# Estimation of the Abduction/Adduction Movement of the Metacarpophalangeal Joint of the Thumb
## 3 Results  
### Experiment A. Prediction Model  
Figure 3 shows the statistics of the data recorded in Experiment A. All joint angles presented wide ranges across postures, covering their reported range of motion [9; 17]. Note that CMC abduction/adduction, MCP flexion/extension and MCP abduction/adduction angles presented more negative than positive values, i.e., the postures recorded needed the thumb more abducted and the MCP extended. Figure 4 shows the correlations between the thumb joint angles for all the postures (660) performed in Experiment A. MCP abduction/adduction joint movement presented the highest correlation of values and was positively correlated with CMC abduction/adduction (0.68) and with MCP flexion/extension (0.27) joint movement. Conversely, MCP abduction/adduction joint movement was negatively correlated with CMC flexion/extension (\(-\)0.30) joint movement. A negative correlation between IP and CMC joints was also present. The MCP flexion/extension appeared to be poorly coordinated with the rest of the thumb joints. Figure 3: Left: Box plot with angles recorded across postures. The horizontal marks in the center of the boxes are the median; the edges of the boxes are the 25 th and 75 th percentile; whiskers extend to 1.5 times the interquartile range. Right: range (maximum minus minimum) and average values (\(\upmu\)) of each joint movement. Positive values represent flexion and adduction angles; negative values represent extension and abduction angles. CMC abduction/adduction (CMCA) and flexion/extension (CMCF); MCP abduction/adduction (MCPA) and flexion/extension (MCPF); and IP flexion/extension (IPF). CMC abduction/adduction is the only joint movement added to the regression model as an independent variable. Interaction between CMC flexion/extension and CMC abduction/adduction is also present in the model. Table 2 shows the coefficients for the prediction model to estimate the MCP abduction/adduction angles. _p_-value indicate that all the coefficients are significant (_p_-value < 0.05). Equation (1) presents the regression equation obtained for the estimation of the MCP abduction/adduction joint movement (with joint angles in degrees). \[\text{MCPA}_{\text{est}}=\ -5.825+\ 0.575\times\text{CMCA}_{\text{i}}+\ 0.015\times\text{CMCF}_{\text{i}}\times\text{CMCA}_{\text{i}} \tag{1}\]  
Figure 5 shows the interaction effect between CMC flexion/extension and CMC abduction/adduction. In particular, it shows that the relationship between CMC flexion/extension and MCP abduction/adduction changes direction based on the CMC abduction/adduction values. The curves of the effect estimated with the other predictor fixed at extreme values are not parallel, thus corroborating interactions between the predictors: for \begin{table}
\begin{tabular}{c c c c c} \hline  & **Coefficient** & **SE** & **tStat** & _p_**-Value** \\ \hline Intercept & \(-5.825\) & 0.428 & \(-13.622\) & 0.000 \\ CMCA & 0.575 & 0.021 & 27.836 & 0.000 \\ CMCF \(\times\) CMCA & 0.015 & 0.001 & 10.584 & 0.000 \\ \hline \end{tabular}
\end{table} 
Table 2: Coefficients, standard error of the coefficients (SE), t-statistic for each coefficient (tStat) and p-value for the t-statistic of the hypothesis test whether the corresponding coefficient, obtained for MCP abduction/adduction estimation, is equal to zero or not (_p_-value). CMCA: CMC abduction/adduction; CMC: CMC flexion/extension. Figure 4: Correlation coefficients between thumb joints movements (CMC abduction/adduction (CMCA) and flexion/extension (CMCF), MCP abduction/adduction (MCPA) and flexion/extension (MCPF) and IP flexion/extension (IPF)) for all the postures performed. high positive values of CMC abduction/adduction, there is a positive relationship between MCP abduction/adduction and CMC flexion/extension, while for high negative values the relationship is negative. The RMSE of the prediction model is 6.29\({}^{\circ}\). Figure 6 shows the RMSE and the recorded versus the estimated MCP abduction/adduction angles, differentiating between grasp types.