On the consistency and relative efficiency of a generalized Robbins-Monro process 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 Yes-No response format is adopted. A well-known earlier work of the variable step-size adaptive methods is the Robbins-Monro process. However, previous studies have paid little attention to other facets of response variables (in addition to the Yes-No response variable) that could be embedded in the Robbins-Monro process. This study concerns a generalization of the Robbins-Monro process by incorporating other response variables, such as response confidence, into the process. We first prove the consistency of the generalized method and explore possible requirements, under which the proposed method achieves (at least) the same efficiency as the original method does. We then conduct a Monte Carlo simulation study to explore some finite-sample properties of the estimator obtained from the generalized method, and compare its performance with the original method.