Ted Dunning writes:
You advocated recently [article to appear in Statistics in Medicine] the normalization of variables to have average deviation of 1/2 in order to match that of a {0,1} binary variable.
This recommendation will disturb lots of people for obvious reasons which may make your recommendation sell better.
But have you considered normalizing the binary variable to {-1, 1} instead of {0,1} before adjusting the mean to zero? This has the same effect but leaves larger communities happier, particularly because much of the applied modeling community has always normalized their binary variables to this range.
My reply: I actually went back and forth on this for awhile. In most of the regression analyses in political science, economics, sociology, epidemiology, etc., that I’ve seen, it’s standard to code binary variables as 0/1. But, yeah, the other way to go would’ve been to standardize by dividing by 1 sd and then give the recommendation to code binary variables as +/- 1. Maybe that would’ve been a better idea. I was trying to decide which way would disturb people less, but maybe I guessed wrong!
The coefficients are still comparable but a change in 1 unit of the binary variable (-1,1) does not correspond to a change in category. Or am I missing something here?
Denis: you aren't missing anything per se. It is just that +1,-1 coding corresponds to effect coding in regression that is popular if you want to mimic ANOVA in a GLM. For those of using ANOVA a lot it seems like a natural way to accomplish this kind of standardization.
Andrew,
I am not quite sure I understand the reason standardizing by 2 standard deviations to "allow the coefficients to be interpreted in the same way as with binary inputs." To my knowledge, for a {0,1} variable, I have never heard of anyone interpreting the coefficient as corresponding to a 2-standard deviation increase. Thus, could you elaborate on why this coincidental interpretation should become the "benchmark"?
Current,
It's not a coincidental interpretation: I chose the 2 sd standardization specifically to make it match up with a binary predictor.
The point is not that the coef for a binary predictor is interpreted as 2 sd; the point is that the coef for a binary predictor corresponds to a typical change–from 0 to 1, which happens to be 2 sd. For continuous predictors, I'd also like the coef to correspond to a typical change. Given the interpretation for binary predictors, it seems to me to make sense to consider 2 sd's as a typical change: from 1 sd below the mean to 1 sd above.