Understanding R output in Logistic Regression

Question

I am following an example here on using Logistic Regression in R. However, I need some help interpreting the results. They do go over some of the interpretations in the above link, but I need more help with understanding a goodness of fit for Logistic Regression and the output that I am given.

For convenience, here is the summary given in the example:

## Call:
## glm(formula = admit ~ gre + gpa + rank, family = "binomial", 
##     data = mydata)
## 
## Deviance Residuals: 
##    Min      1Q  Median      3Q     Max  
## -1.627  -0.866  -0.639   1.149   2.079  
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept) -3.98998    1.13995   -3.50  0.00047 ***
## gre          0.00226    0.00109    2.07  0.03847 *  
## gpa          0.80404    0.33182    2.42  0.01539 *  
## rank2       -0.67544    0.31649   -2.13  0.03283 *  
## rank3       -1.34020    0.34531   -3.88  0.00010 ***
## rank4       -1.55146    0.41783   -3.71  0.00020 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 499.98  on 399  degrees of freedom
## Residual deviance: 458.52  on 394  degrees of freedom
## AIC: 470.5
## 
## Number of Fisher Scoring iterations: 4

1. How well did Logistic Regression fit here?
2. What exactly are the Deviance Residuals? I believe they are the average residuals per quartile. How do I determine if they are bad/good/statistically significant?
3. What exactly is the z-value here? Is it the normalized standard deviation from the mean of the Estimate assuming a mean of 0?
4. What exactly are Signif. codes?

Any help is greatly appreciated! You do not have to answer them all!

Answer 1

• Something quite useful is to use Nagelkerke $R^2$, which is just a generalization of the general R^2 statistic in linear regression. Using the rms package.

   library(rms)
   model <- lrm(y~x)
   summary(model)
  

Also, you could use cross-validation. That means you test for correct predictions in your original data using a criteria (usually 1 if predict > 0.5) and then calculate a rate. (>80% is usually fine, but that depends on the study).

• Deviance is a generalization of the residual sum of squares, and can be used to make some hypothesis testing in logistic regression.

• z-value is the statistic $(B_{j}-\hat{B_{j}})/s.e.(\hat{B_{j}})$ which asymptotically converges to a $\mathcal{N}(0,1)$(Under the null hypothesis $B_{j}=0$).

• Significant codes are just a guide. e.g. * means the associated p-value is $<0.05$.