Percent Nonoverlap: A Way of Thinking About Detection and Discrimination Accuracy and the Value of d′
We defined d′ as the distance between the means of the noise and signal plus noise distributions. We also indicated that the distance from an ROC curve to the diagonal was another measure of d′. The advantage of d′ over other measures of accuracy is that it is a pure measure of detectibility or discriminability. Other measures, such as the probability of a hit, are not independent of decision criteria. Although this presents a strong advantage to using d′, there is also a drawback. We all have a good intuitive understanding of percentage measures. When we say that an observer detected 95% of the signals, we know that, given 100 signals, the observer detected 95 of them. We can also relate this value meaningfully to other percentages of accuracy. We don't have as good an intuitive understanding of d′. If d′ = 1.0, we know that one standard deviation unit separates the means of the signal-plus-noise and noise distributions. But how large is a difference of 1 standard deviation? Is it a modest difference or a huge difference? How can we think about it?
Jacob Cohen ( 1988) introduced the use of percentage nonoverlap to increase our understanding of the magnitude of the difference between means of two normal distributions in standard deviation units. We use percentage nonoverlap to help clarify our understanding of d′.
As was stated previously, when d′ = 0 there is no detection. When d′ = 0 there is no distance between the means of the signal-plus-noise and noise distributions. Another way of saying this is that the two distributions are 100% overlapped; there is 0% nonoverlap. As d′ takes on values greater than zero, some portion of the area covered by both distributions combined will not be overlapped. The percentage nonoverlap provides us with a way of thinking about discriminability and values of d′.
Consider a d′ value of 0.50. The means of the signal-plus-noise and noise distributions are separated by 0.50 standard deviation. This means that a portion of the area covered by both distributions does not overlap. Cohen ( 1988, Table 2.2.1) listed the percentage nonoverlap between two normal distributions separated by distances from 0 to 4 standard deviation units. Referring to his table, we find that 33% of the combined area is nonoverlapped when d′ = 0.50. That is, 33% of the area covered by the signal-plus-noise and noise distributions combined is either noise or signal plus noise, but not both. For a d′ of 1, the percentage of nonoverlap is 55.4 %,and for a d′