Four Theorems on the Psychometric Function

In a 2-alternative forced-choice (2AFC) discrimination task, observers choose which of two stimuli has the higher value. The psychometric function for this task gives the probability of a correct response for a given stimulus difference, [Image: see text]. This paper proves four theorems about the psychometric function. Assuming the observer applies a transducer and adds noise, Theorem 1 derives a convenient general expression for the psychometric function. Discrimination data are often fitted with a Weibull function. Theorem 2 proves that the Weibull “slope” parameter, [Image: see text], can be approximated by [Image: see text], where [Image: see text] is the [Image: see text] of the Weibull function that fits best to the cumulative noise distribution, and [Image: see text] depends on the transducer. We derive general expressions for [Image: see text] and [Image: see text], from which we derive expressions for specific cases. One case that follows naturally from our general analysis is Pelli's finding that, when [Image: see text], [Image: see text]. 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 transducer gradient is zero, which explains why it does not apply to contrast detection. Theorem 4 proves that, when the exponent of a power-function transducer approaches zero, 2AFC performance approaches that of a logarithmic transducer. We show that the power-function exponents of 0.4–0.5 fitted to suprathreshold contrast discrimination data are close enough to zero for the fitted psychometric function to be practically indistinguishable from that of a log transducer. Finally, Weibull [Image: see text] reflects the shape of the noise distribution, and we used our results to assess the recent claim that internal noise has higher kurtosis than a Gaussian. Our analysis of [Image: see text] for contrast discrimination suggests that, if internal noise is stimulus-independent, it has lower kurtosis than a Gaussian.