Reconciling similarity across models of continuous selections

Abstract

Recently-developed models of decision behavior have provided richer accounts of the cognitive processes underlying choice by examining performance on tasks where responses can fall along a continuum, such as identifying the color or orientation of a stimulus. While these have laudably expanded models beyond simple binary and multi-alternative choice paradigms, critical concerns remain about the models' lack of completeness, poor computational tractability, or inability to capture important characteristics of choice responses. Specifically, no model is yet able to predict a truly continuous distribution of responses that exhibit multimodality, where responses have more than one central tendency. This property appears frequently in real data where there is conflict between decision information favoring responses at different locations, such as a predecision cue that indicates a different response than the stimulus. These patterns of data are informative to psychological theory, making it critical to develop a better account of continuous response tasks. In this theoretical note, we use a geometric approach developed by Kvam (2019) to provide a solution to this issue and to reconcile the different existing models under a common framework. The resulting geometric diffusion model (GDM) yields a truly continuous joint distribution of responses and response times, produces multimodal distributions of responses, and offers much greater computational efficiency than models producing discrete approximations of continuous responses. We demonstrate that this model can be extended to account for behavior when alternatives are not arranged in a circle, including a case where responses can fall along a 2-dimensional plane such as identifying the location of an item on a computer screen.

A unified theory of discrete and continuous responding

“…MAAT makes several predictions that are quite different to those of other approaches to modeling continuous and discrete responses (Ratcliff, 2018;Smith, 2016). The first is that the distribution of continuous responses should become more skewed as the target stimulus gets closer to upper or lower anchors (see also Kvam & Turner, 2021, for a discus-sion of how this can be described in terms of representational similarity). As a result, responses near the ends of a scale are faster and more accurate.…”

“…This result violates a core assumption of several continuous models; namely, that stopping rules (i.e., the threshold amount of evidence required to trigger a response) should be consistent across options. Instead, we suggest that the similarity between response options should be a fundamental consideration in selecting a stopping rule for each option in a discrete (or continuous; see Kvam & Turner, 2021) set of alternatives.…”

“…We assumed the position on the arc that a stimulus maps onto is linearly related to its length, although in general this mapping can be nonlinear (as in utility representations or nonlinear similarity between stimuli Kvam & Busemeyer, 2020;Kvam & Turner, 2021). A participant is not being asked to map each stimulus length onto an equivalent arc length (equivalently, angle).…”

“…However, this becomes burdensome for the continuous case. A simpler way to represent the accumulation process is to reduce the large number of accumulators to a smaller number of dimensions and specify a drift rate for each dimension (Kvam & Turner, 2021). Each dimension can then be thought of as being represented by an accumulator.…”

“…There have been a number of proposals for how response boundaries can be used to map a small number of accumulators onto a greater number of responses, which we examine in the supplementary materials. We use the circular boundary model shown in Figure 2, which corresponds to the asymptotic limit of both the multiple threshold race Reynolds, Garton, et al, 2020) and the geometric framework for modeling decision making (Kvam, 2019a;Kvam & Turner, 2021). A single boundary allows discrete and continuous cases to use a commensurate stopping rule, as with the circular diffusion model (Smith & Corbett, 2019;Smith, 2019).…”