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\jmlrvolume{-- 108}
\jmlryear{2026}
\jmlrworkshop{Supplementary Material -- MIDL 2026}
\editors{Accepted for publication at MIDL 2026}

\title[BETA -- Supplementary Material]{Supplementary Material for\\
BETA: Resting-state fMRI Biotypes For tDCS Efficacy in Anxiety Among Older Adults At Risk For Alzheimer's Disease}

% Authors (you can keep exactly consistent with the main paper)
\midlauthor{
\Name{Skylar E. Stolte\nametag{$^{1,2,3}$}\midljointauthortext{Equal Contribution}} \orcid{0000-0002-9300-1322} \Email{skylastolte4444@ufl.edu} \\
\Name{Junfu Cheng\nametag{$^{4}^*$}} \orcid{0009-0003-6590-9485} 
\Email{junfu.cheng@ufl.edu} \\
\Name{Chintan Acharya\nametag{$^{1}$}} \orcid{0009-0000-8761-9137} \Email{chintan.acharya@ufl.edu} \\
\Name{Lin Gu\nametag{$^{5}$}} \orcid{0000-0002-7419-6240} \Email{rin.tani.e8@tohoku.ac.jp} \\
\Name{Andrew O'Shea\nametag{$^{3}$}} \orcid{0009-0006-8697-3330} 
\Email{aoshea@phhp.ufl.edu} \\
\Name{Aprinda Indahlastari\nametag{$^{2,3}$}} \orcid{0000-0002-5994-5514} \Email{aprinda.indahlas@phhp.ufl.edu} \\
\Name{Adam J. Woods\nametag{$^{3,6}$}} \orcid{0000-0002-0786-0236} \Email{adam.woods@utdallas.edu} \\
\Name{Ruogu Fang\nametag{$^{1,3,4}$}\midljointauthortext{Corresponding Author}} \orcid{0000-0003-3980-3532}\Email{ruogu.fang@bme.ufl.edu} \\
\addr $^{1}$ J. Crayton Pruitt Family Department of Biomedical Engineering, University of Florida, United States\\
\addr $^{2}$ Department of Clinical and Health Psychology, University of Florida, United States\\
\addr $^{3}$ Center for Cognitive Aging and Memory, Department of Clinical and Health Psychology, McKnight Brain Institute, University of Florida, USA\\
\addr $^{4}$ Department of Electrical and Computer Engineering, University of Florida, United States\\
\addr $^{5}$ Research Institute of Electrical Communication, Tohoku University, Japan\\
\addr $^{6}$ Behavioral and Brain Sciences, The University of Texas at Dallas, USA\\
}

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\begin{document}

\maketitle


\section{fMRI Data Acquisition}

Neuroimaging data were acquired on harmonized 3~T Siemens scanners across two sites. Data were collected on a Siemens Magnetom Prisma scanner with a 64-channel head coil and a Siemens Magnetom Skyra scanner with a 32-channel head coil. Earplugs and foam padding were used to reduce noise and head motion.

Functional images were acquired using a gradient-echo EPI BOLD sequence (TR = 1800 ms, TE = 2.26 ms, voxel size = 1.0 mm isotropic, FOV = 256 × 256 × 176 mm). Inter-site consistency was monitored using regular FBIRN, GABA, and human phantom scans. The mean signal-to-noise ratio across sites was 12.1 ± 1.54.


\section{Additional Dataset Details}
The State-Trait Anxiety Inventory (STAI) has two subscales, each with a score range of 20 to 80. Change in STAI-S for the Sham and Active tDCS groups are summarized in Supplementary Table~\ref{tab:stai_sham_active}.


\begin{table}[H]
\centering
\small                           % smaller font for a compact look
\setlength{\tabcolsep}{6pt}      % column separation
\renewcommand{\arraystretch}{1.2}% row height
\caption{Sham and Active tDCS State STAI score change from pre-intervention to post-intervention. \textit{n}: number of participants, \textit{STAI-S}: STAI state score, \textit{SD}: Standard deviation, \textit{IQR}: Interquartile range}
\label{tab:stai_sham_active}
\begin{tabular}{
    l   % Measure
    l   % Group
    S[table-format=3.0] % n
    S[table-format=2.2] % Mean
    S[table-format=2.0] % Median
    S[table-format=2.2] % SD
    S[table-format=2.2] % IQR
}
\toprule
\textbf{Measure} & \textbf{Group} & \textbf{n} & \textbf{Mean} & \textbf{Median} & \textbf{SD} & \textbf{IQR} \\
\midrule
Baseline STAI-S  & Sham tDCS   &  99 & 27.33 & 25 & 7.58 &  9.50 \\
Baseline STAI-S  & Active tDCS & 100 & 28.57 & 25 & 8.43 & 14.00 \\
Baseline STAI-S  & All         & 199 & 27.95 & 25 & 8.02 & 12.50 \\
\midrule
Post STAI-S     & Sham tDCS   &  99 & 27.62 & 26 & 8.07 &  9.00 \\
Post STAI-S    & Active tDCS & 100 & 26.68 & 24 & 7.58 & 11.25 \\
Post STAI-S     & All         & 199 & 27.15 & 25 & 7.82 &  9.00 \\
\midrule
STAI-S (Pre-Post)  & Sham tDCS   &  99 &  -0.28 &  0 & 8.64 &  5.50 \\
STAI-S (Pre-Post)   & Active tDCS & 100 & 1.89 &  0 & 7.77 &  7.00 \\
STAI-S (Pre-Post)    & All         & 199 & 0.81 &  0 & 8.27 &  6.00 \\
\bottomrule
\end{tabular}
\end{table}

\section{NeuroSynth-Based ROI Construction}

Anxiety-related regions of interest (ROIs) were defined using meta-analytic data from NeuroSynth. ROIs were derived from \emph{association maps} rather than uniformity maps, as association maps identify brain regions selectively associated with a given term while controlling for base-rate activation across the literature, providing greater specificity for disorder-relevant network definition. The NeuroSynth query term \emph{``anxiety''} was submitted via the NeuroSynth web interface, producing an association map based on an automated meta-analysis of 449 anxiety-related studies.

The association map was thresholded using the default NeuroSynth web interface options, with the threshold interactively adjusted to retain suprathreshold voxels showing positive association with anxiety. The thresholded map was binarized, and spatially contiguous suprathreshold clusters were extracted, with each cluster defined as a distinct ROI. Because ROI generation was performed via the graphical interface rather than a fully scripted pipeline, the exact numeric threshold value was not explicitly recorded; however, all ROIs were fully specified by their voxel extents and spatial coordinates. This procedure yielded 182 ROIs, which were assigned to canonical functional networks (limbic, frontal control, default mode, sensory, and subcortical) based on anatomical location and established network definitions. A complete list of NeuroSynth-derived ROIs, including ROI name, network assignment, voxel count, and MNI centroid coordinates, is provided in Supplementary Table~\ref{tab:neurosynth_rois}.



\begin{longtable}{p{0.55\linewidth} p{0.35\linewidth}}
\caption{NeuroSynth-derived regions of interest (ROIs) and network assignments.}

\setlength{\LTcapwidth}{\textwidth}
\setlength{\LTleft}{0pt}
\setlength{\LTright}{0pt}
\label{tab:neurosynth_rois} \\
\toprule
\textbf{ROI Name} & \textbf{Network} \\
\midrule
\endfirsthead

\toprule
\textbf{ROI Name} & \textbf{Network} \\
\midrule
\endhead

\midrule
\multicolumn{2}{r}{\emph{Continued on next page}} \\
\endfoot

\bottomrule
\endlastfoot

atlas.Cuneal Cortex (Right) & Atlas ROI \\
atlas.Cuneal Cortex (Left) & Atlas ROI \\
atlas.Supracalcarine Cortex (Left) & Atlas ROI \\
atlas.Supracalcarine Cortex (Right) & Atlas ROI \\
atlas.Intracalcarine Cortex (Left) & Atlas ROI \\
atlas.Intracalcarine Cortex (Right) & Atlas ROI \\
atlas.Lingual Gyrus (Right) & Atlas ROI \\
atlas.Lingual Gyrus (Left) & Atlas ROI \\
networks.Visual.Medial (2, -79, 12) & Network \\
Yeo.Visual & Network \\
atlas.Occipital Fusiform Gyrus (Left) & Atlas ROI \\
atlas.Occipital Fusiform Gyrus (Right) & Atlas ROI \\
atlas.Occipital Pole (Right) & Atlas ROI \\
atlas.Occipital Pole (Left) & Atlas ROI \\
networks.Visual.Occipital (0, -93, -4) & Network \\
atlas.Lateral Occipital Cortex, inferior division (Right) & Atlas ROI \\
networks.Visual.Lateral (Right) (38, -72, 13) & Network \\
networks.Visual.Lateral (Left) (-37, -79, 10) & Network \\
atlas.Lateral Occipital Cortex, inferior division (Left) & Atlas ROI \\
atlas.Temporal Occipital Fusiform Cortex (Left) & Atlas ROI \\
atlas.Temporal Occipital Fusiform Cortex (Right) & Atlas ROI \\
atlas.Inferior Temporal Gyrus, temporo-occipital part (Right) & Atlas ROI \\
atlas.Inferior Temporal Gyrus, temporo-occipital part (Left) & Atlas ROI \\

RestingState & Concept / Task \\
Frontoparietal & Concept / Task \\
Attention & Concept / Task \\
Yeo.Dorsal Attention & Network \\

atlas.Superior Parietal Lobule (Right) & Atlas ROI \\
networks.DorsalAttention.IPS (Right) (39, -42, 54) & Network \\
atlas.Superior Parietal Lobule (Left) & Atlas ROI \\
networks.DorsalAttention.IPS (Left) (-39, -43, 52) & Network \\

atlas.Supramarginal Gyrus, anterior division (Left) & Atlas ROI \\
networks.Salience.SMG (Left) (-60, -39, 31) & Network \\
atlas.Supramarginal Gyrus, anterior division (Right) & Atlas ROI \\
networks.Salience.SMG (Right) (62, -35, 32) & Network \\
atlas.Supramarginal Gyrus, posterior division (Right) & Atlas ROI \\

atlas.Inferior Frontal Gyrus, pars opercularis (Right) & Atlas ROI \\
atlas.Frontal Operculum Cortex (Right) & Atlas ROI \\
networks.Salience.AInsula (Right) (47, 14, 0) & Network \\
networks.Salience.AInsula (Left) (-44, 13, 1) & Network \\
Yeo.Salience & Network \\

atlas.Parietal Operculum Cortex (Left) & Atlas ROI \\
atlas.Central Opercular Cortex (Left) & Atlas ROI \\
atlas.Insular Cortex (Left) & Atlas ROI \\
atlas.Insular Cortex (Right) & Atlas ROI \\
atlas.Planum Temporale (Right) & Atlas ROI \\
atlas.Parietal Operculum Cortex (Right) & Atlas ROI \\
atlas.Central Opercular Cortex (Right) & Atlas ROI \\

networks.Sensorimotor.Lateral (Right) (56, -10, 29) & Network \\
Yeo.Somatosensory & Network \\
networks.Sensorimotor.Lateral (Left) (-55, -12, 29) & Network \\
atlas.Precentral Gyrus (Left) & Atlas ROI \\
atlas.Postcentral Gyrus (Left) & Atlas ROI \\
networks.Sensorimotor.Superior (0, -31, 67) & Network \\
atlas.Postcentral Gyrus (Right) & Atlas ROI \\
atlas.Precentral Gyrus (Right) & Atlas ROI \\
atlas.Supplementary Motor Area (Right) & Atlas ROI \\
atlas.Supplementary Motor Area (Left) & Atlas ROI \\

networks.DorsalAttention.FEF (Left) (-27, -9, 64) & Network \\
networks.DorsalAttention.FEF (Right) (30, -6, 64) & Network \\
networks.Salience.RPFC (Right) (32, 46, 27) & Network \\
networks.Salience.RPFC (Left) (-32, 45, 27) & Network \\
networks.Salience.ACC (0, 22, 35) & Network \\
atlas.Anterior Cingulate Cortex & Atlas ROI \\

atlas.Putamen (Right) & Atlas ROI \\
atlas.Putamen (Left) & Atlas ROI \\
atlas.Thalamus (Left) & Atlas ROI \\
atlas.Thalamus (Right) & Atlas ROI \\
atlas.Nucleus Accumbens (Left) & Atlas ROI \\
atlas.Nucleus Accumbens (Right) & Atlas ROI \\
salience & Concept / Task \\

atlas.Pallidum (Right) & Atlas ROI \\
atlas.Pallidum (Left) & Atlas ROI \\

Sad & Concept / Task \\
threatening & Concept / Task \\
Reward & Concept / Task \\
MonetaryReward & Concept / Task \\

Default & Concept / Task \\
Yeo.Default & Network \\
Cognitive Control & Concept / Task \\
Yeo.Frontoparietal & Network \\

atlas.Precuneus & Atlas ROI \\

\end{longtable}



\section{Model Architecture}

We employed a Transformer-based variational autoencoder (VAE) to learn a compact and structured latent representation of whole-brain functional connectivity patterns. The model followed the standard encoder--decoder VAE framework, with self-attention mechanisms replacing conventional convolutional or fully connected hidden layers in order to capture global dependencies among connectivity features.

\subsection{Input Representation}
For each participant, the functional connectivity matrix was vectorized into a one-dimensional feature vector of length $D$, corresponding to the number of unique connectivity edges. This vector served as the direct input to the VAE. Let $\mathbf{x} \in \mathbb{R}^{D}$ denote the input connectivity feature vector.

\subsection{Encoder}
The encoder began with a linear embedding layer that projects the input vector $\mathbf{x}$ into a 512-dimensional feature space. The embedded representation was then processed by a Transformer encoder composed of two stacked Transformer encoder layers, each using four self-attention heads and a feedforward hidden dimension of 512. To accommodate the Transformer architecture, the embedded vector was treated as a single-token sequence by introducing a singleton sequence dimension.

The output of the Transformer encoder was subsequently passed through two parallel fully connected layers to estimate the parameters of the approximate posterior distribution:
\[
q_{\phi}(\mathbf{z} \mid \mathbf{x}) = \mathcal{N}(\boldsymbol{\mu}, \mathrm{diag}(\boldsymbol{\sigma}^2)),
\]
where $\boldsymbol{\mu} \in \mathbb{R}^{50}$ and $\log \boldsymbol{\sigma}^2 \in \mathbb{R}^{50}$ denote the latent mean and log-variance vectors, respectively. The latent dimensionality was fixed at 50 to balance representational capacity and regularization.

\subsection{Latent Sampling}
Latent variables were sampled using the reparameterization trick to enable gradient-based optimization:
\[
\mathbf{z} = \boldsymbol{\mu} + \boldsymbol{\epsilon} \odot \exp\left(\tfrac{1}{2} \log \boldsymbol{\sigma}^2\right),
\quad \boldsymbol{\epsilon} \sim \mathcal{N}(\mathbf{0}, \mathbf{I}).
\]



\subsection{Decoder}
The decoder mirrors the encoder architecture. The latent vector $\mathbf{z}$ was first linearly projected into the same 512-dimensional embedding space and then processed by a two-layer Transformer encoder block (used here as a decoder with self-attention). A final linear output layer mapped the Transformer output back to the original input dimension $D$, producing a reconstruction $\hat{\mathbf{x}} \in \mathbb{R}^{D}$ of the input connectivity features.




\section{Model Training Details}
During optimization, we used the Adam optimizer with a learning rate of $1 \times 10^{-5}$ for a maximum of 200 epochs. Mini-batch training was performed with a batch size of 64 subjects. An initial warmup learning scheduler was used for 5 epochs to ensure a stable gradient during the beginning of the learning process. The model training loss balanced terms for KL Divergence and Mean Squared Error (MSE). The initial variational autoencoder stage learned how to reconstruct the functional correlation data. The final clustering model learned the optimal clustering using a clustering objective that was based off of deep embedded clustering approaches \cite{xie2016unsuperviseddeepembeddingclustering}. 


During both training schemes, the MSE loss refers to the ability to reconstruct the input data. The Variational Autoencoder applies KL divergence as the difference between the latent space and the normal distribution. The Deep Embedded Clustering using KL divergence as the difference between the current clustering assignments and a sharpened version of the clusters, as in \cite{xie2016unsuperviseddeepembeddingclustering}. In addition to the standard reconstruction and KL divergence losses, the VAE objective incorporated a clinical similarity regularization term that penalized squared Euclidean distances between latent representations weighted by pairwise similarity in clinical anxiety measures, encouraging clinically meaningful organization of the latent space.


Importantly, both active-tDCS and sham-tDCS participants were retained throughout training so that the model learned to distinguish genuine treatment effects from the negligible changes seen in the sham group. 
Consistent with best practices in data-driven clustering, the excluded cluster was interpreted as likely reflecting outlier subjects or rare connectivity configurations rather than a stable or generalizable anxiety biotype. All primary conclusions were therefore based on the four retained clusters, which together account for the vast majority of participants and demonstrate robust, interpretable differences in functional connectivity and tDCS response.



\section{Group-level Difference in Functional Connectivity}
This experiment extracted the functional connectivity networks that were statistically different for each cluster compared to the population averages. The key outcome of this analysis was to learn how anxiety presents differently in each cluster. This information could provide potential insight into why the clusters may respond differently to tDCS interventions.
We used one-sample t-testing to extract the functional connectivity differences that were significant for each cluster compared to the entire population. The pipeline used many one-sample t-tests for each cluster and functional connectivity network, so the false discovery rate (FDR) method \cite{benjamini1995controlling} was applied to correct for multiple comparisons.

% More granular description than in main text
Table~\ref{tb1} provides the corresponding quantitative details, including the mean connectivity value for each connection within every biotype and the 95\% confidence interval for the population average. Together, these results demonstrate that each biotype exhibits a unique functional connectivity signature, highlighting the neurobiological heterogeneity across groups.


\begin{table}[h]
\centering
\caption{Average functional connectivity strengths across four subtypes corresponding four clusters. $\uparrow$ and $\downarrow$ mean significantly stronger and significantly weaker connections for the given cluster, respectively. ROI Abbreviations and Full Region Names were listed in Table \ref{tab:roi_names}. The impact of tDCS across all participant in cluster: * = clinically significant anxiety improved with tDCS, ** = all anxiety improved with tDCS.} 
\resizebox{\textwidth}{!}{%
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\textbf{Subtype} 
& \makecell{\textbf{Selective tDCS}\\\textbf{Responder}$^{*}$} 
& \makecell{\textbf{Robust tDCS}\\\textbf{Responder}$^{**}$} 
& \makecell{\textbf{Placebo}\\\textbf{Responder}} 
& \makecell{\textbf{Minimal}\\\textbf{Responder}} 
& \makecell{\textbf{All}\\\textbf{Participants}} \\
\hline

\textbf{$sLOC_{L} - MedFC$} 
& $0.399 \uparrow$ 
& 0.218 
& $0.161 \downarrow$
& $0.0930 \downarrow$ 
& $0.219\pm0.0380$ \\
\hline

\textbf{$Ver8 - MidFG_{R}$} 
& $-0.0136 \downarrow$ 
& 0.0220 
& $-0.0381 \downarrow$ 
& $0.197 \uparrow$ 
& $0.0216\pm0.0290$ \\
\hline

\textbf{$IFG_{tri_{R}} - Ver8$} 
& 0.0332 
& 0.0111 
& $-0.0153 \downarrow$ 
& $0.197 \uparrow$ 
& $0.0301\pm0.0310$ \\
\hline

\textbf{$TP_{R} - Cereb8_{R}$} 
& $0.171 \uparrow$ 
& 0.0885 
& 0.0679 
& $-0.0672 \downarrow$ 
& $0.0783\pm0.0310$ \\
\hline

\textbf{$sLOC_{L} - ICC_{L}$} 
& $0.371 \uparrow$ 
& 0.206 
& $0.181 \downarrow$ 
& $0.118 \downarrow$ 
& $0.217\pm0.0350$ \\
\hline

\textbf{$SFG_{L} - AG_{R}$} 
& $0.461 \uparrow$ 
& $0.450 \uparrow$ 
& $0.317 \downarrow$ 
& 0.404 
& $0.398\pm0.0340$ \\
\hline

\textbf{$sLOC_{L} - FO_{R}$} 
& $0.264 \uparrow$ 
& 0.155 
& 0.127 
& $0.0533 \downarrow$ 
& $0.149\pm0.0320$ \\
\hline

\textbf{$AG_{R} - PreCG_{R}$} 
& $0.407 \uparrow$ 
& $0.361 \uparrow$ 
& $0.246 \downarrow$ 
& 0.323 
& $0.325\pm0.0320$ \\
\hline

\textbf{$MidFG_{L} - AG_{R}$} 
& $0.364 \uparrow$ 
& $0.384 \uparrow$ 
& $0.230 \downarrow$ 
& $0.254 \downarrow$ 
& $0.308\pm0.0360$ \\
\hline

\textbf{$IFG_{oper_{L}} - Cereb8_{R}$} 
& $0.214 \uparrow$ 
& 0.0975 
& 0.0628 
& $-0.0225 \downarrow$ 
& $0.0949\pm0.0330$ \\
\hline

\textbf{$AG_{L} - sLOC_{R}$} 
& $0.635 \uparrow$ 
& $0.661 \uparrow$ 
& $0.515 \downarrow$ 
& $0.477 \downarrow$ 
& $0.580\pm0.0370$ \\
\hline

\textbf{$IC_{L} - IFG_{oper_{L}}$} 
& $0.557 \uparrow$ 
& $0.574 \uparrow$ 
& $0.445 \downarrow$ 
& 0.499 
& $0.513\pm0.0350$ \\
\hline

\textbf{$SFG_{R} - IFG_{oper_{L}}$} 
& $0.655 \uparrow$ 
& $0.668 \uparrow$ 
& $0.517 \downarrow$ 
& $0.530 \downarrow$ 
& $0.592\pm0.0400$ \\
\hline

\textbf{$SFG_{R} - IFG_{tri_{R}}$} 
& 0.714 
& $0.729 \uparrow$ 
& 0.645 
& $0.640 \downarrow$ 
& $0.678\pm0.0380$ \\
\hline

\textbf{$Ver8 - pSMG_{R}$} 
& 0.0243 
& 0.0319 
& $0.00341 \downarrow$ 
& $0.191 \uparrow$ 
& $0.0453\pm0.0280$ \\
\hline

\textbf{$MidFG_{R} - PO_{R}$} 
& -0.0702 
& $-0.0340 \uparrow$ 
& $-0.160 \downarrow$ 
& -0.103 
& $-0.0842\pm0.0320$ \\
\hline
\end{tabular}

}
\label{tb1}
\end{table}

\begin{table}[h]
\centering
\caption{ROI Abbreviations and Full Region Names}
\small
\setlength{\tabcolsep}{12pt}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{ll}
\toprule
\textbf{Abbrev.} & \textbf{Region Name} \\
\midrule
sLOC\textsubscript{L}   & Lateral Occipital Cortex, superior division Left \\
MedFC                   & Frontal Medial Cortex \\
ICC\textsubscript{L}    & Intracalcarine Cortex Left \\
SFG\textsubscript{L}    & Superior Frontal Gyrus Left \\
AG\textsubscript{R}     & Angular Gyrus Right \\
FO\textsubscript{R}     & Frontal Operculum Cortex Right \\
PreCG\textsubscript{R}  & Precentral Gyrus Right \\
IFG\textsubscript{tri R} & Inferior Frontal Gyrus, pars triangularis Right \\
Ver8                    & Vermis 8 \\
MidFG\textsubscript{L}  & Middle Frontal Gyrus Left \\
AG\textsubscript{L}     & Angular Gyrus Left \\
sLOC\textsubscript{R}   & Lateral Occipital Cortex, superior division Right \\
TP\textsubscript{R}     & Temporal Pole Right \\
IFG\textsubscript{oper R} & Inferior Frontal Gyrus, pars opercularis Right \\
Cereb8\textsubscript{R} & Cerebellum 8 Right \\
MidFG\textsubscript{R}  & Middle Frontal Gyrus Right \\
pSMG\textsubscript{R}   & Supramarginal Gyrus, posterior division Right \\
PO\textsubscript{R}     & Parietal Operculum Cortex Right \\
IC\textsubscript{L}     & Insular Cortex Left \\
SFG\textsubscript{R}    & Superior Frontal Gyrus Right \\
\bottomrule
\end{tabular}
\label{tab:roi_names}
\end{table}



\section{Cluster-by-Cluster Results Across Baseline Severity Levels}
In cluster level pre--post evaluation, for each cluster, change scores were tested against zero using one-sample \textit{t}-tests when Shapiro--Wilk normality was not violated ($p < 0.05$) and Wilcoxon signed-rank tests otherwise. Subgroups in each cluster defined by baseline STAI S in clusters with $n = 1$ or $n = 0$ were not tested.
For the Minimal/Mild baseline anxiety group, Active--Sham differences were evaluated using Welch’s two-sample \textit{t}-test (unequal variances) and the nonparametric Mann--Whitney \textit{U} test, accompanied by assumption checks via Shapiro--Wilk normality tests and Levene’s median-based variance test. Effect sizes were computed using Cohen’s $d$, Hedges’ $g$, and Glass’s $\Delta$. 
Because sample sizes were limited in the Moderate/Severe baseline group, condition differences were assessed using a two-sample permutation test on mean differences (10{,}000 permutations, two-sided), supplemented by the same normality and variance diagnostics and effect size estimates. The permutation test evaluates whether the observed Active--Sham difference could arise by chance by repeatedly shuffling group labels and recomputing the mean difference across thousands of random reallocations. 
Together, this approach ensured that each comparison was evaluated with statistical methods appropriate for the underlying sample size, distribution, and variance structure. The change of STAI-S across each cluster was illustrated in Figure \ref{fig:stai_subtypes}.
\begin{figure}[!t]
\centering
\includegraphics[width=\textwidth]{figures/state_stai_score_change_clusters.pdf}
\caption{
Pre–post changes in STAI-State scores across four clusters.
Each panel shows baseline and post-intervention STAI-State scores for participants in the
Selective tDCS Responder, Robust tDCS Responder, Placebo Responder, and Minimal Responder subtype clusters.
Individual trajectories are displayed with light lines, overlaid with violin distributions and boxplots representing the score distributions at each time point.
Colored trajectories indicate responder status (blue = non-responder; red = responder).
Overall patterns highlight subtype-specific differences in anxiety reduction following the intervention.
}
\label{fig:stai_subtypes}
\end{figure}

\paragraph{Selective tDCS Responder Subtype.} 
Individuals with moderate/severe baseline status show a very large improvement margin, with active tDCS producing gains more than ten points greater than sham. Selective tDCS Responder subtype cluster exhibits the strongest clinical improvement. This clinical response aligns with the Selective tDCS Responder subtype’s densely interconnected fronto-parietal and medial-frontal network architecture, characterized by strengthened links among superior frontal, angular, occipital, and precentral regions—an organization that appears especially receptive to modulation by tDCS.

\paragraph{Robust tDCS Responder Subtype.}
Individuals show clear and statistically significant gains under active tDCS compared with sham, including reliable improvement in the minimal and mild baseline groups. Robust tDCS Responder subtype cluster demonstrates a moderate but consistent benefit across individuals receiving active stimulation (7.0 ± 11.47; $n = 5$). The observed pre–post reductions were consistent with the Robust tDCS Responder subtype’s selectively strengthened fronto-parietal and frontal–temporal connections, a network configuration that may permit effective engagement by tDCS-driven neuromodulation.

\paragraph{Placebo Responder Subtype.}
Individuals demonstrate greater improvement during sham stimulation + cognitive training (CT) than active tDCS + CT, indicating sensitivity to CT rather than stimulation-dependent benefit. Placebo Responder subtype shows significant improvement under active tDCS (7.3 ± 6.95; $n = 6$) and notable reductions even in the sham group (12.8 ± 13.00; $n = 4$). Improvement in the absence of stimulation suggests strong expectancy or placebo effects.

\paragraph{Minimal Responder Subtype.}
Individuals exhibit little to no improvement under either sham or active tDCS, reflecting limited responsiveness across condition. Minimal Responder subtype cluster, in contrast, exhibits minimal clinical benefit.

\subsection{Selective tDCS Responder Subtype}

\paragraph{Minimal/Mild Baseline Group.}
Participants showed minimal pre--post improvement in both conditions, with no evidence of an Active benefit.

\begin{table}[h!]
\centering
\caption{Selective tDCS Responder Subtype (Minimal/Mild): Descriptive Statistics}
\begin{tabular}{lccc}
\toprule
Condition & $n$ & Mean Change & SD \\
\midrule
Sham   & 14 & $-0.50$ & 6.50 \\
Active & 20 & $0.10$ & 5.56 \\
\bottomrule
\end{tabular}
\end{table}

\begin{table}[h!]
\centering
\caption{Selective tDCS Responder Subtype (Minimal/Mild): Statistical Tests}
\begin{tabular}{lcc}
\toprule
Test & Statistic & $p$-value \\
\midrule
Welch's \textit{t}-test    & $t=0.28$ & 0.781 \\
Mann--Whitney \textit{U}  & $U=148$ & 0.805 \\
Cohen's $d$               & 0.07 & --- \\
\bottomrule
\end{tabular}
\end{table}

\paragraph{Moderate/Severe Baseline Group.}
Active participants showed a numerically larger reduction than Sham, but permutation testing indicated the effect was not statistically reliable.

\begin{table}[h!]
\centering
\caption{Selective tDCS Responder Subtype (Moderate/Severe): Descriptive and Permutation Results}
\begin{tabular}{lcccc}
\toprule
Condition & $n$ & Mean Change & SD & --- \\
\midrule
Sham   & 2 & 7.00 & 2.83 & \\
Active & 3 & 18.00 & 7.00 & \\
\midrule
Observed Difference (A--S) & \multicolumn{4}{c}{$+11.00$} \\
Permutation $p$-value      & \multicolumn{4}{c}{$0.1956$} \\
Cohen's $d$                & \multicolumn{4}{c}{$2.10$ (unstable)} \\
\bottomrule
\end{tabular}
\end{table}

\paragraph{All Participants.}
When pooling Minimal/Mild and Moderate/Severe baseline groups, Active stimulation produced a slightly larger mean reduction in STAI-State scores than Sham in the Selective tDCS Responder subtype, but this difference was not statistically reliable. Both Welch's two-sample \textit{t}-test and the Mann--Whitney \textit{U} test were nonsignificant, and the effect size was small.

\begin{table}[h!]
\centering
\caption{Selective tDCS Responder Subtype (All Participants): Active--Sham Statistical Tests}
\begin{tabular}{lcc}
\toprule
Test & Statistic & $p$-value \\
\midrule
Welch's \textit{t}-test    & $t = 0.69$   & 0.4968 \\
Mann--Whitney \textit{U}  & $U = 196$  & 0.7525 \\
Cohen's $d$               & 0.21         & --- \\
\bottomrule
\end{tabular}
\end{table}

% -------------------------------------------------------------

\subsection{Robust tDCS Responder Subtype}

\paragraph{Minimal/Mild Baseline Group.}
This subtype showed the strongest and only statistically reliable Active--Sham difference among the Minimal/Mild participants. Active stimulation produced significantly greater symptom reduction than Sham across both parametric and nonparametric tests based on Mann-Whitney \testit{U} test.

\begin{table}[h!]
\centering
\caption{Robust tDCS Responder Subtype (Minimal/Mild): Descriptive Statistics}
\begin{tabular}{lccc}
\toprule
Condition & $n$ & Mean Change & SD \\
\midrule
Sham   & 29 & $-2.38$ & 7.93 \\
Active & 34 & $-6.68$ & 9.57 \\
\bottomrule
\end{tabular}
\end{table}

\begin{table}[h!]
\centering
\caption{Robust tDCS Responder Subtype (Minimal/Mild): Statistical Tests}
\begin{tabular}{lcc}
\toprule
Test & Statistic & $p$-value \\
\midrule
Welch's \textit{t}-test     & $t=2.65$ & $\mathbf{0.0105}$ \\
\textbf{Mann--Whitney \textit{U}}   & $U=649$  & $\mathbf{0.0309}$ \\
Cohen's $d$                & 0.59     & --- \\
\bottomrule
\end{tabular}
\end{table}

\paragraph{Moderate/Severe Baseline Group.}
Only Active participants were present ($n=5$), so no condition comparisons could be conducted.

\paragraph{All Participants.}
Across all participants in the Robust tDCS Responder subtype, Active stimulation produced a substantially larger mean reduction in STAI-State scores than Sham. This Active--Sham difference was statistically reliable in the Mann--Whitney \textit{U} test, with a medium-to-large effect size.

\begin{table}[h!]
\centering
\caption{Robust tDCS Responder Subtype (All Participants): Active--Sham Statistical Tests}
\begin{tabular}{lcc}
\toprule
Test & Statistic & $p$-value \\
\midrule
Welch's \textit{t}-test     & $t = 2.95$   & $\mathbf{0.0046}$ \\
\textbf{Mann--Whitney \textit{U}}   & $U = 753$  & $\mathbf{0.0197}$ \\
Cohen's $d$                & 0.73         & --- \\
\bottomrule
\end{tabular}
\end{table}

% -------------------------------------------------------------

\subsection{Placebo Responder Subtype}

\paragraph{Minimal/Mild Baseline Group.}
This subtype showed moderately larger improvement under Active stimulation, though results did not reach statistical significance.

\begin{table}[h!]
\centering
\caption{Placebo Responder Subtype (Minimal/Mild): Descriptive Statistics}
\begin{tabular}{lccc}
\toprule
Condition & $n$ & Mean Change & SD \\
\midrule
Sham   & 32 & $-3.25$ & 6.75 \\
Active & 18 & $-7.11$ & 8.98 \\
\bottomrule
\end{tabular}
\end{table}

\begin{table}[h!]
\centering
\caption{Placebo Responder Subtype (Minimal/Mild): Statistical Tests}
\begin{tabular}{lcc}
\toprule
Test & Statistic & $p$-value \\
\midrule
Welch's \textit{t}-test     & $t=-0.90$ & 0.135 \\
Mann--Whitney \textit{U}   & $U=259$ & 0.129 \\
Cohen's $d$                & 0.52 & --- \\
\bottomrule
\end{tabular}
\end{table}

\paragraph{Moderate/Severe Baseline Group.}
Active participants showed slightly smaller improvements than Sham, but permutation tests confirmed no reliable difference.

\begin{table}[h!]
\centering
\caption{Placebo Responder Subtype (Moderate/Severe): Descriptive and Permutation Results}
\begin{tabular}{lcccc}
\toprule
Condition & $n$ & Mean Change & SD \\
\midrule
Sham   & 4 & 12.75 & 12.99 \\
Active & 6 & 7.33 & 11.47 \\
\midrule
Observed Difference (A--S) & \multicolumn{3}{c}{$-5.42$} \\
Permutation $p$-value      & \multicolumn{3}{c}{$0.3946$} \\
Cohen's $d$                & \multicolumn{3}{c}{$-0.56$} \\
\bottomrule
\end{tabular}
\end{table}

\paragraph{All Participants.}
When Minimal/Mild and Moderate/Severe participants were combined, Active and Sham conditions produced very similar mean pre--post changes in STAI-State in the Placebo Responder subtype. Neither Welch's two-sample \textit{t}-test nor the Mann--Whitney \textit{U} test showed a significant Active--Sham difference, and the overall effect size was small and slightly negative.

\begin{table}[h!]
\centering
\caption{Placebo Responder Subtype (All Participants): Active--Sham Statistical Tests}
\begin{tabular}{lcc}
\toprule
Test & Statistic & $p$-value \\
\midrule
Welch's \textit{t}-test     & $t = -0.49$  & 0.6249 \\
Mann--Whitney \textit{U}   & $U = 426.5$  & 0.9395 \\
Cohen's $d$                & $-0.12$      & --- \\
\bottomrule
\end{tabular}
\end{table}

% -------------------------------------------------------------

\subsection{Minimal Responder Subtype}

\paragraph{Minimal/Mild Baseline Group.}
Participants in the Minimal Responder Subtype exhibited modest reductions in STAI-State scores in both Sham and Active conditions, with no statistically significant difference between stimulation types. Both Welch's \textit{t}-test and Mann–Whitney \textit{U} test indicated nonsignificant effects.

\begin{table}[h!]
\centering
\caption{Minimal Responder Subtype (Minimal/Mild): Descriptive Statistics}
\begin{tabular}{lccc}
\toprule
Condition & $n$ & Mean Change & SD \\
\midrule
Sham   & 12 & $-4.00$ & 8.47 \\
Active & 12 & $-1.17$ & 6.58 \\
\bottomrule
\end{tabular}
\end{table}

\begin{table}[h!]
\centering
\caption{Minimal Responder Subtype (Minimal/Mild): Statistical Tests}
\begin{tabular}{lcc}
\toprule
Test & Statistic & $p$-value \\
\midrule
Welch's \textit{t}-test     & $t=0.92$ & 0.371 \\
Mann--Whitney \textit{U}   & $U=82$ & 0.581 \\
Cohen's $d$                & 0.34 & --- \\
\bottomrule
\end{tabular}
\end{table}

\paragraph{Moderate/Severe Baseline Group.}
Only Sham participants were present ($n=3$, mean change $=5.33$), so no Active--Sham comparison was possible.

\paragraph{All Participants.}
Across all participants in the Minimal Responder subtype, both Sham and Active conditions showed only modest pre--post changes, and Active stimulation did not outperform Sham. Welch's two-sample \textit{t}-test and the Mann--Whitney \textit{U} test were nonsignificant, with a small effect size.

\begin{table}[h!]
\centering
\caption{Minimal Responder Subtype (All Participants): Active--Sham Statistical Tests}
\begin{tabular}{lcc}
\toprule
Test & Statistic & $p$-value \\
\midrule
Welch's \textit{t}-test     & $t = 0.54$   & 0.5960 \\
Mann--Whitney \textit{U}   & $U = 117$  & 0.5822 \\
Cohen's $d$                & 0.19         & --- \\
\bottomrule
\end{tabular}
\end{table}


\subsection{Demographic and Psychological Profiles of Four Clusters}
We identified four clusters reflecting distinct demographic and psychological profiles (Table \ref{tab:cluster_demo_full}). Based on prior work showing strong depression–anxiety comorbidity \cite{kessler2008co,lamers2011comorbidity}, we included BDI-II subscales to capture shared affective variance that may influence connectivity patterns. Two clear demographic and psychological profiles patterns emerged among the four clusters. The Minimal Responder and Selective tDCS Responder subtypes cluster groups had more female participants and showed higher baseline STAI and BDI-II scores. In contrast, the Robust tDCS Responder and Placebo Responder subtypes were the groups with largest participant numbers, with moderate symptoms and balanced sex ratios. 

\begin{table}[ht]
\centering
\small
\setlength{\tabcolsep}{8pt}
\renewcommand{\arraystretch}{1.25}
\caption{
Demographic, clinical, and connectivity characteristics across the four data-driven clusters.
\textit{n} = number of participants; \textit{SD} = standard deviation.
BDI-II subscales include: Somatic (physical/vegetative symptoms), 
Affective (emotional symptoms), and Cognitive (self-critical or distorted thinking patterns).
}
\label{tab:cluster_demo_full}
\begin{tabular}{lccccc}
\toprule
\textbf{Measure}
    & {\scriptsize \makecell{\textbf{Selective tDCS}\\\textbf{Responder}\\\textbf{Subtype }}}
    & {\scriptsize \makecell{\textbf{Robust tDCS}\\\textbf{Responder}\\\textbf{Subtype }}}
    & {\scriptsize \makecell{\textbf{Placebo}\\\textbf{Responder}\\\textbf{Subtype }}}
    & {\scriptsize \makecell{\textbf{Minimal}\\\textbf{Responder}\\\textbf{Subtype }}}
    \\
\midrule
n                          & 39    & 68    & 60    & 29        \\
Proportion (\%)            & 19.60 & 34.17 & 30.15 & 14.57     \\

Age (Mean\, (SD))                 & 70.49 (3.46)  & 70.24 (4.25)  & 72.30 (4.81)  & 72.62 (5.43)  \\

STAI-State Pre (Mean\, (SD))      & 29.15 (8.87)  & 26.94 (6.98)  & 28.28 (8.62)  & 28.55 (8.20)  \\

STAI-Trait (Mean\, (SD))          & 28.56 (8.41)  & 27.91 (6.28)  & 28.37 (7.57)  & 29.07 (7.21)  \\

BDI Total (Mean\, (SD))           & 2.59 (3.23)   & 2.76 (3.76)   & 3.65 (4.08)   & 4.72 (4.17)   \\

BDI Somatic (Mean\, (SD))         & 1.64 (1.90)   & 1.78 (2.15)   & 2.38 (2.54)   & 3.10 (2.64)   \\

BDI Affective (Mean\, (SD))       & 0.38 (0.75)   & 0.44 (0.84)   & 0.55 (0.98)   & 0.79 (1.01)   \\

BDI Cognitive (Mean\, (SD))       & 0.56 (1.21)   & 0.54 (1.51)   & 0.72 (1.40)   & 0.83 (1.26)   \\

Male (n)                   & 10    & 23    & 22    & 6          \\
Female (n)                 & 29    & 45    & 38    & 23         \\
Male (\%)                  & 25.64 & 33.82 & 36.67 & 20.69      \\
Female (\%)                & 74.36 & 66.18 & 63.33 & 79.31      \\

Mean Red-Box Connectivity  & 0.3241 & 0.2889 & 0.2065 & 0.2365  \\
\bottomrule
\end{tabular}
\end{table}
\newpage
\printbibliography

\end{document}
