Scaling regression coefficients Take 2: Gelman (2008) approach

Question

I am asking a follow-up question about interpreting regression coefficients that have been scaled following Gelman's (2008, 2009) recommendations.

Original recommendation to divide continuous predictor by 2 SD.
http://www.stat.columbia.edu/~gelman/research/published/standardizing7.pdf II)

Updated recommendation (2009): divide continuous predictor by 1 SD AND re-code binary input values from (0,1) to (-1,+1)). Updated recommendation (1 SD, recode binary): http://andrewgelman.com/2009/06/09/standardization/

When comparing a dichotomous and a continuous predictor within a binary logistic regression, I am unclear on proper interpretation of the "SCALED" coefficients

(1) Logit coefficient (2) Odds ratio (coefficients)

Example of coefficients for a binary logistic regression model

Binary IV: Logit=1.08, OR=2.96, p=0.002 Continuous IV: Logit=0.32, OR=1.37, p=0.008

Binary IV has been recoded to take -1, +1 Continuous IV has been rescaled by dividing the variable by 1 SD Binary DV has not been recoded

QUESTION:

What is the optimal format of the scaled regression coefficient (Logit, OR, probability?)?

Could someone offer an appropriate interpretation of the above values? Is it acceptable to conclude that the binary predictor variable shows a "stronger" association with the DV than does the continuous variable?

Answer 1

Gelman added a bit more detail to the explanation of the rescaling here. The -1/+1 recoding accommodates an interpretation of a 1 SD change to a change from 0 to 1 in the binary variable. If coded as 0/1, a change from 0 to 1 would correspond to a change of appx. 2 SD. With the recoding you can then rescale all inputs, whether binary og continuous, by dividing with 1 SD.

Also note that all regression inputs, not just the predictors, should be rescaled to get the convenient parameter interpretation. It's unclear from your question whether that is what you have done.

But assuming you have rescaled all inputs, a change from 0 to 1 in the binary variable changes your predicted value more than a 1 SD change in the continuous value (assuming 1.08 and 0.32 are your parameter estimates).