Modeling across-trial variability in the Wald drift rate parameter
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Modeling across-trial variability in the Wald drift rate parameter Helen Steingroever1 · Dominik Wabersich2 · Eric-Jan Wagenmakers1
© The Author(s) 2020
Abstract The shifted-Wald model is a popular analysis tool for one-choice reaction-time tasks. In its simplest version, the shifted-Wald model assumes a constant trial-independent drift rate parameter. However, the presence of endogenous processes— fluctuation in attention and motivation, fatigue and boredom—suggest that drift rate might vary across experimental trials. Here we show how across-trial variability in drift rate can be accounted for by assuming a trial-specific drift rate parameter that is governed by a positive-valued distribution. We consider two candidate distributions: the truncated normal distribution and the gamma distribution. For the resulting distributions of first-arrival times, we derive analytical and sampling-based solutions, and implement the models in a Bayesian framework. Recovery studies and an application to a data set comprised of 1469 participants suggest that (1) both mixture distributions yield similar results; (2) all model parameters can be recovered accurately except for the drift variance parameter; (3) despite poor recovery, the presence of the drift variance parameter facilitates accurate recovery of the remaining parameters; (4) shift, threshold, and drift mean parameters are correlated. Keywords Cognitive modeling · Evidence accumulation · One-choice decision tasks · Reaction time modeling · Decision-making · Inverse Gaussian distribution Human decision-making has been studied using a large variety of experimental paradigms. One of the most elementary tasks requires that participants respond immediately after detecting the onset of a stimulus. The key dependent variable in these tasks is reaction time (RT), the time from stimulus onset to participants’ execution of the motor response (usually a key press). Examples of such tasks include simple RT tasks (chapter 2 in Luce, 1986; Smith, 2000), go/no-go tasks (Heathcote, 2004; Schwarz, 2001), temporal-cueing tasks (Jepma et al., 2012), the psychomotor vigilance test (Ratcliff & Van Dongen, 2011), the brightness detection task (Ratcliff & Van Dongen, 2011), the braking task (Ratcliff & Strayer, 2014), and the driving-around task (Ratcliff & Strayer, 2014). Data from these RT tasks can be analyzed with the shifted-Wald model (SW; Fig. 1). The SW model is based on the Wald distribution (Wald, 1947; also known as the inverse
Helen Steingroever
[email protected] 1
Department of Psychology, University of Amsterdam, PO Box 15906, Amsterdam, 1001 NK, The Netherlands
2
Department of Psychology, University of T¨ubingen, T¨ubingen, Germany
Gaussian distribution) which represents the density of the first-arrival times of a Wiener diffusion process toward a single absorbing boundary. The basic model has three parameters that correspond closely to the three parameters of the Ratcliff diffusion model (Ratcliff, 1978; Forstmann et al., 2016; Ratcliff et al., 2016): (1) t
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