In my opinion, the $\beta'_j$ measure you propose doesn't have a useful role in understanding logistic regression and you would be better of using traditional generalized linear model summaries.
Here is a simple logistic regression example to show that the $\beta_j'$ measures are misleading:
> x1 <- seq(from=-1, to=1, length=1000)
> beta <- 0.5
> p <- exp(x1*beta) / (1+exp(x1*beta))
> y <- rbinom(1000, prob=p, size=1)
> x2 <- rep(0,1000)
> x2[500] <- 1
> y[500] <- 1
> fit <- glm(y~x1+x2, family=binomial, maxit=100, epsilon=1e-18)
Warning message:
glm.fit: fitted probabilities numerically 0 or 1 occurred
In this example x1 is highly significant whereas x2 is not significant at all.
x2 has little or no predictive power:
We can see this either from the t-statistics and Wald p-values:
> summary(fit)
Call:
glm(formula = y ~ x1 + x2, family = binomial, maxit = 100, epsilon = 1e-18)
Deviance Residuals:
Min 1Q Median 3Q Max
-1.4218 -1.1601 0.9551 1.1401 1.3907
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 3.481e-02 6.400e-02 0.544 0.587
x1 5.234e-01 1.117e-01 4.686 2.78e-06 ***
x2 3.153e+01 6.711e+07 0.000 1.000
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 1386.0 on 999 degrees of freedom
Residual deviance: 1362.2 on 997 degrees of freedom
AIC: 1368.2
Number of Fisher Scoring iterations: 30
or from analysis of deviance and likelihood ratio tests:
> anova(fit, test="Chisq")
Analysis of Deviance Table
Model: binomial, link: logit
Response: y
Terms added sequentially (first to last)
Df Deviance Resid. Df Resid. Dev Pr(>Chi)
NULL 999 1386.0
x1 1 22.4111 998 1363.6 2.201e-06 ***
x2 1 1.3513 997 1362.2 0.245
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Nevertheless x2 has a much higher $\beta_j'$ measure than x1:
> beta1 <- coef(fit)[2]
> beta2 <- coef(fit)[3]
> beta1*sd(x1)
[1] 0.3026638
> beta2*sd(x2)
[1] 0.9971314
Here the value of $\beta_2'$ would actually be infinite if we were able to fit the glm in exact arithmetic rather than on a floating point computer.
Measuring predictive power in a meaningful way requires comparing the predicted probabilities to the actual outcome values.
One useful summary is the area under the receiver operating curve (auROC):
The fitted model with x1 has auROC of 59%, regardless of whether x2 is in the model as well:
> library(limma)
> auROC(y, fitted(fit))
[1] 0.5870262
> fit1 <- glm(y~x1, family=binomial)
> auROC(y,fitted(fit1))
[1] 0.5861539
x2 alone is hardly better than random:
> fit2 <- glm(y~x2, family=binomial)
> auROC(y, fitted(fit2))
[1] 0.5009823