## 2 Materials and Methods
### Statistical Analysis  
A generalized linear mixed model (GLMM) approach was employed to evaluate the effect of the shaking exercise device on upper limb functional recovery in patients with chronic stroke [34]. This method was selected due to the repeated measures nature of the data and the inclusion of both fixed and random effects to account for within-subject correlation and between-subject variability. If the GLMM analysis revealed a significant interaction effect between the intervention group and study periods, pairwise comparisons between groups at each period were performed to verify that the groups were significantly different. This would provide insight into the temporal pattern of disability recovery and the potential effectiveness of the vibration exercise device during the intervention period. A stratified analysis was conducted using baseline FMA-UE scores to examine the potential impact of impairment severity on intervention effectiveness. Patients were categorized into the following subgroups based on their initial disability level: (A) severe paralysis: \(\leq\)20, (B) moderate paralysis: 21-45, and (C) mild paralysis: 46-66 [32]. This stratification enabled (1) a comparison of intervention effects among subgroups, (2) an assessment of the vibration exercise device's effectiveness relative to initial upper limb motor impairment, (3) a more precise evaluation of the intervention's impact within homogeneous clinical severity groups, and (4) specific model adjustments for each severity group. The primary outcome variable was the FMA-UE total score, which measures motor function recovery [35]. The fixed effects in the model included the intervention periods, the intervention group, and their interaction. Additionally, the model controlled for several covariates: age, sex, type of diagnosis, affected side, initial FMA score at baseline, and the dose of Botulinum Toxin A treatment. The GLMM can be expressed as follows:  
FMA-UE total = \(\beta_{0}\) + \(\beta_{1}\)\(\times\) periods + \(\beta_{2}\)\(\times\) group + \(\beta_{3}\)\(\times\) (periods \(\times\) group) + \(\beta_{4}\)\(\times\) age + \(\beta_{5}\)  
\(\times\) sex + \(\beta_{6}\)\(\times\) diagnosis + \(\beta_{7}\)\(\times\) affected side + \(\beta_{8}\)\(\times\) BoNT-A dosage + \(u_{\mathrm{i}}\) + \(\epsilon\)  
where \(u_{\mathrm{i}}\) represents the random intercept for each subject, accounting for within-subject correlation, and \(\epsilon\) is the residual error. The model was fitted using the Restricted Maximum Likelihood (REML) method to provide unbiased estimates of the variance components. The significance of fixed effects was evaluated using Wald z-tests. Model assumptions, including normality of residuals and homoscedasticity, were thoroughly checked. The normality of residuals was assessed using the Shapiro-Wilk test,and visual inspections of histogram and Q-Q plots. Homoscedasticity was evaluated using residuals vs. fitted values plots and confirmed by the Breusch-Pagan test. We performed both intent-to-treat (ITT) and per-protocol (PP) analyses. In the ITT analysis, we included all participants initially assigned, regardless of adherence or missing data, to maintain the benefits of randomization. Where data were missing from our study, we applied multiple imputation techniques to estimate and replace the missing values, ensuring comprehensive data analysis. This approach aligns with the recommended practices for handling missing data in clinical research, as detailed by [PERSON] et al., who emphasized the importance of addressing such gaps to maintain the validity of statistical inference [36]. In the PP analysis, we included only participants who completed the study without significant deviations from the protocol to assess the effectiveness of the intervention for those who adhered to the protocol and had complete data. Post hoc comparisons of changes in between-group differences were performed using the Mann-Whitney U-test. In addition, intragroup comparisons were performed using the Wilcoxon signed-rank test.