Extending signal detection theory to the analysis of generalization and other tasks using continuous responses to multiple items

Generalization studies typically use a design in which multiple stimuli vary along a single stimulus dimension and a given outcome or response is associated with a single value in the dimension. This is similar to the method of constant stimuli used to characterize psychometric curves in psychophysics, although in many cases measuring continuous rather than discrete responses to each stimulus. Here, we propose a generalization of the signal detection model for the psychometric curve, that deals with continuous responses. As in the traditional model, we assume normally-distributed decision variables with means and variances that change depending on the presented stimulus. We also assume that a monotonic link function transforms such variables into the measured responses, which are perturbed by random normal noise. The model is a generalization of traditional signal detection models, which are obtained by assuming a staircase link function. We propose an algorithm that uses a combination of quantile functions and monotone spline regression to estimate the parameters of this model from data, and show that the inclusion of a flexible link function allows the model to fit continuous data better than ROC analyses previously proposed for continuous data. Potential applications include the adaptive estimation of generalization curves and application to continuous neural data such as fMRI activity estimates.