Eye Gaze

The DDM models two-choice decision making as a noisy process accumulating evidence over time (Fig 2). This process approaches one of two boundaries with a certain speed (drift-rate, influenced by the amount of evidence conveyed by the stimuli). When one of the two boundaries is crossed, the associated response is executed. The distance between the two boundaries is called the decision threshold; larger thresholds lead to slower, but more accurate responding. Estimation of these underlying decision processes was accomplished using DDM analysis of test phase choices. We fit each participant's choices and RT distributions with a DDM model assuming that the proportional difference in reward values for the two options sets the drift rate (Cavanagh, Wiecki, et al., 2011 (link); Ratcliff & Frank, 2012 (link)), i.e. the rate of evidence accumulation for one option over the other. We used hierarchical Bayesian estimation of DDM parameters, which optimizes the tradeoff between random and fixed effect models of individual differences, such that fits to individual subjects are constrained by the group distribution, but can vary from this distribution to the extent that their data are sufficiently diagnostic (Wiecki, Sofer, & Frank, 2013 (link)). This procedure produces more accurate DDM parameter estimates for individual and groups, particularly given low trial numbers or when assessing coefficients between psychophysiological measures and behavior.
Estimation of the Hierarchical DDM (HDDM) was performed using recently developed software (http://ski.clps.brown.edu/hddm_docs) (Wiecki et al., 2013 (link)). Bayesian estimation allowed quantification of parameter estimates and uncertainty in the form of the posterior distribution. Markov chain Monte-Carlo (MCMC) sampling methods were used to accurately approximate the posterior distributions of the estimated parameters. Each DDM parameter for each subject and condition was modeled to be distributed according to a normal (or truncated normal, depending on the bounds of parameter intervals) distribution centered around the group mean with group variance. Prior distributions for each parameter were informed by a collection of 23 studies reporting best-fitting DDM parameters recovered on a range of decision making tasks (Matzke & Wagenmakers, 2009 (link)), see the supplement of (Wiecki, Sofer, & Frank, 2013 (link)) for visual depictions of these priors. A model using non-informative priors (e.g. uniform distributions that assign equal probability to all parameter values over a large interval) resulted in highly similar results. There were 5000 samples drawn from the posterior; the first 200 were discarded as burn-in following the conventional approach to MCMC sampling whereby initial samples are likely to be unreliable due to the selection of a random starting point.
We estimated regression coefficients in separate HDDM models to determine the relationship between single trial variations in psychophysiological measures (eye gaze, pupil dilation) and model parameters (drift rate, decision threshold):
$$\mathit{parameter}=\beta 0+\beta 1\left(\mathit{psychophysiology}\right)$$
In these regressions the coefficient β1 weights the slope of parameter (drift rate, threshold) by the value of the psychophysiological measure (proportional gaze dwell time, pupil change from baseline) on that specific trial, with an intercept β0. We extended this regression approach to formally compare three competing models of the influence of value and gaze dwell time on drift rate. Each of these models contained multiple regression coefficients in order to test independent and interactive influences of value and gaze dwell time on drift rate. In each of these models, the continuous influence of value (10%, 20%, 40%, 50%, 60%) was used instead of the condition-specific differences (win-win, lose-lose, and win-lose), although we do plot the condition-specific effect of gaze in one instance for descriptive purposes. For descriptive clarity, these models are formally explained in the Results section. This regression approach was also used to model the influence of pupil dilation on decision threshold in the corrected win-win and lose-lose conditions.
Bayesian hypothesis testing was performed by analyzing the probability mass of the parameter region in question (estimated by the number of samples drawn from the posterior that fall in this region; for example, percentage of posterior samples greater than zero). Statistical analysis was performed on the group mean posteriors. The Deviance Information Criterion (DIC) was used for model comparison, where lower DIC values favor models with the highest likelihood and least number of parameters (Gelman, 2004 ). While alternative methods exist for assessing model fit, DIC is widely used for model comparison of hierarchical models (Spiegelhalter, Best, Carlin, & van der Linde, 2002 ), a setting in which Bayes factors are not easily estimated (Wagenmakers, Lodewyckx, Kuriyal, & Grasman, 2010 (link)).

An Eyelink 1000 eye-tracker (SR Research) monitored the gaze location of participants' right eyes during reading. The eye tracker has a spatial resolution better than 30! arc and a 1000 Hz sample rate. Participants viewed the stimuli binocularly on a monitor 63 cm from their eyes; approximately 3 characters equaled 1° of visual angle.