Calculating bias from hit rate and false alarm rate
On the previous page, we explored how sensitivity (d′) can be determined from hit rate and false alarm rate. Now we will consider how to do the same for bias (c).
Try moving the threshold around, and observe what effect this has on the proportion of hits for the key[signal-plus-noise distribution] and the proportion of false alarms for the noise distribution. In general, as the threshold moves to the left, and a more liberal bias, we increase the hits and false alarms. As the threshold moves to the right, and a more conservative bias, we decrease both hits and false alarms. This helps develop our intuition that c is related to the sum of the hit rate and the false alarm rate, using the inverse cumulative distribution function of the normal distribution, Φ−1:
As we did for sensitivity, using the equation above we can now determine the bias for each point in ROC space. In the graph below, intensity of color (saturation) is used to represent c along a continuum. In order to help us see the resulting pattern, iso-bias contours have been added to show sets of points with the same values of c:
As with sensitivity, bias has a non-linear relationship with hit rate and false alarm rate. Furthermore, any c can occur at any hit rate and any false alarm rate, and vice versa.
Visualizing the relationship between c, HR, and FAR
Similar to what we did sensitivity, we can plot a single iso-bias curve through our data point. This curve shows all of the combinations of hit rate and false alarm rate that have the same bias as our actual data.
You can change the performance by altering 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 change simultaneously. Observe how the iso-bias 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-Bias Curve will adjust as well.
The resulting similarities and differences with sensitivity are instructive. The iso-bias curve always spans from the top left to the bottom right corner of ROC space. As a result, neither hit rate alone nor false alarm rate alone tells you anything about bias — you need both! Furthermore, altering either the hit rate or the false alarm rate necessarily changes the bias.
However, there is one way to manipulate the performance in just such a way that the results change while keeping bias constant. This happens when we manipulate sensitivity, which is independent from bias. Adjust the value of d′ by moving the distributions in our model and observe that our data point shifts smoothly along the iso-bias curve in ROC space, but the curve itself remains stationary.