Fig 1.
Gaze-weighted linear accumulator model.
In the GLAM, preference formation during the decision process is dependent on the allocation of visual gaze (A). For each item in the choice set, an average absolute decision signal is computed (dashed lines in A). The magnitude of this signal is determined by the momentary allocation of visual gaze: While an item is currently not looked at, its signal is discounted by parameter γ (γ ≤ 1; discounting is illustrated by gray arrows) (A). To determine a relative decision signal Ri for each item in the choice set, absolute evidence signals are transformed in two steps (B): First, the difference between each average absolute decision signal
and the maximum of all others is determined. Second, the resulting differences are scaled through a logistic transform, as the GLAM assumes an adaptive representation of the relative decision signals that is especially sensitive to differences close to 0 (where the absolute signal for an item is very close to the maximum of all others). The resulting relative decision signals Ri can be used to predict choice and RT, by determining the speed of the accumulation process in a linear stochastic race (C). The stochastic race then provides first-passage time distributions pi, describing the likelihood of each item being chosen at each time point.
Fig 2.
In the hierarchical model, individual subject parameters γi, vi. σi, and τi (subject plate) are drawn from Truncated Normal group level distributions with means μ and standard deviations σ (outside of the subject plate). Weakly informative Truncated Normal priors are placed on the group level parameters. RT and choice data xi,t for each trial t is distributed according to the subject parameters and the GLAM likelihood (Eq (8); inner trial plate).
Table 1.
The first two rows of a pandas DataFrame ready to be used with GLAM.
Fig 3.
Individual differences in the data.
A-C: distributions of individuals’ mean RT (A), probability of choosing the highest-valued item in a trial (B), and behavioural influence of gaze allocation on choice behaviour (C). D-F: associations between individuals’ probability of choosing the highest-valued item and mean RT (D), individuals’ behavioural influence of gaze allocation on choice behaviour and their mean RT (E), individuals’ behavioural influence of gaze allocation on choice behaviour and their probability of choosing the highest-valued item (F). Red lines indicate linear regression fits with confidence bands surrounding them. Pearson’s r coefficients with corresponding P-values are reported for each association in D-F.
Table 2.
Output from compare_models function for the first two subjects.
Fig 4.
Individual differences in the strength of the association of gaze allocation and choice behaviour.
A: Distribution of γ estimates resulting from the in-sample individual model fits. B: Association of γ estimates and individuals’ values on the behavioural gaze bias measure. The red line indicates a linear regression fit, with surrounding 95% confidence bands. Pearson’s r correlation with P-value is given.
Fig 5.
Comparison of individuals’ simulated observed response behaviour with the out-of-sample predictions of a GLAM variant with (A-C) and without gaze bias (D-F): Individuals’ mean RT (A, D), probability of choosing the best item (B, E), and influence of gaze allocation on choice probability (C, F). Points indicate individual participant means.
Fig 6.
Aggregate view of the simulated data for Example 2.
(A) Mean RT binned by trial difficulty (the difference between the highest item value in a choice set and the maximum value of all other items). (B) The probability that an item is chosen based on its relative value (the difference of the item’s value and the maximum value of all other items in the choice set). (C) The probability of choosing an item based on its relative gaze (the difference between the gaze towards this item and the maximum gaze towards a different item). (D) The probability of choosing an item based on its relative gaze, when correcting for the influence of its value. Bars correspond to the pooled data, while coloured lines indicate individual groups.
Fig 7.
Pairwise comparison of posterior group-level parameter estimates between groups.
Each row corresponds to one model parameter. The leftmost column shows the estimated posterior distributions for each parameter and group. Pairwise differences between the group posterior distributions are shown in all other columns. For each posterior distribution of the difference, the mean and 95% HPD are indicated, as well as the proportion of samples below and above zero (in red). All three groups differ on the γ parameter (row B). No evidence for differences on any of the other model parameters is found (the 95% HPD of the pairwise differences between groups all include zero).
Fig 8.
Results from a basic parameter recovery.
The lower row (E-H) shows deviations between known generating parameter values and recovered MAP estimates (circles) and their 95% HPDs (horizontal error bars) for each participant. Green (red) colour indicates that the true value is within (outside) the 95% HPD. Most parameters were recovered with small deviations. Panels A-D show distributions of deviations across individuals. Distributions are mostly centered around zero, indicating no systematic under- or overestimation (bias) across individuals.