How do you reconcile a statistically insignificant logistic regression with a significant AUC? Suppose I run logistic regression with one continuous predictor. The Wald tests of significance show a p-value that is not significant at the 5% level. Looking at the receiver operating characteristics curve though shows an AUC that is significant, with DeLong's method. 
How can I interpret these results?
 A: The AUC in this case with a single continuous predictor has a simple interpretation: among all the pairs of observations with different outcomes, it's the fraction of pairs that had the correct order of the predictor value. It's closely related to the non-parametric Mann-Whitney U test for comparing a continuous predictor between two groups. It doesn't care about the shape of the relation between the predictor value and outcome probability.
The logistic regression model, on the other hand, tests a much more restrive hypothesis: that the log-odds of group membership is linearly related to the value of the continuous predictor. If there is a relation between the continuous predictor and group membership probability that does not meet this linearity assumption, it could well fail a Wald test in logistic regression but pass a non-parametric test for a significant AUC.
You might want to test the linearity assumption in your logistic regression and try  transformations or spline modeling of the continuous predictor to improve the logistic regression performance.
