= Working out 2 group signal detection diagnostics =

The area under Receiver Operating Characteristic (ROC) curves is a way of describing the magnitude of how well separated two groups are with respect to some diagnostic. 

For two groups where the mean of A > mean of B it is a plot of sensitivity at each of T thresholds (Probability of being in group A given you have a value less than or equal to t) against 1 - specificity (probability of being in group B given you have a value less than or equal to t) as seen [[attachment:rocplot.jpg|here]].

The sensitivity is the blue horizontal area and 1-specificity is the diagonal orange area evaluated at a score of 1. These can also be evaulated using frequencies. You can see that as sensitivity increases so does 1-specificity (with the threshold increasing ie line moving to the right). When the line is at the far left both sensitivity and 1-specificity are zero and when the line is on the far right they are both equal to unity.

||||||||<style="TEXT-ALIGN: center">  || ||||<style="TEXT-ALIGN: center">True ||
||||||||<25% style="VERTICAL-ALIGN: top; TEXT-ALIGN: center">   || ||  A || B ||
||||||||<25% style="VERTICAL-ALIGN: top; TEXT-ALIGN: center"> Score  || <= t ||  a || b ||
||||||||<25% style="VERTICAL-ALIGN: top; TEXT-ALIGN: center">    || > t ||  c || d ||

In the above table the sensitivity is the proportion of observations in group A which have values less than or equal to t and 1-specificity is the proportion of observations in group B which have a value less than or equal to a particular score t.

These probabilities are evaluated at each observed score, t and plotted with sensitivity on the y-axis and 1-specificity on the x-axis. The points may be joined together to form a curve and the area under the curve evaluated using, for example, the trapezium rule. The area under the ROC curve may be computed on the raw data [[FAQ/roc| using the LOGISTIC REGRESSION procedure in SPSS]] or a [[FAQ/rocplot| SPSS macro.]]