Figure 1.
Patent Race game. A) After a fixation screen, subjects were presented with the Patent Race game, with information regarding their endowment, the endowment of the opponent, and the potential prize. B) Subjects input the decision (self-paced) by pressing a button mapped to the desired investment amount from the initial endowment. C) After 2–6 s, the opponent’s choice was revealed. If the subject’s investment was strictly more than those of the opponent, the subject won the prize; otherwise, the subject lost the prize. In either case, the subject kept the portion of the endowment not invested.
Figure 2.
Histogram of posterior means of the participant- and session-level parameters (δ ij , λ ij , ϕ ij , ρ ij for all i and j) from the full model with posterior means of the parameters from the shared parameter model (δ, λ, ϕ, ρ) indicated with the red dashed line. The average posterior standard deviations (averaged across all participants and sessions) corresponding to each grouping of parameters is δ: 0.12, λ: 0.17, ϕ: 0.05, ρ: 0.08.
Table 1.
Posterior inference for the data application presented on the odds ratio (δ, ϕ, ρ) and relative risk (λ) scales
| Effect | Parameter | Mean (SD) | Posterior quantile | ||
|---|---|---|---|---|---|
| 0.025 | 0.500 | 0.975 | |||
| Age (years) | δ | 0.87 (0.23) | 0.47 | 0.85 | 1.38 |
| λ | 1.06 (0.07) | 0.93 | 1.06 | 1.21 | |
| ϕ | 2.38 (1.93) | 0.64 | 1.81 | 7.77 | |
| ρ | 2.25 (1.45) | 0.81 | 1.86 | 6.44 | |
| Male vs. female | δ | 1.81 (1.07) | 0.57 | 1.58 | 4.41 |
| λ | 0.90 (0.13) | 0.66 | 0.89 | 1.19 | |
| ϕ | 8.15 (11.62) | 0.61 | 4.60 | 39.24 | |
| ρ | 3.68 (3.64) | 0.42 | 2.47 | 13.61 | |
| Session: 1 vs. 2 | δ | 1.15 (0.43) | 0.52 | 1.08 | 2.18 |
| λ | 1.16 (0.10) | 0.96 | 1.16 | 1.37 | |
| ϕ | 0.65 (0.19) | 0.36 | 0.63 | 1.08 | |
| ρ | 0.76 (0.22) | 0.39 | 0.75 | 1.25 | |
| Strong vs. weak role | δ a | 4.67 (2.18) | 2.10 | 4.18 | 10.23 |
| λ a | 0.50 (0.05) | 0.42 | 0.50 | 0.60 | |
| ϕ a | 0.26 (0.08) | 0.14 | 0.25 | 0.44 | |
| ρ a | 0.30 (0.09) | 0.15 | 0.29 | 0.52 | |
Figure 3.
Scatterplot of the posterior mean estimates for the participant- and session-level parameters from the full model and the unique parameter model. The asterisk indicates the corresponding posterior mean estimated from the shared parameter model.
Table 3.
Simulation study results
| Variability | Model | δ | λ | ϕ | ρ | WAIC (EP) |
|---|---|---|---|---|---|---|
| No | Shared | 0.00 (1.00) | 0.90 (1.00) | 0.75 (1.00) | 0.60 (1.00) | 12,471 (4) |
| Unique | 13.27 (1.00) | 53.84 (0.70) | 17.14 (0.94) | 34.02 (0.88) | 13,012 (209) | |
| Full | 5.23 (1.00) | 9.61 (1.00) | 1.51 (1.00) | 3.01 (1.00) | 12,814 (105) | |
| Yes | Shared | 19.05 (0.14) | 27.74 (0.33) | 14.73 (0.04) | 18.16 (0.05) | 12,839 (5) |
| Unique | 10.35 (1.00) | 41.16 (0.83) | 10.59 (0.83) | 31.82 (0.68) | 11,537 (234) | |
| Full | 7.31 (1.00) | 10.81 (0.98) | 3.12 (0.95) | 5.87 (0.94) | 11,350 (141) |
Figure 4.
Histograms of posterior means of the participant- and session-level parameters (δ ij , λ ij , ϕ ij , ρ ij for all i and j) obtained from fitting the different models on the simulated dataset with no variability. True values of the parameters for generating this simulated dataset (δ, λ, ϕ, ρ) are indicated by the black dashed lines.
Figure 5.
True versus estimated (posterior means) participant- and session-level parameters (δ ij , λ ij , ϕ ij , ρ ij for all i and j) obtained from fitting the different models on the simulated dataset with variability. The identity line (y = x), which indicates the ideal situation where estimated parameters are exactly the same as true parameters, is shown in blue. R 2 = coefficient of determination.
Figure 6.
True versus estimated session effects (i.e., differences of posterior means) on parameters (δ ij , λ ij , ϕ ij , ρ ij for all i) between the two sessions obtained from fitting either the unique parameter model or the full model on the simulated dataset with variability. The identity line (y = x), which indicates the ideal situation where estimated session effects are exactly the same as true session effects, is shown in blue. R 2 = coefficient of determination.