# Movement Time for Pointing Tasks in Real and Augmented Reality Environments
## 3 Results
### Regression Models  
Regression analyses were performed for each of the gender and handedness conditions for both horizontal and vertical movement tasks using Equation (5). The regression modeling was based on all the MT data using the W\({}_{\mathrm{ave}}\), distance, and ID in Table 2. The results are shown in Table 3. The \(R^{2}\) values of the models developed were between 0.73 and 0.79. All the VIF values of \(b\), \(c\), and \(d\) were 1, indicating that there was no correlation between any two of the independent variables.  
Figure 5: MT of female (**a**) and male (**b**) participants performing the horizontal pointing task.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline  & & & **Regression** & \multicolumn{2}{c}{**Coefficient**} \\ \hline
**Gender** & **Hand** & \(a\) & \(b\) & \(c\) & \(d\) & \(R^{2}\) \\ \hline Horizontal & Movement & & & & & \\ Female & D & 633.8 & 134.4 & 118.7 * & \(-\)970.1 & 0.76 \\  & ND & 660.6 & 144.1 & 109.8 * & \(-\)1025.9 & 0.73 \\ Male & D & 365.7 & 141.6 & 227.3 & \(-\)846.8 & 0.75 \\  & ND & 454.8 & 143.4 & 217.7 & \(-\)910.1 & 0.77 \\ Vertical & Movement & & & & & \\ Female & D & 1028.6 & 66.8 & 68.0 * & \(-\)855.6 & 0.74 \\  & ND & 1051.4 & 74.5 & 85.7 * & \(-\)908.5 & 0.73 \\ Male & D & 878 & 62.8 & 133.5 & \(-\)755.2 & 0.79 \\  & ND & 924.3 & 67.6 & 143.8 & \(-\)803.4 & 0.78 \\ \hline \end{tabular} Note: D: dominant; ND: nondominant. All the regression coefficients, except those marked with * and *, were significant at \(p<0.001\) for a two-tailed \(t\) test of \(\mu=0\). * \(p<0.05\); * \(p>0.05\).  
\end{table}
Table 3: Results of regression modeling of the MT.  
An IDadj was defined using the following equation to incorporate the components of both the inclined angle and \(T\):  
\[\text{ID}_{\text{adj}}=\text{ID}+c^{\prime}\text{ sin}\theta+d^{\prime}\text{ }T, \tag{6}\]  
where \(c^{\prime}\) and \(d^{\prime}\) are constants to be determined.  
Both \(c^{\prime}\) and \(d^{\prime}\) may be calculated by dividing \(c\) and \(d\), respectively, by the \(b\) in Table 3. Table 4 shows the \(c^{\prime}\), \(d^{\prime}\), and \(1/b\). Equation (5) may then be rewritten as follows:  
\[\text{MT}=a+b\text{ ID}_{\text{adj}}, \tag{7}\]  
where \(a\) and \(b\) are constants to be determined.  
Figure 6: MT of female (**a**) and male (**b**) participants performing the vertical pointing task.  
Equation (6) shows that ID may be adjusted because of the incline angle and T. By substituting the \(c^{\prime}\) and \(d^{\prime}\) in Table 4 into Equation (6), we obtained the corresponding ID\({}_{\text{adj}}\). An increase in ID (ID\({}_{\text{increase}}\)) using the following equation may be defined:  
\[\text{ID}_{\text{increase}}\left(\%\right)=\ \left(\frac{ID_{adj}-ID}{ID} \right)\times 100\%. \tag{8}\]  
ANOVA results indicated that both \(T\) and movement direction significantly affected the ID\({}_{\text{increase}}\) (\(p<0.0001\)). The ID\({}_{\text{increase}}\) of the virtual condition (197.5%) was significantly higher than that (25.3%) of the physical condition. The ID\({}_{\text{increase}}\) of the vertical movement (159.4%) was significantly higher than that (63.4%) of the horizontal movement. These facts imply that \(T\) is the dominant factor affecting the increase of ID, and the incline angle was also a factor to be considered. The results indicate that incorporating the \(T\) factor increased the difficulty of the pointing tasks more for the vertical movements than for the horizontal movements.