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We generalize the circular 2D diffusion model of Smith (Psychological Review, 123, 425-451: 2016) to provide a new model of speeded decision-making in multielement visual displays. We model decision-making in tasks with multielement displays as evidence accumulation by a vector-valued diffusion process in a hypersphere, whose radius represents the decision criterion for the task. We show that the methods used to derive response time and accuracy predictions for the 2D model can be applied, with only minor changes, to predict performance in higher-dimensional spaces as well. We apply the model to the double-target deficit paradigm of Duncan (Psychological Review, 87, 272-300: 1980) in which participants judge whether briefly presented four-element displays contain one- or two-digit targets among letter distractors. A 4D version of the hyperspherical diffusion model correctly predicted distributions of response times and response accuracy as a function of task difficulty in single-target and double-target versions of the task. The estimated drift rate parameters from the model imply that the mental representation of the decision alternatives, which we term the "decision template" for the task, encodes configural stimulus properties that reflect the number of targets in the display. Along with its application to multielement decision-making, the model has the potential to characterize the speed and accuracy of multiattribute decisions in studies of cognitive categorization, visual attention, and other areas.

Speeded multielement decision-making as diffusion in a hypersphere: Theory and application to double-target detection