Hello everyone,



Many thanks to those who replied to my question on interpretation of the coefficients of a multiple linear regression model that includes centred predictors.  If you recall, my model took the form:



Yhat = alpha1+B1*(X1-X1bar) + B2*(X2-X2bar)



Let’s call this:



Yhat = alpha1+B1*Z1 + B2*Z2



Where Z1=(X1-X1bar) and Z2=(X2-X2bar)



As Montgomery and Peck (1992) say “If the range of data includes Z1=Z2=0 then the intercept, alpha1, is the mean of Y when Z1=Z2=0.  Otherwise the intercept has no physical interpretation”.  Thus, in my example above, the centring of X1 and X2  makes alpha1 interpretable and it means that we interpret alpha1 as the average value of Y when X1  is at its mean value and X2 is at its mean value.



As I mentioned yesterday, the values of the coefficients B1 and B2 are the same regardless of whether the variables X1 and X2 are centred or not, thus the interpretation of these coefficients is the same in either situation i.e. for a model having centred or uncentred predictors, B1 = the average change in Y associated with a unit change in X1 when X2 is held constant and B2= the average change in Y associated with a unit change in X2 when X1 is held constant.



Thanks again to all those who replied.  I have listed the responses below.



Kind Regards,

Kim









Kim,



  You can meaningfully interpret the alpha in eqn(2).  That is the advantage.  I am not sure centering is good when x1 & x2 are correlated( multicollinear).

   The interpretation of b1 or b2  is same as the interpretation from multivariate linear regression. ( ie., Suppose if you have two individuals with same x2 unit but differ a unit in x1 between them there will be a difference of b1 in Yhat between them.)



Regards







Dear Kim,



I hope this is helpful...



You should find that the beta coefficients are exactly the same in both models, but the alpha coefficient is different, as you explain.



Not only are the coefficients the same, the interpretation is the same: Holding one of the variables constant (it does not matter what that constant is), the change in the average outcome will change by the other variable's coefficient if the other variable increases by 1.



So, you can hold one of the variables at whatever constant value you like, including its mean value if you wish, to interpret the relationship between the other variable and the average outcome.



All the best.







On Mon, Jul 2, 2018 at 11:34 AM, Kim Pearce <[log in to unmask]<mailto:[log in to unmask]>> wrote:

Hello everyone,



I wonder if someone has any views on the following.



Say we have a multiple linear regression model talking the usual form:



Yhat = alpha + B1*X1 * B2*X2                                    (1)



We centre X1 and X2 (i.e. subtract the mean of the X1 values from each individual value of X1 and subtract the mean of the X2 values from each individual value of X2)



We arrive at the following:



Yhat = alpha1+B1*(X1-X1bar) + B2*(X2-X2bar)                     (2)



The predicted values and residuals are the same for (1) and (2) and both models (1) and (2)  fit the data equally well.



It is also obvious that, with a little rearrangement:  alpha = alpha1 - B1*X1bar - B2*X2bar



Now my question is about interpretation...in (1) we have the standard interpretation, that alpha = the average value of Y when X1 and X2 are equal to 0 and B1 = the average change in Y associated with a unit change in X1 when X2 is held constant and B2= the average change in Y associated with a unit change in X2 when X1 is held constant.



I would like to ask how (2) is interpreted.  It seems clear that:



alpha1 = the average value of Y when X1  is at its mean value and X2 is at its mean value.



But how do we interpret B1 and B2 in model (2) ?  Is B1, for example, interpreted as the average change in Y associated with a unit change in X1 when X2 is at its mean value?



Many thanks for your advice on this.



Kind Regards,

Kim















Dr Kim Pearce PhD, CStat, Fellow HEA

Senior Statistician

FMS Graduate School

Room 3.14

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Ridley Building 1

Newcastle University

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Newcastle Upon Tyne

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