Hello everybody,
I was wondering about the computation and interpretation of interaction terms of continuous variables that are used in a multiple regression.
Normally, one would mean-center (or z-standardize) the two constituent continuous variables and then multiply these values to create the interaction term. Then all three variables (continuous variable A, continuous variable B, and the interaction term AxB) can be used as regressors in a multiple regression.
However, I am a bit confused regarding negative values in the continuous variables (that result for instance from mean-centering/z-standardization). If one has only positive values, then two high positive values in both A and B would result in a high value in AxB (e.g., 10 x 10 = 100), and lower values in A and B would result in very small values in AxB (e.g., 0.2 x 0.2 = 0.04). This would result in a regressor where low values in the interaction regressor indicate low values of both continuous variables, and high values indicate high values of both.
BUT: With negative values, this relationship appears to change fundamentally. Let's say we have now two low values of both continous variables that are negative, then the interaction term would be a high positive value (e.g., -10 x -10 = 100). But this has the same value as both values in the positive domain (e.g., 10 x 10 = 100). Shouldn't they get different weights in order to differentiate effects of high/high vs. low/low? In the same fashion, high/low and low/high receive equal weights (e.g. -10 x 10 = -100; 10 x -10 = -100) and cannot be differentiated. That confuses me a lot. Or do I miss something here?
Best, Stefan
Hey Stefan! Been awhile...
You've got it pretty much down already. Intuitively, the interaction term then becomes effectively the index of how deviant the joint combination of the two other variables is. There's no need to differentiate high\high vs low\low in that omnibus term since all that's interested in is testing to see if the variability due to joint extremeness (independent of the actual shape) is significantly different from zero. Imagine a X type interaction (with the lower values dipping down to the negative zone) vs one driven by positive slopes of differing strength. You can easily see how the overall interaction test could be equal despite the fact that the pattern of high/low, low/high, low/low and high/highs are totally different. That lack of differentiation is exactly what's desired. The sign of the index is less important than it's relative relationship to the other variables. In your example. even if you hadn't mean centered the variables (leaving everything positive and along the lines of your first AxB case), the test of the interaction term would be unimpacted would have had the exact same value.
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