Click here to download Manuscript: Weibull function PLoS ONE R2 FINAL 003.doc

In a 2-alternative forced-choice (2AFC) discrimination task, observers choose which of two stimuli has the higher value. The psychometric function (PF) for this task gives the probability of a correct response for a given stimulus difference, x. This paper proves four theorems about the PF. Assuming the observer applies a transducer and adds noise, Theorem 1 derives a convenient general expression for the PF. Discrimination data are often fitted with a Weibull function. Theorem 2 proves that the Weibull “slope” parameter, , can be approximated by Noise Transducer , where Noise is the of the Weibull function that fits best to the cumulative noise distribution, and Transducer depends on the transducer. We derive general expressions for Noise and Transducer , from which we derive expressions for specific cases. One case that follows naturally from our general analysis is Pelli’s finding that, when d (x)b, Noise b. We also consider two limiting cases. Theorem 3 proves that, as sensitivity improves, 2AFC performance will usually approach that for a linear transducer, whatever the actual transducer; we show that this does not apply at signal levels where the transduce