What causes perfect prediction but no significant predictors in logistic regression? I want to do a logistic regression with R.
I have 18 continuous covariates and a sample consisting of 100 observations.
When I enter all of the covariates into the glm() model, none of them are significant, but the model predicts the outcome perfectly on the test data!
My questions are:


*

*Is the sample size large enough for running glm() with this many covariates?

*What might be other causes of the problem?

*How can I run such a model properly?

 A: You did not state the number of events and non-events.  A rough rule of thumb is that to use ordinary maximum likelihood estimation (i.e., without shrinkage - penalization) requires 15 times the number of events and the number of non-events as the number of candidate predictors.  You are far from having an adequate sample size in your case even if $Y$ is split 50-50.  I suggest doing data reduction (masked to $Y$, e.g. variable clustering, principal components, or redundancy analysis) or fitting the full list of variables and solving for the amount of shrinkage needed to yield a reliable model.  You can see case studies of these methods in my Handouts at http://biostat.mc.vanderbilt.edu/CourseBios330.
Ordinary unpenalized variable selection methods in no way deal with this problem correctly.
A: It would help if you could provide some additional information in response to the comments and @Frank Harrell's point regarding how many successes and failures you have.  
My first guess would be that you have some multicollinearity, that is, your continuous covariates are correlated with each other.  The effect of this is that, while your betas estimates are still potentially unbiased, the standard errors will be inflated.  That means that they will be less 'significant', but may still do a good job of predicting the response (note @Roland's good point about overfitting, however).  
Given your N and the number of covariates you have, you may want to watch out for quasi-separation as well.  
