It seems that in a lot of cases people use a multiplication of centered variables

http://www.theanalysisfactor.com/when-not-to-center-a-predictor-variable-in-regression/

http://stats.stackexchange.com/questions/64949/interaction-term-using-centered-variables-hierarchical-regression-analysis-what

http://www.talkstats.com/showthread.php/6302-Centering-for-interactions-in-multiple-regression

The main reason for this besides interpretation being that it eliminates collinearity. Because in the end people want to know to what extent a predictor predicts or is correlated to the predicted variable and not how it is related to other predicting variables.

As you have observed and remarked this means also multiplication of two negative values, resulting in a positive value.

The course example below demonstrates in a nice way why people might do it like this and consider it useful.

http://www.med.mcgill.ca/epidemiology/joseph/courses/EPIB-621/centered_var.pdf

As the example on page two shows multiplication of two regular variables for an interaction leads to an interactionterm that is sort of correlated with its two variables (high st values means high interaction value and high tc value means high interaction value).

However multipication of the two centerd variables changes something, it seems to eliminate that correlation. As the example shows the values of the interaction now

correspond with BOTH high values and low values AND negative and positive values in the variables that make up the interaction, while the interaction retains the same value in half of the cases (37.5 & -62.5). This also seems to be a consequence of multiplying negative terms; as the example shows because multiplication of 15 and 2.5 & -2.5 and -15 results in the same number forthe interaction while the original multiplied seperate variable values have low values once (negative) and high values once (positive), thus getting rid of a certain relation with the interaction.

I'm not entirely sure if it would work the same way if you center afterwards because you then just subtract 1 mean and the multiplication of negative values probably never took place, which seems to be part of the idea or at least acording to the above example.

It also reduces your variance inflation factor:

http://mlrv.ua.edu/2009/vol35_1/robinson_schumacke_rproof.pdf

Thanks Gregory,

that helps immensely. I've fully understood all the other reasons for centering variables re: interpretation and convergence in random-coefficient models, but I either missed or it was glossed over it's benefits to reduce collinearity.

I've run it both ways (multiplying centered variables and centering the product of two non-centered variables). Both give the same beta-coefficents and SE, but didn't check other diagnostics.

Thanks for the links too, very helpful.

cheers,

If you use software that does Design of Experiments, the software will code your values for (-1 to 1), run the analysis, then present coefficients based upon the original range of the variables.

Before you center your original variables, make sure the mean and median are near each other. If not, centering the variable will not do you much good. Once you have centered the original variables, then make your quadratic terms and interactions.