Modeling continuous outcome color decisions with the circular diffusion model: Metric and categorical properties.

Abstract

The circular diffusion model is extended to provide a theory of the speed and accuracy of continuous outcome color decisions and used to characterize eye-movement decisions about the hues of noisy color patches in an isoluminant, equidiscriminability color space. Heavy-tailed distributions of decision outcomes were found with high levels of chromatic noise, similar to those found in visual working memory studies with high memory loads. Decision times were longer for less accurate decisions, in agreement with the slow error property typically found in difficult 2-choice tasks. Decision times were shorter, and responses were more accurate in parts of the space corresponding to nameable color categories, although the number and locations of the categories varied among participants. We show that these findings can be predicted by a theory of across-trial variability in the quality of the evidence entering the decision process, represented mathematically by the drift rate of the diffusion process. The heavy-tailed distributions of decision outcomes and the slow-error pattern can be predicted by either of 2 models of drift rate. One model is based on encoding failures and the other is based on a nonlinear transformation of the stimulus space. Both models predict highly inaccurate stimulus representations on some trials, leading to heavy-tailed distributions and slow errors. The color-category effects were successfully modeled as stimulus biases in a similarity-choice framework, in which the drift rate is the vector sum of the encoded metric and categorical representations of the stimulus.

Reconciling similarity across models of continuous selections.

“…In that case, the CDM can be simulated and performs about as well as the GSR. Of course, even the simulated CDM cannot han-dle multimodality in response distributions (again, noting that this is for a single drift; a mixture of drifts / certain types of drift variability across trials can create multimodality Smith et al, 2020) and so the GSR appears to be the most efficient model for such cases. Across all of the simulations, spanning both complex and simple implementations of all the models, the GSR is consistently faster to simulate than the SCDM.…”

“…For the two-dimensional case, a characterization of drift rate variability is provided by Smith (2019), who derived the analytic distribution of response times and responses for a model with bivariate normally distributed drift rates. A bivariate normal may not be appropriate for all cases, and instead we may run into cases where the analytic distributions are very difficult or impossible to derive (e.g., the long-tailed distributions in Smith et al, 2020). In any case, drift rate variability results in a joint distribution of responses and response times where responses are fastest near the drift direction and slower as responses fall farther away from it.…”

A unified theory of discrete and continuous responding

“…In this case, participants may set a greater threshold for response options with similar competitors in an attempt to improve accuracy at the cost of longer response times (Usher et al, 2002). Importantly, prominent theories of continuous responding, including the circular diffusion model (Smith, 2016(Smith, , 2019Smith et al, 2020) and spatially continuous diffusion model (Ratcliff, 2018;Ratcliff & McKoon, 2020) have only a single threshold or an invariant function (e.g., in the color experiments of Ratcliff, 2018, there are higher thresholds for non-primary / nonsecondary colors) specifying the thresholds for all response options. Consequently, a set of unequally spaced response options, say A, B, and C, with C being further away (i.e., A-B--C) must have the same thresholds for each option.…”

“…Responses on the hue wheel are a prevalent methodology in visual working memory tasks (Zhang & Luck, 2009, 2011 and have driven the development of continuous-response models (Smith, 2016;Ratcliff, 2018;Kvam, 2019a;Smith et al, 2020). It is particularly notable in these tasks that participants' responses tend group near 'cardinal' hues, red, green, yellow, blue, magenta, and cyan, even when the stimuli are uniformly distributed across the hue wheel.…”

“…In the past, models have mainly focused on discretevalued (choice) responses, but continuous-valued responses can theoretically confer more information about internal representations. Recent developments of models for tasks involving continuous responses (Kvam, 2019a;Ratcliff, 2018;Smith, 2016) have enhanced our understanding of the cognitive processes underlying these tasks, and have begun to be applied successfully in domains such as orientation estimation (Kvam, 2019b;Ratcliff, 2018), color identification (Ratcliff, 2018;Smith et al, 2020), numeracy (Ratcliff & McKoon, 2020), and pricing (Kvam & Busemeyer, 2020). Critically,relations between continuous and discrete response measures have not been thoroughly explored.…”