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A question about scaling

Brian Kundsen writes:

I recently read your article “Scaling regression inputs by dividing by two standard deviations“. I am writing my thesis, and am trying to decide whether and how to standardize regression coefficients. A few quick questions that come up after having read your article.

First, if one uses the typical standardization (i.e. divide independent and dependent variables by one SD after subtracting mean), I think the interpretation is that if the independent variable changes by 1 standard deviation, the dependent variable changes by “beta” standard deviations. For your technique (i.e. only standardize the independent variables and divide by two sd’s), is there also some interpretation of the standardized coeffs?

Second, related to my first question, if I use your technique but also standardize the dependent variable, how does one interpret the standardized coefficient?

My reply:

Under my scaling, the interpretation of the coefficient is that it corresponds to a change of 2 sd’s, which happens to be the approximately same as a change from 0 to 1 for a binary predictor (if p falls between 0.3 and 0.7, say). The idea is that binary predictors are intuitive, so that the divide-by-2-sd’s rule puts continuous predictors on this interpretable scale.

The other alternative, which I kinda wish I’d done, is to scale by dividing by 1 sd, and then to redefine binary inputs to take on the values -1 and +1 (instead of 0 and 1). This would work just as well, it’s just a slightly different convention.

If you scale the outcome as well, then you can interpret the coefficient as the number of standard deviation changes in y that correspond to a 1 standard deviation change in x. This puts multiple regression coefficients on the same scale as correlations.

P.S. When searching on my webpage, I found that I had three articles with the word “scaling” in the title.