aren't that useful in themselves - you have to look at their joint impact on the outcome Y (e.g., a graph is a good way to do this) over a range of plausible values for X and M.

Despite the wacky numbers the centred and uncensored models both make the same predictions (thanks to their joint properties when combined with the intercept).

There is also some complicated maths to prove that the test of the interaction effect X*M is unaffected by the collinearity in the model.

see Article Clarifying the role of mean centring in multicollinearity of...

Thanks Thom Baguley for the clarifying example!

Rainer Duesing, while I have recommended your post, I do have a different view from you on one point. You wrote:

I would (nearly) always use centered predictors, since it makes interpretation easier.

I, on the other hand, would generally use center the variables only if necessary to avoid computational problems that cause a variable to be kicked out of the model.

Why? Because when it comes to generating fitted values of Y at selected combinations of the interacting variables (or simple slopes of one variable at selected levels of the other), I do not have to do mental arithmetic to convert things into terms of the centered variables--I can just specify the desired values on the original scales.

I think that we would be better off if we got away from being held hostage by the usual table of regression coefficients. Each of those coefficients represents a comparison of two fitted values, or of two simple slopes, etc. But the comparisons represented by the coefficients are just a fraction of all possible comparisons that might be of interest. And with modern statistical software, it is easy to generate the comparisons of interest to us, many of which are not captured by any of the usual coefficients.

While I'm on my soapbox, there is another way that regression coefficients can distort our thinking: They can make us think that our observational studies are more like true experiments than they actually are. With real-world observational data, the explanatory variables are typically correlated with each other to the extent that looking at the effect of one variable on the fitted value of Y while holding all other variables constant is not a very realistic thing to do. I sometimes think it would be better to compare the fitted values of Y for two realistic combinations of a set of (correlated) explanatory variables. As an example, I remember using an automobile data set to illustrate something, and the difference in miles-per-gallon (mpg) between US-made and Japanese-made cars was of interest. IIRC, I compared mpg for US cars with the median weight & length for US cars to mpg for Japanese cars with the median weight & length for Japanese cars. Using the medians may not be the best choice, but it is at least an attempt to obtain typical combinations of variables that would differ between the groups in the real world. (If anyone can suggest other better ways of doing that, I'd be happy to hear them.)

End of rant. YMMV. ;-)

Cheers,

Bruce

Sticking more to the original point of the thread than I did in my previous post, here are a couple of comments that might be helpful. The first addresses the question of when you can safely ignore multicollinearity. And the second suggests that perhaps we are generally a bit excessive in our concern over multicollinearity. (It's also quite funny, IMO.)

• https://statisticalhorizons.com/multicollinearity
• https://davegiles.blogspot.com/2011/09/micronumerosity.html

HTH.

Latter results. If centering changes the results it is probably the right thing to do. Inteterested in other opinions?

Bruce Weaver thanks for the comment and your links. The point about the mental artihmetics: maybe this is something how you are used to "imagine" the model for yourself. In psychology there are seldom really meanigful zero values nor are there very often clear cut defined cut-off points of interest, so that I started using centered variables (nearly always if interactions are included) to get a "feeling" for the model and for simple slope analyses percentile values. But I do not want to contradict you. In a recent paper (under review) we used an psychological inventar with clearly defined and psychopathological meaningful cut-off values. In such a case I would also prefer using uncentered variables and the specific points of interest, instead of generic percentile values to assess the combination of predictors and their impact on the DV.

And yes, multicollinearity is not so much of a problem as it has been stated still for a while ago. It depends in which variables/predictors you are interested in, similar to the link you shared.

Hyungkun Park just one quick comment on your choice of using cluster robust sd.

If you are worried about heteroscedasticity you should use a sandwich estimator which will address it directly.

Cluster robust standard errors can be used when you have a decent number of clusters (20+). A small number of clusters is known to lead to downward-biased SEs.

Esarey & Menger show that cluster-adjusted t-statistics are appropriate when the number of clusters is small. http://www.justinesarey.com/cluster-paper.pdf

http://faculty.econ.ucdavis.edu/faculty/cameron/research/Cameron_Miller_JHR_2015_February.pdf

https://blogs.worldbank.org/impactevaluations/beware-of-studies-with-a-small-number-of-clusters