What is the relationship between the Harrel's C and the AUC? I model survival outcomes using a Cox proportional hazards model and want to evaluate model fit.
Harrel's concordance index C is defined as the proportion of observations that the model can order correctly in terms of survival times. When censoring is observed the statistic only includes those patient pairs for which valid comparisons can be made. Sometimes C is called the AUC.
The area under the receiver operating characteristic curve, also called AUC (or AU-ROC), is defined as the area under the curve of sensitivity and 1-specificity; this statistic is equal to the concordance of predicted and observed classes, see here for example. 
At a time point t the Cox model may be used to obtain a ROC and its AUC if there is no censoring. I suppose if there is censoring, the area under the ROC can still be obtained with some additional caclulations (time-dependent AUC).
However, what is the relationship between Cox-Survival (Harrel's) C (AUC) and the AU-ROC? When are they equivalent, when different?
 A: In the case of a binary outcome and a continuous predictor, the AUC of the ROC or c-index is simply a function of how well the ordered values of the continuous predictor correlate to the corresponding event status.
In a Cox model or other time to event method, those persons with a higher predictor value (hazard ratio) should have a shorter time to event.  In addition, censoring and timing of censoring affect which pairs of data are usable in the final accounting of the censored c-index, while no such requirement is imposed on the simple c-index (except for ties).
To more directly answer your question, the censored c-index has no obligatory correlation to the c-index. The censored c-index’s requirement for accurate time ordering is simply not measured in the simple c-index. The effect of censoring and time mean that not all values are used in a censored c-index. For these reasons, the two measure discrimination differ lu and are not expected to be the same.
As proof of this concept, I created 1000 simulations of 20 subjects with random follow-up, event status and prognostic index values demonstrating no correlation between the two:

A: AUROC is the same as concordance probability (Harrel's C) for the binary outcome. If the outcome is not binary, or it is censored, then in general whatever measure you compute for this outcome would not be called AUC/ROC.  
