A threshold theory account of psychometric functions with response confidence under the balance condition

The study of thresholds for discriminability has been of long-standing interest in psychophysics. While threshold theories embrace the concept of discrete-state thresholds, signal detection theory discounts such a concept. In this paper we concern ourselves with the concept of thresholds from the discrete-state modelling viewpoint. In doing so, we find it necessary to clarify some fundamental issues germane to the psychometric function (PF), which is customarily constructed using psychophysical methods with a binary-response format. We challenge this response format and argue that response confidence also plays an important role in the construction of PFs, and thus should have some impact on threshold estimation. We motivate the discussion by adopting a three-state threshold theory for response confidence proposed by Krantz (1969, Psychol. Rev., 76, 308-324), which is a modification of Luce's (1963, Psychol. Rev., 70, 61-79) low-threshold theory. In particular, we discuss the case in which the practice of averaging over order (or position) is enforced in data collection. Finally, we illustrate the fit of the Luce-Krantz model to data from a line-discrimination task with response confidence.