## 3 Results
### Analysis of Secondary Outcomes and Data Imputation  
An intention-to-treat analysis was performed for secondary outcomes. The collection of MAS, ROM, and VAS data for the eight patients at one participating clinic was not feasible owing to the constraints imposed by the Japanese insurance system. To overcome this limitation and ensure the continuity and completeness of our dataset for secondary outcome analysis, missing MAS, ROM, and VAS data were imputed using multiple techniques. \begin{table}
\begin{tabular}{l|c|c|c|c|c|c} \hline \hline
**Variable** & \multicolumn{2}{c}{**Severe (\(n=34\))**} & \multicolumn{2}{c}{**Moderate (\(n=50\))**} & \multicolumn{2}{c}{**Mild (\(n=9\))**} \\ \cline{2-7}  & **Shaking** & **Control** & **Shaking** & **Control** & **Shaking** & **Control** \\ \hline Sex (Female/Male) & 8/5 & 13/8 & 16/14 & 13/7 & 3/2 & 4/0 \\ Age & 66 [52; 74] & 59 [54; 70] & 56.5 [53; 66.5] & 56.5 [52.75, 64] & 58 [57; 61] & 69 [58; 73] \\ Diagnosis (CI:ICH; Other) & 7:6:0 & 7:13:1 & 8:3:2 & 8:11:1 & 1:4:0 & 2:1:1 \\ Affected Side (L/R) & 9/4 & 11/10 & 15/15 & 7/13 & 3/2 & 1/3 \\ Month of Onset & 167 [94; 248] & 138 [92; 176] & 134 [103; 174] & 116 [80; 169] & 122 [110; 137] & 88 [79; 136] \\ FMA-UE Total & 15 [11; 18] & 11 [10; 18] & 30.5 [24.25, 34.75] & 26 [23; 37.5] & 49 [49; 52] & 50.5 [49; 52] \\ BoNT-A treatment (times) & 14 [11; 23] & 13 [10; 17] & 15 [10; 21] & 10 [6; 15] & 18 [14; 27] & 14 [6; 22] \\ BoNT-A (unit) & 275 [200; 300] & 250 [250], 400 & 250 [200; 400] & 200 [197.5, 250] & 250 [100; 250] & 175 [138; 250] \\ \hline \hline \end{tabular} Data are presented as median [1 st, 3 rd quartile], CI; cerebral infarction, ICH; intra cerebral hemorrhage, L; left, R; right, FMA-UE; Fugl-Meyer assessment of the upper extremity, BoNT-A; botulinum toxin type A. Pink; Shaking group, Light blue; Controls.  
\end{table} 
Table 4: Baseline data for shaking exercise and control groups for upper limbs of patients with stroke according to severity. \begin{table}
\begin{tabular}{l l c c c c c c c} \hline \hline
**Severity** & **Group** & **Pre** & **Post** & **Delta** & **U** & \(p\)** & **r** \\ \hline \multirow{3}{*}{Severe (\(n=34\))} & Shaking & 15.0 & [11.0, 18.0] & 16.0 & [12.5, 18.3] & 1.0 & [\(-\)0.25, 1.0] & 16.00 & 0.22 & 0.43 \\  & Control & 11.0 & [10.0, 18.0] & 13.0 & [9.75, 17.0] & 0.0 & [0.0, 1.25] & 21.50 & 0.09 & 0.00 \\ \hline \multirow{3}{*}{Moderate (\(n=50\))} & Shaking & 30.5 & [24.3, 34.8] & 33.5 & [29.0, 37.5] & 1.0 & [0.0, 4.0] & 59.50 & 0.01 & * & 0.25 \\  & Control & 26.0 & [23.0, 37.5] & 27.0 & [23.0, 37.3] & 0.0 & [0.0, 1.0] & 25.50 & 0.50 & 0.00 \\ \hline \multirow{3}{*}{Mild (\(n=9\))} & Shaking & 49.0 & [49.0, 52.0] & 51.5 & [47.7, 57.5] & 2.0 & [0.5, 3.5] & 1.50 & 0.25 & 0.77 \\  & Control & 50.5 & [49.0, 52.0] & 51.0 & [51.0, 52.0] & 1.5 & [\(-\)0.5, 2.75] & 3.50 & 0.63 & 0.36 \\ \hline \hline \end{tabular} The severity levels are divided into Severe, Moderate, and Mild based on initial FMA scores. Shaking refers to the intervention group using the jgggling device, and Control refers to the control group. The change in FMA score (delta) from pre to post intervention. The Wilcoxon signed-rank test for comparing the changes within the Shaking and Control groups. Data are presented as medians and 25 th–75 th percentile values. * \(p<0.05\) for comparison. Pink; Shaking group, Light blue; Controls.  
\end{table} 
Table 5: Stratified analysis results of FMA-UE score changes in stroke patients: comparison of shaking device intervention and control groups by severity level. To examine the effects of the intervention on chronic pain using the VAS, Wilcoxon signed-rank tests were performed to compare the pre- and post-intervention scores within each group (Table 6). In the control group, a significant difference in the VAS scores was observed from pre- to post-intervention (W = 23.00, \(p\) = 0.02, [PERSON]'s d = 0.00). Conversely, no significant changes were observed in the intervention group (W = 8.00, \(p\) = 0.60, [PERSON]'s d = 0.00).