For each subject in my sample, I need to compute a sensitivity index, d-prime, defined in Signal Detection Theory as
d' = z(HR) - z(FAR)
where HR and FAR are the hit- and false-alarm-rates respectively. I am confused about why and how each of these rates need to be standardised.
These rates are each expressed as a percentage, i.e. they are scalars, whereas z-scores are, as I understand, computed for a vector of scalars (a distribution).
It is also unclear to me whether the standardisation is to be made with respect to the other scores in the sample (other subjects), or just with respect to a standard normal distribution, N(0,1).
What is in fact the cost (the mistake) in defining a measure of performance just as the HR-FAR difference, i.e. with non-standardised rates? For example, a study by Patel et al. 2008 define in this manner performance in a music task where normal and anomalous chord sequences have to be discriminated:
Patel et al. seem to use this measure as a sensitivity index (d'), although they do not actually call it as such.
It seems like in both cases (scores standardised or not), chance-level responding still leads to a score of 0 - is this is not a sanity check that both measures are equally valid?
Patel, A. D., Iversen, J. R., Wassenaar, M., & Hagoort, P. (2008). Musical syntactic processing in agrammatic Broca’s aphasia. Aphasiology, 22(7–8), 776–789. https://doi.org/10.1080/02687030701803804