# Receiver Operating Characteristic Space: Visualizing Performance

## ROC space

The hit rate and false alarm rate allow us to summarize performance with two values. We can use these values to visualize performance as a point in a space, with the *x*-coordinate defined by FAR, and the *y*-coordinate defined by HR. This space is called receiver/relative operating characteristic space (ROC space) (Swets, 1996). (The term “receiver” dates back to the original development of this approach in the context of separating signal and noise in the output of radar receivers used to detect enemy planes during World War II.)

A point in ROC space uniquely defines a combination of HR and FAR:

The graph of ROC space is *live*, so you can drag the point on the graph to move it, thereby changing the HR and FAR. Furthermore, the graph and the *live* table are linked, so when you drag the point, the table will update as well, and vice versa.

Move the point in ROC space around and observe that while each point defines a unique combination of HR and FAR, you can find multiple points that have the same accuracy. This is another way of observing the same thing we discussed on the previous page, namely that accuracy alone does not uniquely describe signal detection performance.

## Iso-accuracy contours

We might then wonder which points in ROC space share the same accuracy. We can visual this by coloring each point in ROC space according to it’s accuracy, and drawing a *contour* through the space for every 5% increase in accuracy from 0% up to 100%. The resulting iso-accuracy contours help us see how accuracy varies across the ROC space, in the same way that contour lines on a topographic map help us understand how elevation changes across the landscape.

We can now immediately see that all of the points in ROC space along each line with a slope of one result in the same accuracy. Hopefully this helps us further appreciate that knowing the accuracy of performance generally tells us very little about either the hit rate or the false alarm rate!

## Comparing patterns of outcomes

We can explore ROC space further by comparing a pair of points.

For example, we might wonder if we can have one point with a hit rate of 0.9 and an accuracy of 0.75, and a second point with a false alarm rate of 0.1 and the same accuracy of 0.75. Is this possible? Where are the two points in ROC space? Is there any sort of symmetry apparent in their placement? Try to move the two points in the ROC space below to meet this description.

The point with the up arrow, ↑, is described in the *live* table before the ROC space, while the point with the down arrow, ↓, is described in the table after the ROC space. The ROC space is *live*, so the points can be dragged to move them.

Try some other positions… What about two points that are symmetric about the dotted diagonal from lower left to upper right? What about points along an *imaginary* diagonal from upper left to lower right?

## Comparing performance

In the study of human performance, ROC space is typically used to visual performance across multiple blocks of trials. You can try that below. The current point will update throughout a block of trials, but when you reset and start a new block, the point will be left behind, and a new point will follow the next block of trials. Complete a few blocks of trials with the coherence set to different levels, including some levels that make the task a real challenge.

Use the

, , and sliders to configure a block of trials.Then use

to start the block, use if you need a break, and after the block is over use to move to the next block.Each block will have an associated point in ROC space, numbered in order. If you want to remove all of the points, just reload the page.

After you’ve completed a few blocks of trials with different settings for coherence, do you see any trends revealed in the pattern of points within ROC space?

We now have a task, and we can now describe performance in terms of hits, misses, correct rejections, false alarms, hit rate, false alarm rate, and as a point in ROC space. It is time that we think about the processes that lead to this performance.