# Understanding R output in Logistic Regression

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!

• possible duplicate of Interpretation of R's output for binomial regression May 3 '14 at 19:47
• That link definitely answers some of my questions, but not all. Specifically, I am still unsure about goodness of fit as the example didn't consider goodness of fit because the data as univariate. May 3 '14 at 19:57
• Here's a really nice resource for the theory behind the glm function in R: people.bath.ac.uk/sw283/mgcv/tampere/glm.pdf. Since you're using family="binomial", I believe deviance is just -2*log(likelihood). You've left the world of Sums of Squares and have wandered into the land of likelihood. Things will feel a little strange, but that resource will walk you through the analogs to ordinary regression. May 4 '14 at 2:17
• Regarding the idea of goodness of fit, people often recommend so-called $\text{pseudo }R^2$'s. This is somewhat controversial, see: Which pseudo-R2 measure is the one to report for logistic regression (Cox & Snell or Nagelkerke)? May 4 '14 at 3:10

• 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 $<0.05$.