Psychometric functions from multiple responses : Dedicated to the memory of Colin L. Mallows

By using three or more ordered response categories and varying the stimulus feature being judged over a large range, it is possible to generate a family of psychometric functions (PMFs), each based on a different partition of the responses. An earlier paper showed how, when it is treated as a probability distribution, the traditional single PMF based on binary-choice data can be decomposed into sensory and decision components, expressed as two independent random variables that are summed to create the PMF. Here we extend this development to the multiple-response procedure, and use it to elucidate the relations among the spreads and shapes of the resulting family of PMFs, which can be described by their first four cumulants. For example, we determine conditions under which the PMFs can have the same spread and shape, differing only by translation on the stimulus axis. Whereas PMFs depend on both sensory and decision processes, differences among the PMFs in a family depend only on the decision processes. Application of this multiple-PMF method to several decision models, whose evaluations depend on the PMF cumulants, shows it to have greater power than the single-PMF method for understanding the perceptual process. Although this work was inspired by experiments on the perception of temporal order, it can be applied to experiments where features of stimuli other than their occurrence times are being compared, such as the pitch of tones or the brightness of lights.