11. Étude de Cas - Aspects Statistques IX

Signal Detection Indexes: Sensitivity (d') and Response Criterion (C)

What does it mean that individual x has p=0.78 correct responses? According to the Signal Detection Theory (SDT; Green & Swets, 1966), proportions are composite scores. It is assumed that stimulus events are processed by a two-stage process: during an early sensory stage, the detected signal generates an internal response, which depends on the state of the observer’s sensory system. The output of this system is invariant when other factors than sensitivity change, and therefore, does not rely on strategies or motivational factors. A later decision process is influenced by strategic and payoff factors. It represents the subject’s tendency to select one response rather than another. At this stage, a response criterion is set and the system must determine whether the value it receives from the sensory system resulted from a trial in which the signal was present or absent. Importantly, the two processes  - sensory and decisional - are assumed independent: each can be changed or impaired without affecting the other.

(a) Detection tasks (yes/no tasks)
Consider a situation in which an individual is required to indicate whether a particular signal, the target, is present or absent. Is such “yes/no” tasks, the combination between the stimulus configuration and the subject’s behavior results in four types of responses: “hits” (H) are true positives, that is, the target was present and the subject said “present”; “false alerts” (FA) are false positives, that is, the target was absent but the subject said “present”; “correct rejections” (CR) are true negatives, that is, the target was absent and the subject said “absent”; finally, “misses” (M) are false negatives, that is, the target was present but the subject said “absent”. According to the SDT, the proportion of H and FA can be used to compute one value reflecting the sensitivity index, d’, and one value reflecting the response criterion, C. Equations 1 and 2 (MacMillan & Creelman, 2005) allow to compute these indexes:

d'=z(H)-z(FA)             (eq.1)                        C=-0.5(z(H)+z(FA))    (eq.2)

where H is the proportion of H and corresponds to H/(H+M), and FA is the proportion of FA corresponding to FA/(CR+FA). Thus, z(H) and z(FA) denote the standard normal distribution transform of H and FA, respectively. The sensitivity index, d’, describes the degree of target-nontarget detectability. If d’=0, the subject failed to distinguish targets from nontargets and, thus, performed at chance level. Conversely, increasing d’ suggests increased detectability. If C=0 then, independently from the response correctness, the proportions of “yes” and “no” responses given by the individual were the same; if C>0 then the subject adopted a conservative criterion leading him/her to give more “no” than “yes” responses, and thus, tending to reduce the number of FA; if C<0 then the subject adopted a liberal criterion leading him/her to give more “yes” than “no” responses, and thus, tending to reduce the number of M.


(b) Two-alternative forced choice (2AFC) tasks
Now consider a situation in which an individual is required to make a 2AFC and indicate which of two particular signals was presented. Is such tasks, the target is always present and subjects are required to make a discrimination judgment. Thus, there is only two kinds of reponses: correct responses and errors. The four categories described in the previous section do not exist anymore, this is why eq.1 and eq.2 are not adequate. Furthermore, since there are not yes/no responses, performance is relatively free of response bias and, therefore, the C index is no more computed  (MacMillan & Creelman, 2005). Equation 3 allows to compute the d'2AFC :

d'=z(p)√2             (eq.3)

where p is the proportion of correct responses. The d'2AFC describes the degree to which the subject can discriminate the target. If d’=0, the subject failed to discriminate which of the two alternatives was presented and, thus, performed at chance level.


(c) Signal Detection and the Single Case
The SDT was created on the basis of the single-subject model, a reason why classical psychophysical reasearch is especially carried out on single-subject performance (Green & Swets, 1966). Some authors (MacMillan & Creelman, 2005) insist that signal detection analyses are appropriate only in single-subject studies. There are, however, some requirements: (a) the number of trials must be  large enough (N>40 and even N>100); (b) if 0.0 or 1.0 proportions suggest that the test was either too difficult or too easy and that SDT analyses are inappropriate.



REFERENCES
- Green D., Swets J. (1966). Signal detection theory and psychophysics. New York, John Wiley & Sons.
- MacMillan N.A., Creelman C.D. (2005). Detection theory: A user's guide. London, LEA.