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?
