A unified theory of discrete and continuous responding

Abstract

Understanding the cognitive processes underlying choice requires theories that can disentangle the representation of stimuli from the processes that map these representations onto observed responses. We develop a dynamic theory of how stimuli are mapped onto discrete (choice) and continuous response scales. It proposes that the mapping from stimuli to the input to an evidence accumulation process is accomplished using multiple reference points or "anchors". Evidence is accumulated until a threshold amount for a particular response is obtained, with the relative balance of support for each anchor at that time determining the response. We tested this Multiple Anchored Accumulation Theory (MAAT) using the results of two experiments requiring discrete or continuous responses to line length and color stimuli. We manipulated the number of options for discrete responses, the number of different stimuli, and the similarity amongst them, and compared the outcomes to continuous response conditions. We show that MAAT accounts for several key phenomena: more accurate choices and more skewed response distributions near the ends of a response scale; a decrease in accuracy and response speed as the number of discrete choice options increases; and longer response times and lower accuracy when discrete responses were more similar to one another. Our empirical and modeling results suggest that discrete and continuous response tasks can share a common evidence representation, and that the decision process is sensitive to the perceived similarity among the response options.

A unified theory of discrete and continuous responding.

“…This study was not preregistered. Study materials including data, analysis code, and additional figures can be found on the Open Science Framework at https://osf.io/6d29q (Kvam & Heathcote, 2022).…”

Winner Takes All! What Are Race Models, and Why and How Should Psychologists Use Them?

“…Building on Smith’s (2016) seminal circular-diffusion model, Kvam showed that with enough thresholds, or a nonlinear joint threshold, race models can be extended to accommodate continuous responding. Kvam et al (2022) used this approach to provide a unified account of discrete and continuous responding with respect to line-length and color-matching judgments.…”

“…The diagrams in (b), (c), and (d) represent threshold-counting models, alternatives to the MTR models. Threshold-counting models have been used to account for both discrete and continuous responding (Kvam et al, 2022). The diagram in (b) represents a threshold-counting model with three evenly spaced thresholds for each accumulator: A response is triggered when three thresholds have been crossed.…”

Decisions among shifting choice alternatives reveal option-general representations of evidence

“…The evidence state is updated as stimulus information comes in, and a decision is made when the state crosses a spatial threshold as shown in Figure 1 (bottom-right). This vector-based representation of choice options allows it to accommodate similarities among choice options that are critical to accounting for choices and response times in both discrete and continuous tasks (31,35), but the most important property of this model is that the representation of decision evidence is distinct from representations of the choice options. When the options change over time, the vectors representing response options change, and support is re-mapped onto the new options (light blue changes to dark blue choice boundary in Figure 1).…”

“…We opted to summarize the data in terms of accuracy, mean response times, and the 10/30/50/70/90 th response time quantiles for correct and incorrect responses in each of the 11 conditions. Quantile-based methods for model fitting have been used effectively for many years in dynamic models of RTs (35), and provide a short but rich characterization of the pattern of response times within each condition. In addition, they can be computed for every input data set while yielding precise parameter estimates and good model recovery, as we show below.…”