Calculating sensitivity from hit rate and false alarm rate
It is helpful to think further about the relationship between our behavioral measures and our model parameters. Each point in ROC space describes a particular pattern of performance in terms of hit rate and false alarm rate, and that particular pattern of performance can be accounted for by SDT using a certain combination of sensitivity and bias. We’ve already seen how to calculate the hit rate and false alarm rate from d′ and c, but we can go the other direction as well, calculating the model parameters from the performance measures. Here we will focus on sensitivity, and then on the next page we will focus on bias.
If you experiment with the model above, you will discover that in order for the proportions of hits and false alarms to be equal, the distributions must be in the same location. If you then move the signal distribution to the right and sensitivity increases, there will be more hits and fewer false alarms. Indeed, using the inverse cumulative distribution function of the normal distribution, Φ−1, d′ can be determined from the difference between hit rate and false alarm rate:
Using the equation above, we can now determine the sensitivity for each point in ROC space. In the graph below, intensity of color (saturation) is used to represent d′ along a continuum. In order to help us see the resulting pattern, iso-sensitivity contours have been added to show sets of points with the same values of d′:
What is immediately clear is that sensitivity has a non-linear relationship with hit rate and false alarm rate. Furthermore, any d′ can occur at any hit rate and any false alarm rate, and vice versa.
Visualizing the relationship between d′, HR, and FAR
Showing the sensitivity for every location in ROC space is instructive, but when we are focused on performance, a more typical approach is to plot a single iso-sensitivity curve through our data point. This curve shows all of the combinations of hit rate and false alarm rate that have the same sensitivity as our actual data.
Try manipulating performance by changing values in the outcome table, adjusting the locations of the distributions or threshold in the model, or by directly moving the data point in ROC space. All of the other representations of performance will change simultaneously. Observe how the iso-sensitivity curve responds:
The values are linked between the live table, ROC space, and model. As the Hit Rate or False Alarm Rate are changed, d′ and c will also change, and thus the Iso-Sensitivity Curve will adjust as well.
Note that the iso-sensitivity curve always spans from the bottom left to the upper right corner of ROC space. As a result, neither hit rate alone nor false alarm rate alone tells you anything about sensitivity — you need both! Furthermore, altering either the hit rate or the false alarm rate necessarily changes the sensitivity.
However, there is one way to manipulate the performance in just such a way that the results change while keeping sensitivity constant. This happens when we manipulate bias, which is independent from sensitivity. Adjust the value of c by changing the threshold in our model and observe that our data point shifts smoothly along the iso-sensitivity curve in ROC space, but the curve itself remains stationary.