Interpreting regression coefficients based on Andrew Gelman's re-scaling method

Question

I have two predictors in a binary logistic regression model: One binary and one continuous. My primary goal is to compare the coefficients of the two predictors within the same model.

I have come across Andrew Gelman's suggestion to standardize continuous regression input variables:

I) Original proposal (2008): divide continuous predictor by 2 SD

Original manuscript: 
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/

Proper interpretation of the resulting coefficients still is elusive to me:

SCENARIO 1: BOTH PREDICTORS ARE SIGNIFICANT IN THE SAME MODEL

Outcome: non-transformed binary Y Continuous predictor: XCONT (divided by 1sd) Binary predictor: XBIN (recoded to take values -1 or 1)

  > orfit1c=with(data=mat0, glm(YBIN~XCONT+XBIN, 
   family=binomial(link="logit")))
  > summary(orfit1c)

   Call:
   glm(formula = YBIN ~XCONT + XBIN, family = binomial(link = "logit"))

   Deviance Residuals: 
           Min       1Q   Median       3Q      Max  
       -0.9842  -0.6001  -0.5481  -0.5481   1.9849  

       Coefficients:
                    Estimate Std. Error z value Pr(>|z|)    
        (Intercept)  -1.8197     0.1761 -10.331  < 2e-16 ***
        XCONT         0.3175     0.1190   2.667  0.00765 ** 
        XBIN          1.0845     0.3564   3.043  0.00234 ** 
        ---
       Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

       (Dispersion parameter for binomial family taken to be 1)

       Null deviance: 398.99  on 409  degrees of freedom
       Residual deviance: 385.88  on 407  degrees of freedom
       AIC: 391.88

SCENARIO 2: NEITHER SIGNIFICANT IN SAME MODEL (BUT when they are entered separately in two different models, their coefficients are both significant)

       Call:
      glm(formula =YBIN2 ~ XCONT2 + XBIN2, family = binomial(link = 
       "logit"))

               Deviance Residuals: 
           Min       1Q   Median       3Q      Max  
          -1.0090  -0.6265  -0.5795  -0.5795   1.9573  

            Coefficients:
                   Estimate Std. Error z value Pr(>|z|)    
       (Intercept)  -1.7562     0.1835  -9.570   <2e-16 ***
       XCONT2         0.2182     0.1318   1.656   0.0977 .  
       XBIN2        0.6063     0.3918   1.547   0.1218    
                         ---
       Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

          (Dispersion parameter for binomial family taken to be 1)

          Null deviance: 398.99  on 409  degrees of freedom
          Residual deviance: 390.01  on 407  degrees of freedom
          AIC: 396.01

Question: For the original scaling method, it was explained that "a one-unit change in a continuous predictor covers two standard deviations of that predictor"

For the updated scaling method, am I correct in my interpretation of scenario #1 that:

(1) a one unit change in continuous predictor covers 1 standard deviation of the XCONT

(2) and this change in 1 SD in XCONT is equivalent to a 1 unit change (i.e. absence or presence) of the binary predictor (XBIN).

(3) accordingly, 1 SD change in XBIN predicts a 1 unit increase in YBIN whereas 1/3 unit increase in YCONT predicts a 1 unit increase in YBIN?

QUESTIONS

• Does interpretation outlined in 1-3 need any correction? As the binary outcome variable was not re-coded, can I still say that 1 unit change in X predicts a 1 unit change in binary outcome (0 or 1)?

• What else could be said about the results, specifically when I attempt to compare the two coefficients for the continuous and binary predictors?

Answer 1

(1) is somewhat awkwardly stated, and I'm not sure exactly what you mean. I would interpret the XCONT parameter estimate as:

Observations one standard deviation above the mean of XCONT have YBIN ~32 percent more often.

When you interpret the XBIN parameter estimate, keep in mind that you're interpreting at the average of XCONT.

Richard McElreath works through an example of this rescaling in Statistical Rethinking.