## 2 The Data
### Statistical Analyses  
In the statistical context, entropy measures complexity between signal data or time series (TS) that links the amount of information to a probability distribution [16; 17]. One option for analyzing and modeling the entire TS is to apply summary statistics, for example, the processor average. We have previously outlined a dose-response model testing the intensity and polarity-dependent effect of HD-tDCS, and compared the effect of anodal and cathodal stimulation polarity at different intensities (1 mA, 2 mA, and 3 mA) in VV, electroencephalogram (EEG), and WBA [12]. Furthermore, an entropy-based study was performed and discussed following the same protocol using only the EEG results in a deeper discussion, whereas significant statistical results comparing Cathodal vs. Anodal montage in the intensity of 2 mA was noticeable [14]. Nonetheless, a discussion about the different responses across  
Figure 1: Illustration of the WBA study protocol. The left-hand side figure exhibits the platforms. In the middle, an illustration of a participant positioned on this equipment. The right-hand side shows the position of the high definition transcranial stimulation (HD-tDCS) electrodes. the Sham-Cathodal and Sham-Anodal continues to be a gap when it comes to tDCS clinical studies [18]. We adopted the Transfer Entropy (TE) in the data, which considers a measurement of directional information transfer among two random processes, and it is always positive. For instance, considering two random processes, \(W1_{t}\) and \(W2_{t}\), indexed in the time \(t\in N\). Through the Shannon entropy, H(. ), and considering \(L\) the length of an embedded vector or history length (underlying state of the Markov process, further information, please see [19]), the TE is calculated as:  
\[TE_{W1\to W2}=H(W2_{t}|W2_{t-1:t-L})-H(W2_{t}|W2_{t-1:t-L},W1_{t-1:t-L}). \tag{1}\]  
The time series process was analyzed using entropy index for all data acquired towards the complexity of the Fz component of the platforms after baseline or the stimulation protocol application, as shown in the data pre-processing procedure in Figure 2.  
We evaluated the effect of HD-tDCS applied in the TPJ on postural control, observing the intensity for each combination of condition and intensity. Thus, using the transfer entropy [15; 20] as an entropy measure has enabled to encompass whether the past state of one Fz signal could improve the prediction of the other Fz signal on each platform (right-side to the left, and vice-versa), addressing the causal inference among the Fz components. Moreover, with the transfer entropy, one measures the causal direction and enables to test such statement, and if existent, one can compare whether a dominant side impacts the other or reverse. Therefore, we sought to compare the TS causality summarized across the different moments, using Mixed-Effect Models (considering random intercept for each individual that can present a personal characteristic). Furthermore, as a Longitudinal study (in which the same individual is observed multiple times and on different days), to distinguish between stimulus types versus intensity and to quantify the differences in regularity between the platforms. As the hypotheses were defined in advance, we used a global test between comparison treatments complexity. A significance level of 5% was considered in all tests. Statistical analyses were performed using R software for Statistical Computing and Analysis. The descriptive results of the figures were presented as the difference from the baseline. We described Transfer Entropy (TE) and Generalized Linear Mixed-Effect Models (GLMM). This study adopted the implemented TE function from the _RTransferEntropy_ package, and the implemented GLMM from the _lme4_ and _lmerTest_ packages to test the linear contrasts of fixed effects (estimate the \(p\)-values), all in R software [21].