# Will ROC curve for a model always be symmetric if we have enough training data?

ROC curve usually looks like the following figure:

If we have enough data, could we safely assume that ROC curve for a model will always be symmetric around the line y = 100 - x? If not, is there any way to make the ROC curve shift up (the red line)?

I am asking this question because my final goal is to optimize sensitivity at a minimum specificity. However, using sensitivity as my evaluation metric seems to be very similar to using AUC. So, I am wondering whether using AUC is highly correlated with using sensitivity as an evaluation metric because you can not intentionally tweak the shape of ROC curve.

I also tried using sample weight, but it seems to only impact the cutoff of the classification model. Please check my attempts with SVM by changing the weight of the positive class.

• It's ROC curve, not AUC curve. AUC is a single statistic you compute from the curve. Commented Feb 28, 2017 at 6:04
• No, it won't always be symmetric. It depends on the distribution of the "score" that generates the curve. Commented Mar 8, 2017 at 21:58

Take as an example a dataset with a binary dependent variable $y$ and a single binary independent variable $x$. Assume that $y=1$ whenever $x=1$ and that $y$ is randomly selected when $x=0$, taking value 1 with probability 0.5 and taking value 0 with probability 0.5. Assume $x=1$ for 1/3 of the observations and $x=0$ for 2/3 of the observations.
Basically you can identify half of the positives with no errors using the observations with $x=1$, and the rest of the selection is more or less random. The ROC curve from your model will not be symmetric regardless of the size of the training set.