technical note: d-prime proportion correct in choice experiments (signal detection theory)

If you don’t understand the title of this post, you almost certainly will regret reading further.

We’re doing an experiment in which one target is presented along with m distracters. The participant tries to determine which is the target, and must respond with their best guess regarding which is it. Together the m distracters + 1 target = “number of alternatives”.

In the plots shown are the predictions from vanilla signal detection theory for the relationship between probability correct, d-prime, and number of alternatives. Each distracter is assumed to have a discriminability of d-prime from the target.

The two plots are essentially the inverse of each other.

Note that many studies use two-interval forced choice wherein the basic stimulus containing distracters are presented twice, one with the signal and the participants has to choose which contained the signal. In contrast, here I’m showing predictions for an experiment wherein the target with all its distracters is only presented once, and the participant reports which location contained the target.

I should probably add a lapse rate to these models, and generate curves using a reasonable lapse rate like .01.

I’ll post the R code using ggplot that I made to generate these later; email me if I don’t or you want it now. UPDATE: the code, including a parameter for lapse rate.

reference: Hacker, M. J., & Ratcliff, R. (1979). A revised table for d’ for M-alternative forced choice, 26(2), 168-170.
#To determine the probability of target winning, A, use the law of total probability:
# p(A) = Sum (p(A|B)p(B)) over all B
# Here, B will be all possible target TTC estimates and p(A|B) will be probability distracters
# are all lower than that target TTC estimate, B

# x is TTC estimate of distracter
# Probability that distracter TTC estimate less than target is pnorm(x): area under curve
# less than x.
# m: number of objects, m-1 of which are distracters
# p(A|B)*p(B) = pnorm(x)^(m-1) * dnorm(x-dprime)
# Hacker & Ratcliff, 1979 and Eliot 1964 derive this as did I
# Jakel & Wichmann say that "numerous assumptions necessary for mAFC" where m>2 but not clear
# whether talking about bias only or also about d'