One main research topic in classical psychophysics is to explore the relation between physical stimuli and psychological sensation, and the relevant study of threshold and underlying representations is of theoretical interest. A prominent example is the work of models of detection and discrimination under the Fechnerian framework (Falmagne, 1985). A closely related empirical issue within the Fechnerian framework is to develop a method (or experimental procedure) using a binary response format to find the stimulus intensity corresponding to a certain threshold level in the psychometric function, which typically displays a sigmoidal curve ranging in the open interval ]0,1[ for the proportion of “yes” responses (in a detection task) or “louder”, “longer”, etc. responses (in a discrimination task).1

In the past, some commonly used procedures for threshold estimation include the method of constant stimuli, the method of limits, and the method of adjustment (for a review, see Falmagne, 1985, Treutwein, 1995). These procedures have some limitations, however. Specifically, for the method of constant stimuli, the set of stimuli has to be determined first and appropriately assigned to the trials, and it usually requires many trials. For the method of limits and the method of adjustment, while the stimuli are not pre-assigned and in some cases can be adjusted continuously, there is a lack of theoretical justification for their statistical properties.

Alternatively, one can estimate a threshold directly using adaptive (also known as ‘up-down’) methods, in which the stimulus assigned to the current trial depends on a participant’s response in the previous trial(s). Treutwein (1995) categorized adaptive methods into parametric and non-parametric adaptive methods, with the main difference being that the latter do not require strict model assumptions about the form of the psychometric function. Hence, it is more feasible to use the non-parametric adaptive methods than to use the parametric ones when the form of the psychometric function is unknown.

The non-parametric adaptive methods can be further classified into fixed step-size and variable step-size adaptive methods. An early example of fixed step-size adaptive methods was developed by Dixon and Mood (1948) to analyze dosage mortality data. This method was applied to psychophysical experiments to estimate the 50% threshold (point) of the psychometric function. Later work that follows mostly aimed to generalize Dixon and Mood’s method to the estimation of other percentiles of threshold in a psychometric function. For instance, Derman (1957) proposed an adaptive method to estimate any percentiles of threshold. Levitt (1971) proposed a transformed up-down method for certain percentiles of threshold (e.g., 70.7%, 79.4%). Kaernbach (1991) proposed an alternative to the transformed up-down method in which the step sizes for the ‘up’ and ‘down’ directions are different, contingent on the specific percentile of threshold to be estimated, and thus is applicable to estimating any percentiles of threshold. Kaernbach (2001) later proposed a modification of that method to deal with three response categories. Recently, Cheng et al. (2022) expanded the work of fixed step-size adaptive methods by incorporating response confidence into existing algorithms (see also Hsu & Chin, 2014).

A well-known earlier work of the variable step-size adaptive methods is the Robbins–Monro process (or algorithm, method), named after the seminal work of Robbins and Monro (1951). This process was originally developed to find the root of a monotone function. Robbins and Monro (1951) proved that the proposed process converges in probability to the required root under certain conditions. Kesten (1958) proposed a modification to facilitate the Robbins–Monro process, known as the ‘accelerated stochastic approximation’ (procedure); it takes the number of sign changes during the process into account. Kesten (1958) proved that the modification also converges in probability to the required root.

In practice, estimates obtained from the fixed step-size adaptive methods are affected by the relative step sizes which are usually unknown to experimenters (García-Pérez, 1998). Compared to the fixed step-size adaptive methods, the variable step-size adaptive methods are more reliable, and their convergences are less affected by the initial step size (Faes et al., 2007). Treutwein (1995) pointed out that, compared with other parametric methods, the accelerated stochastic approximation has a nearly optimal performance, albeit its fewer assumptions about the psychometric function. For this reason, Treutwein (1995) advocated the use of the Robbins–Monro process and its accelerated version.

However, previous studies have paid little attention to other facets of response variables 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. It is organized as follows. Section 2 gives a brief review of the Robbins–Monro theorem, which is followed by the introduction of the generalized method. The property of consistency of the generalized method will be proved. Section 3 presents the setup of a simulation design aiming to explore the finite-sample properties of the estimator obtained from the generalized method with either the response time or the response confidence as the variable of interest. Section 4 describes the simulation results, focusing on the unbiasedness and efficiency (in terms of the mean squared error (MSE)). We compare the performance of the generalized method with the performance of the original method. Section 5 discusses the issue of relevant efficiency. The final section contains some concluding remarks.