Condition Level Model

Data

Condition Level Data at Early Phase
Subj ROI beta cond TRAIT STATE
MAX101 L ACC 0.845514 -0.5 -1.164152 -1.011495
MAX101 L BLBM Amygdala 2.732654 -0.5 -1.164152 -1.011495
MAX101 L BST 3.940520 -0.5 -1.164152 -1.011495
MAX101 L CeMe Amygdala 2.331533 -0.5 -1.164152 -1.011495
MAX101 L Crus II 0.611231 -0.5 -1.164152 -1.011495
MAX101 L Hippocampus body 1.607156 -0.5 -1.164152 -1.011495
MAX101 L Hippocampus tail 1.549801 -0.5 -1.164152 -1.011495
MAX101 L Hypothalamus 2.862488 -0.5 -1.164152 -1.011495
MAX101 L IFG-1 0.759511 -0.5 -1.164152 -1.011495
MAX101 L IFG-2 0.893275 -0.5 -1.164152 -1.011495
MAX101 L IFG-3 0.904597 -0.5 -1.164152 -1.011495
MAX101 L IFG-4 3.831213 -0.5 -1.164152 -1.011495
MAX101 L IFG-5 1.538472 -0.5 -1.164152 -1.011495
MAX101 L IFG-6 0.857912 -0.5 -1.164152 -1.011495
MAX101 L Lobule IX 0.819383 -0.5 -1.164152 -1.011495
MAX101 L PAG 1.970929 -0.5 -1.164152 -1.011495
MAX101 L PCC 4.679768 -0.5 -1.164152 -1.011495
MAX101 L PCC/precuneus 4.341770 -0.5 -1.164152 -1.011495
MAX101 L Ventral striatum 1.019936 -0.5 -1.164152 -1.011495
MAX101 L ant. Caudate 0.464388 -0.5 -1.164152 -1.011495
MAX101 L ant. Hippocampus 1.426054 -0.5 -1.164152 -1.011495
MAX101 L ant. MCC 0.838646 -0.5 -1.164152 -1.011495
MAX101 L ant. Putamen 0.889415 -0.5 -1.164152 -1.011495
MAX101 L ant. Thalamus 0.582706 -0.5 -1.164152 -1.011495
MAX101 L ant. dorsal Insula 0.657682 -0.5 -1.164152 -1.011495
MAX101 L ant. ventral Insula 0.666038 -0.5 -1.164152 -1.011495
MAX101 L dlPFC 0.771227 -0.5 -1.164152 -1.011495
MAX101 L lat. OFC 1.621197 -0.5 -1.164152 -1.011495
MAX101 L med. OFC 1.361375 -0.5 -1.164152 -1.011495
MAX101 L mid/post Insula 0.636562 -0.5 -1.164152 -1.011495
MAX101 L post. Caudate 0.658339 -0.5 -1.164152 -1.011495
MAX101 L post. Putamen 1.028448 -0.5 -1.164152 -1.011495
MAX101 L post. Thalamus 1.203706 -0.5 -1.164152 -1.011495
MAX101 L pre-SMA 1.439372 -0.5 -1.164152 -1.011495
MAX101 M PCC 2.766386 -0.5 -1.164152 -1.011495
MAX101 M vmPFC1 2.587934 -0.5 -1.164152 -1.011495
MAX101 M vmPFC2 1.636236 -0.5 -1.164152 -1.011495
MAX101 R ACC 0.683580 -0.5 -1.164152 -1.011495
MAX101 R BLBM Amygdala 3.354218 -0.5 -1.164152 -1.011495
MAX101 R BST 5.167851 -0.5 -1.164152 -1.011495

Model Description

Model expressed in lme4 format:

y ~ 1 + cond + state + trait + (1 + cond | SUB) + (1 + cond + state + trait | ROI)

mathematics format:

\[Y_{s,r} \sim \text{Student}(\nu,\mu_{s,r},\sigma^{2}_{s,r})\]
\[\mu_{s,r} = \alpha + \alpha_{\text{SUB}} + \alpha_{\text{ROI}} + (\beta_{\text{cond}} +\beta_{\text{SUB}_{\text{cond}}}+ \beta_{\text{ROI}_{\text{cond}}})*\text{cond} + (\beta_{\text{state}} + \beta_{\text{ROI}_{\text{state}}})*\text{state} + (\beta_{\text{trait}} + \beta_{\text{ROI}_{\text{trait}}})*\text{trait} + \epsilon \]

Where,

  • \(\mu_{s,r}\): mean ROI response in the given subject for a particular phase and block type.
  • cond: Threat (0.5) vs. Safe (-0.5)
  • state: subject’s state score
  • trait: subject’s trait score

\[ \begin{bmatrix} \alpha_{\text{ROI}} \\ \beta_{\text{ROI}_{\text{cond}}} \\ \beta_{\text{ROI}_{\text{state}}} \\ \beta_{\text{ROI}_{\text{trait}}} \end{bmatrix} \sim \text{Multivariate t} \begin{pmatrix} \nu_{\text{ROI}}, \text{ } \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \text{ } \mathbf{S}_{\text{ROI}} \end{pmatrix} \]

\[ \mathbf{S}_{\text{ROI}} = \begin{bmatrix} \sigma_{\alpha_{ROI}} & & & \\ & \sigma_{\beta_{ROI_{cond}}} & & \\ & & \sigma_{\beta_{ROI_{state}}} & \\ & & & \sigma_{\beta_{ROI_{trait}}} \end{bmatrix} R_{\text{ROI}} \begin{bmatrix} \sigma_{\alpha_{ROI}} & & & \\ & \sigma_{\beta_{ROI_{cond}}} & & \\ & & \sigma_{\beta_{ROI_{state}}} & \\ & & & \sigma_{\beta_{ROI_{trait}}} \end{bmatrix} \]

and

\[ \begin{bmatrix} \alpha_{\text{SUB}} \\ \beta_{\text{SUB}_{\text{cond}}} \\ \end{bmatrix} \sim \text{Multivariate t} \begin{pmatrix} \nu_{\text{SUB}}, \text{ } \begin{bmatrix} 0 \\ 0 \end{bmatrix}, \text{ } \mathbf{S}_{\text{SUB}} \end{pmatrix} \]

\[ \mathbf{S}_{\text{ROI}} = \begin{bmatrix} \sigma_{\alpha_{SUB}} & \\ & \sigma_{\beta_{SUB_{cond}}} \end{bmatrix} R_{\text{SUB}} \begin{bmatrix} \sigma_{\alpha_{SUB}} & \\ & \sigma_{\beta_{SUB_{cond}}} \end{bmatrix} \]

\[\alpha \sim \text{Student t}(3, 0, 10)\] \[\beta_{i} \sim \text{Student t}(3, 0, 10)\] \[\sigma^{2}_{s,r} \sim \text{Half Student}(3, 0, 10)\] \[\nu \sim \text{Gamma}(3.325, 0.1)\] \[\sigma_{\alpha_{j}} \sim \text{Half Student}(3,0,10)\] \[\sigma_{\beta_{j_{i}}} \sim \text{Half Student}(3, 0, 10)\] \[\nu_{j} \sim \text{Gamma}(3.325, 0.1)\] \[\mathbf{R}_{j} \sim \text{LKJcorr}(2)\] notice that \(i=\text{cond, state, trait}\), \(j=\text{SUB, ROI}\); and \(\text{SUB}=1,2,...,N\), \(\text{ROI}=1,2,...,M_{n}\). \(N\) is the number of participants; \(M_{n}\) is the number of ROI for \(nth\) participant.

P+ Plot of Condition at Early Phase

             
 
 
             
 
             
 

Trial Level Model

Data

Trial Level Data
Subj Trial ROI cond Rating Response_early Response_late
MAX105 1 R_med._OFC -0.5 -0.7535452 2.405743 2.292440
MAX105 2 R_med._OFC -0.5 -0.7535452 3.470698 1.276179
MAX105 3 R_med._OFC -0.5 -0.7535452 2.928458 1.497804
MAX105 4 R_med._OFC -0.5 -0.7535452 1.691090 1.959200
MAX105 5 R_med._OFC -0.5 -0.7535452 1.086177 1.859716
MAX105 6 R_med._OFC -0.5 -0.7535452 1.247588 1.378844
MAX105 7 R_med._OFC -0.5 -0.7535452 1.712305 1.245857
MAX105 8 R_med._OFC -0.5 -0.7535452 0.954379 1.651502
MAX105 9 R_med._OFC -0.5 -0.7535452 1.849904 1.556020
MAX105 10 R_med._OFC -0.5 -0.7535452 1.410534 2.379080
MAX105 11 R_med._OFC -0.5 0.5976393 2.748708 2.821240
MAX105 12 R_med._OFC -0.5 -0.7535452 1.819616 2.137524
MAX105 13 R_med._OFC -0.5 0.5976393 1.262307 2.520669
MAX105 14 R_med._OFC -0.5 -0.7535452 0.752120 1.474834
MAX105 15 R_med._OFC -0.5 -0.7535452 1.110723 0.668991
MAX105 16 R_med._OFC -0.5 0.5976393 3.917190 1.312629
MAX105 1 L_med._OFC -0.5 -0.7535452 2.628091 1.602966
MAX105 2 L_med._OFC -0.5 -0.7535452 2.652362 2.190268
MAX105 3 L_med._OFC -0.5 -0.7535452 0.918932 1.774820
MAX105 4 L_med._OFC -0.5 -0.7535452 1.777620 1.433157
MAX105 5 L_med._OFC -0.5 -0.7535452 3.077369 1.531268
MAX105 6 L_med._OFC -0.5 -0.7535452 1.049030 1.713100
MAX105 7 L_med._OFC -0.5 -0.7535452 1.375446 1.439981
MAX105 8 L_med._OFC -0.5 -0.7535452 2.980231 2.849461
MAX105 9 L_med._OFC -0.5 -0.7535452 2.084510 2.422930
MAX105 10 L_med._OFC -0.5 -0.7535452 1.593774 1.035980
MAX105 11 L_med._OFC -0.5 0.5976393 1.040766 1.216895
MAX105 12 L_med._OFC -0.5 -0.7535452 0.928876 2.502893
MAX105 13 L_med._OFC -0.5 0.5976393 4.460567 1.819581
MAX105 14 L_med._OFC -0.5 -0.7535452 1.198841 1.216494
MAX105 15 L_med._OFC -0.5 -0.7535452 2.293487 1.930950
MAX105 16 L_med._OFC -0.5 0.5976393 1.298570 1.439921
MAX105 1 R_lat._OFC -0.5 -0.7535452 1.277495 4.975390
MAX105 2 R_lat._OFC -0.5 -0.7535452 6.846640 3.396472
MAX105 3 R_lat._OFC -0.5 -0.7535452 4.198430 2.742396
MAX105 4 R_lat._OFC -0.5 -0.7535452 1.592378 1.566960
MAX105 5 R_lat._OFC -0.5 -0.7535452 3.375294 2.968270
MAX105 6 R_lat._OFC -0.5 -0.7535452 1.299896 1.695815
MAX105 7 R_lat._OFC -0.5 -0.7535452 1.468730 4.316290
MAX105 8 R_lat._OFC -0.5 -0.7535452 3.886243 2.386790

Model

lme format:

y ~ 1 + cond + rating + ( 1 + cond + rating | gr(Subj, dist = “student”)) + ( 1 + cond + rating | gr(ROI, dist = “student”))

mathematics format:

\[ y \sim \text{LogNormal}(\mu, \sigma) \]

where

\[ \mu = \alpha + \alpha_{[\text{subj}]} + \alpha_{[\text{roi}]} + (\beta_{\text{cond}} + \beta_{\text{cond}, [\text{subj}]} + \beta_{\text{cond}, [\text{roi}]}) \times \text{Cond} + (\beta_{\text{rating}} + \beta_{\text{rating}, [\text{subj}]} + \beta_{\text{rating}, [\text{roi}]}) \times \text{Rating} \]

  • cond: Threat (0.5) vs. Safe (-0.5)
  • rating: subject’s rating after standardization.

and

\[ \begin{bmatrix} \alpha_{[j]} \\ \beta_{\text{cond}, [j]} \\ \beta_{\text{rating}, [j]} \end{bmatrix} \sim \text{Multivariate t} \begin{pmatrix} \nu_{[j]}, \text{ } \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}, \text{ } \mathbf{S}_{[j]} \end{pmatrix} \]

where

\[ \begin{aligned} \mathbf{S}_{[j]} & = \begin{pmatrix} \sigma_{\alpha_{[j]}}^2 & \rho_{1,2}\sigma_{\alpha_{[j]}}\sigma_{\beta_{\text{cond}, [j]}} & \rho_{1,3}\sigma_{\alpha_{[j]}}\sigma_{\beta_{\text{rating}, [j]}} \\ & \sigma_{\beta_{\text{cond}, [j]}}^2 & \rho_{2,3}\sigma_{\beta_{\text{cond}, [j]}}\sigma_{\beta_{\text{rating}, [j]}} \\ & & \sigma_{\beta_{\text{rating}, [j]}}^2 \end{pmatrix} \\ & = \begin{pmatrix} \sigma_{\alpha_{[j]}}^2 & & \\ & \sigma_{\beta_{\text{cond}, [j]}}^2 & \\ & & \sigma_{\beta_{\text{rating}, [j]}}^2 \end{pmatrix} \mathbf{R}_{[j]} \begin{pmatrix} \sigma_{\alpha_{[j]}}^2 & & \\ & \sigma_{\beta_{\text{cond}, [j]}}^2 & \\ & & \sigma_{\beta_{\text{rating}, [j]}}^2 \end{pmatrix} \end{aligned} \]

and

\[\alpha \sim \text{Student t}(3, \mu_{y}, 2.5)\] \[\beta_{i} \sim \text{Student t}(3, 0, 2.5)\] \[\sigma \sim \text{Half Student}(3, 0, 2.5)\] \[\nu_{[j]} \sim \text{Gamma}(3.325, 1)\] \[\sigma_{\alpha_{[j]}} \sim \text{Half Student}(3, 0, 2.5)\] \[\sigma_{\beta_{i, [j]}} \sim \text{Half Student}(3, 0, 2.5)\] \[\mathbf{R}_{[j]} \sim \text{LKJcorr}(2)\]

notice that \(\mu_{y}\) is the sample mean of response; \(i=\text{cond, rating}\), \(j=\text{subj, roi}\); and \(\text{subj}=1,2,...,N\), \(\text{roi}=1,2,...,M_{n}\). \(N\) is the number of participants; \(M_{n}\) is the number of ROI for \(nth\) participant.

P+ Plot of Condition at Early Phase