When performing linear regression, if the interaction is considered, in order to prevent multicollinearity, it is necessary to standardize the interactive variables and then multiply the two variables. (x1,x2)
But if x1,x2 is originally a positive number, then after standardized, it seems that the original multiplication result will change.
Such as, when both variables are at their minimum, their product is the same as when both variables are at their maximum.( For example, when x1 and x2 are not standardized, the value is 0 to 1, and after standardized, the value of s_x1 and s_x2 is -0.5 to 0.5, when both are minimum, s_x1=-0.5, s_x2=-0.5, s_x1*s_x2=0.25. when both are maximum, s_x1=0.5, s_x2=.05, s_x1*s_x2=0.25) They are clearly not the same without standardized.(when both are minimum, x1*x2=0, when both are maximum, x1*x2=1). (Just like before standardize, the image looks like a linear function after multiplying the two, but after standardize, it looks like a quadratic function)
Does this lead to a change in the meaning of the interaction term?
For example, x1 and x2 are the amount of funds and personal ability respectively, and Y is the work result. When both funds and personal abilities are high, additional work results will be harvested, and when funds and personal abilities are low, additional work results will be reduced. But after standardized, it seems that "high financial and high personal ability " and "low financial and low personal ability" have the same impact on work results.
I'm confused what's going on here.
No standardization is needed, and I typically would not recommend it. To remove non-essential collinearity (collinearity that is due to the variables having non-zero means), you can simply center the variables at the mean, then compute the interaction (product) term of centered variables. This will not change the regression weight for the interaction term relative to an uncentered analysis but it will change the regression weights of the lower-order terms and will often make them more interpretable.
Collinearity thus does not affect the analysis of the interaction term and is therefore often irrelevant/not a concern. Centering mostly serves to make the lower-order terms more interpretable when x1 and x2 have no meaningful zero points. See
Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2003). Applied multiple regression/correlation analysis for the behavioral sciences (3rd ed.). Lawrence Erlbaum Associates Publishers.
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