Confused about sensitivity, specificity and area under ROC curve (AUC) Just read a unpublished paper for review purpose. The reported results like
Leave-one-out cross validation sensitivity is 95%. Leave-one-out cross validation specificity is 100%. Leave-one-out cross validation AUC is 1.
Is this possible?
If AUC is 1, does that mean both sensitivity and specificity should be 100%? Just can't get my head around how they got this kind of results.
If it is possible then how to interpret the results?
 A: An alternative interpretation of AUC is that it gives the probability that a randomly selected positive is ranked above a randomly selected negative (see, e.g., Wikipedia). Thus, AUC is 1 if and only if all positives are ranked above all negatives. However, it is still possible to obtain sensitivity or specificity under 1 by selecting a suboptimal cutoff. 
For example, let us consider a test case with 4 subjects and an algorithm predicting probabilities (of being positive) as follows:
\begin{equation}
\begin{array}{c|c}
\textrm{Truth} & \textrm{Probability} \\ \hline
\textrm{P} & 0.9 \\
\textrm{P} & 0.4 \\
\textrm{N} & 0.2 \\
\textrm{N} & 0.1
\end{array}
\end{equation}
All positives are ranked above all negatives, thus the AUC is 1. However, if the cutoff probability is set as $0.5$, one of the positives is classified as negative, and thus the sensitivity is only $50\%$. Indeed, depending on the cutoff, the sensitivities and false positive rates ($1-$ specifities) will be as follows.
\begin{equation}
\begin{array}{c|c|c}
\textrm{Cutoff in range} & \textrm{Sensitivity} & 1-\textrm{ Specificity} \\ \hline
(-\infty,0.1] & 1 & 1 \\
(0.1,0.2] & 1 & 0.5 \\
(0.2,0.4] & 1 & 0 \\
(0.4,0.9] & 0.5 & 0 \\
(0.9,\infty) & 0 & 0 
\end{array}
\end{equation}
This is illustrated in the following figure. The possible (Sensitivity, $1-$ Specificity) combinations are drawn as red circles and the (interpolated) ROC curve as green line. The entire unit square is under the curve, and thus the area under the curve is 1.

A: AUC=1 means for all specificities in [0,1] the sensitivity is 1. It is IMPOSSIBLE to get a sensitivity of 95%. So that paper is reporting a fundamental error. Hope this helps.
