The generalized Robbins–Monro process and its application to psychophysical experiments for threshold estimation

In classical psychophysics, the study of threshold and underlying representations is of theoretical interest, and the relevant issue of finding the stimulus intensity corresponding to a certain threshold level is an important topic. In the literature, researchers have developed various adaptive (also known as ‘up-down’) methods, including the fixed step-size and variable step-size methods, for the estimation of threshold. A common feature of this family of methods is that the stimulus to be assigned to the current trial depends upon the participant’s response in the previous trial(s), and very often a binary response format is adopted. A well-known earlier work of the variable step-size adaptive methods is the Robbins–Monro process (and its accelerated version). However, previous studies have paid little attention to other facets of response variables (in addition to the binary response variable) that could be jointly embedded into the process. This article concerns a generalization of the Robbins–Monro process by incorporating an additional response variable, such as the response time or the response confidence, into the process. We first prove the consistency of the estimator from the generalized method. We then conduct a Monte Carlo simulation study to explore some finite-sample properties of the estimator from the generalized method with either the response time or the response confidence as the variable of interest, and compare its performance with the original method. The results show that the two methods (and their accelerated version) are comparable. The issue of relative efficiency is also discussed.