zROC Space: Transforming ROC space

A z-transformed ROC space

In each of our graphs of ROC space up till now, hit rate and false alarm rate lie on a nice, evenly spaced grid, while our iso-sensitivity curve and iso-bias curve have been, well, beautifully curvaceous. This is great for working with the rates, but it makes things messy and non-linear when working with the curves. What if there was a simple way to reverse this situation?

Well, it turns out there is! The key is to work in the native units of d′ and c, instead of in units of hit rate and false alarm rate. And the key to that is revealed by a quick look at our SDT model to remind ourselves that the proportions of outcomes are determined by the areas under the distributions defined by the model parameters. Which brings us to the z-transformation (i.e. the inverse cumulative distribution function of the normal distribution, Φ−1).

Another path to the same conclusion is to note that while d′ and c have a complex non-linear relationship with hit rate and false alarm rate, they have simple additive relationships with their z-transformations:

As a result, if we use z-transformed hit rate and false alarm rate, our iso-sensitivity curve and iso-bias curve are straight lines in what is called zROC space:

This example is set up for model exploration. You can flip back and forth between ROC space and zROC space with the zROC-ROC switch.

Play with it for a little while and see that changing the values of d′ and c move the iso-sensitivity curve and iso-bias curve around (i.e. their y-intercept changes), but their slope remains the same (at one and negative one respectively).

Iso-contours in zROC space

Another way to visualize the effect of the transformation is to look at the iso-bias, iso-sensitivity, and iso-accuracy contours in zROC space as compared to ROC space:

The utility of zROC space will be become clearer on the next page when we discuss distributions with unequal variance. So let us proceed.