Drift diffusion modeling: Model comparisons

General approach

We used a Bayesian Hierarchical Modeling approach to estimate individual and group parameters simulta-neously. We used JAGS in R to sample from the posterior distributions. In Bayesian parameter estimation, parameter estimates are represented as prior distributions and then updated into posterior distributions based on the observed data. The advantage of a hierarchical approach is that it can account for individ-ual variation while simultaneously pooling individindivid-ual estimates into group-level distributions. The joint posterior parameter distributions were estimated using Monte Carlo Markov Chain methods implemented in JAGS, called from R. We ran 25 chains, each with 4,000 recorded samples, which were drawn from the posterior distributions after a burn-in period of 500 samples. Model selection was based on deviance information criteria, with lower values indicating better fit; and on posterior predictive checks in which we assessed how well the model aligned to behavioral results (across learning conditions).

The experiment had a risk–reward (between-participants)× timepressure (within-participants) design that we fit simultaneously. Moreover, half of the participants completed the choice task after incidental learning and the other half after explicit learning. We fit the two learning conditions separately.

We inspected the quality of the posterior distributions by visually inspecting the mixing of the chains and autocorrelation, and on the basis of the Gelman-Rubin statistic. We compared model fits using DIC values (lower values indicate better model fit) and creating posterior predictions for choices and response times using the mean parameter values obtained from the “winning” models, on both the condition and the participant–level.

We compared five different models with the simple DDM as the baseline model. The second model only takes into account the value differences between the options, with larger EV differences leading to higher drift rates. The third model predicts drift from gaze differences between the two options alone.

The fourth model tests for an additional main effect of gaze on the drift rate. The fifth model also allows for an interaction between gaze and value (similar to Cavanagh et al., 2014), including additive effects of value and gaze. Model selection was done by fitting the models across learning conditions (to obtain one DIC per model). Parameter estimates were obtained by re-fitting the models separately for each learning condition. We did not include an aDDM–like model in which the drift rate solely depends on the interaction between value and gaze, without additive effects—the reason for this is that we wanted to partial out and compare “EV sensitivity”/“EV usage” for the three risk–reward conditions, which is not possible in an interaction–only model. The best–fitting model was an extended DDM in which the drift rate depended on an additive effect of gaze differences and EV differences (i.e. each of which was quantified by a free parameter, or regression coefficient).

Drift diffusion model specifications

Model 1: Simple DDM We estimated the main diffusion parameters for each condition, α, τ, and δ (β is fixed at .5).

Model 2: Value differences impact drift rate

We estimated α and τ, and fix β at .5. In addition, we estimated the drift rate using a regression approach, such that the drift is described by an intercept δ0 and a regression coefficient, δEV. The intercept is participant–dependent and therefore represents individual differences in the ability to detect the higher EV option.

δ = δ0+δEV× (EVH− EVL) (1)

Model 3: Gaze differences impact drift rate

We estimated α and τ, and fix β at .5. In addition, we estimated the drift rate using a regression approach, such that the drift is described by an interceptδ0 and a regression coefficient,δgaze. The gaze coefficient models to what extent choices depend on pure gaze towards a particular option. Gaze is entered as the proportion of total gaze to the higher vs. lower EV gamble.

δ = δ0+δgaze× (gazeH− gazeL) (2)

Model 4: Value and gaze differences impact drift rate

We estimated α and τ, and fix β at .5. In addition, we estimated the drift rate using a regression approach, such that the drift is described by an interceptδ0and two regression coefficients,δEV andδgaze. The value coefficient models peoples’ sensitivity to differences in expected values. The gaze coefficient models to what extent choices depend on pure gaze towards a particular option. Gaze is entered as the proportion of total gaze to the higher vs. lower EV gamble.

δ = δ0+δEV× (EVH− EVL) +δgaze× (gazeH − gazeL) (3)

Model 5: Value, gaze and their interaction impact drift rate (aDDM)

As model 3, but with an interaction between value and gaze (δinteraction). Cavanagh et al. (2014) modeled the interaction without separate additive effects, but here we aimed to compare “EV use” across risk–reward conditions.

δ = δ0+δEV× (EVH− EVL) +δgaze× (gazeH − gazeL) + δinteraction× (gazeH× EVH − gazeL× EVL)(4)

Results: DICs and posterior predictions

Model DIC

Model 1: Simple DDM 73124.41

Model 2: Drift: Valuediff. 72027.83

Model 3: Drift: Gazediff. 72138.86

Model 4: Drift: Valuediff. + Gazediff. 71274.90 Model 5: Drift: Valuediff. + Gazediff. + Value× Gaze 71385.54

Table C3. Deviance Information Criterion (DIC) for five different formalizations of the Drift Diffusion model.

Drift: Valuediff. + Gazediff; Posterior predictions for Incidental

0.0 0.2 0.4 0.6 0.8 1.0

Negative Positive Uncorrelated

p (choose EV) Best - Data

Best - Model Fast - Data Fast - Model A

0 1 2 3 4

Negative Positive Uncorrelated

Response time (s)

B

NegativePositiveUncorrelated

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 0.0

0.5 1.0

0.0 0.5 1.0

0.0 0.5 1.0

Participant

p (choose EV)

C

NegativePositiveUncorrelated

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 0

5 10

0 5 10

0 5 10

Participant

Response Time (s)

D

Drift: Valuediff. + Gazediff; Posterior predictions for Explicit

0.0 0.2 0.4 0.6 0.8 1.0

Negative Positive Uncorrelated

p (choose EV) Best - Data

Best - Model Fast - Data Fast - Model A

0 1 2 3 4

Negative Positive Uncorrelated

Response time (s)

B

NegativePositiveUncorrelated

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0.0

0.5 1.0

0.0 0.5 1.0

0.0 0.5 1.0

Participant

p (choose EV)

C

NegativePositiveUncorrelated

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0

5 10

0 5 10

0 5 10

Participant

Response Time (s)

D