Analysis of a two-alternative force-choice signal detection theory model

A basic problem in psychophysics is recovering the mean internal response and noise amplitude from sensory discrimination data. Since these components cannot be estimated independently, several indirect methods were suggested to resolve this issue. Here we analyze the two-alternative force-choice method (2AFC), using a signal detection theory approach, and show analytically that the 2AFC data are not always suitable for a reliable estimation of the mean internal responses and noise amplitudes. Specifically, we show that there is a subspace of internal parameters that are highly sensitive to sampling errors (singularities), which results in a large range of estimated parameters with a finite number of experimental trials. Four types of singular models were identified, including the models where the noise amplitude is independent of the stimulus intensity, a situation often encountered in visual contrast discrimination. Finally, we consider two ways to avoid singularities: (1) inserting external noise to the stimuli, and (2) using one-interval forced-choice scaling methods (such as the Thurstonian scaling method for successive intervals).