May, K. A. and Solomon, J. A. (2013). Four theorems on the psychometric function. PLoS One, 8(10), doi: 10.1371/journal.pone.0074815
Abstract
In a 2alternative forcedchoice (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, Δx. 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, β, can be approximated by [Formula: see text], where [Formula: see text] is the β of the Weibull function that fits best to the cumulative noise distribution, and [Formula: see text] depends on the transducer. We derive general expressions for [Formula: see text] and [Formula: 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 [Formula: see text], [Formula: 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 powerfunction transducer approaches zero, 2AFC performance approaches that of a logarithmic transducer. We show that the powerfunction exponents of 0.40.5 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 β 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 β for contrast discrimination suggests that, if internal noise is stimulusindependent, it has lower kurtosis than a Gaussian.
Publication Type:  Article 

Subjects:  B Philosophy. Psychology. Religion > BF Psychology 
Departments:  School of Health Sciences > Optometry & Visual Science 
URI:  http://openaccess.city.ac.uk/id/eprint/2861 

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