# An Agent-Specific Stochastic Model of Generalized Reaching Task Difficulty
## 4 Statistical Analysis on the Index of Difficulty \(ID_{agent}\) for Differently Aged Subjects  
The walking trajectories of 34 subjects have been analyzed at the slow, comfortable, and fast speed levels. The subjects are grouped into \"Young\" (21-37 years, N = 20) and \"Old\" (50-73 years, N = 14) age groups. Each subject performs an average of nine walking trials per speed level. The second stride of each trial is sampled to avoid non-steady gait effects and averaged for each subject and each speed condition. The stochastic model of task difficulty \(ID_{agent}(x,\ \varphi)\) has been evaluated for the considered motor task, demonstrating the proposed agent-specific index's capability of capturing statistical differences in compared populations. The cumulative index \(ID_{agent}(x,\ \varphi)\) profiles for each subject are shown for the \"Young\" and \"Old\" groups at perceived walking speed levels \"Slow\" and \"Fast\" (Figure 6). Values refer to probability level \(\varphi=0.96\) (i.e., \(z=2.133\)). Figure 5: Cumulative Index of Difficulty \(ID_{agent}(x,\varphi=0.96)\) and local affordance limits \(W(x,\varphi=0.96)\) evaluated at each point of the ankle path during a forward stride of the swing foot, from flat foot to heel strike. The specific agent is a 61-year-old healthy subject walking at an average speed of 1.584 m/s. In both speed conditions, the elderly population shows a steeper increase in the Index of Difficulty during the first part of the trajectory, compared to the young population; this is a consequence of lower values of \(W(x,\varphi)\) for old subjects, who show a smaller adaptation capability and, therefore, a lower affordance during the pre-swing phase of the trajectory. Similarly, greater \(ID_{agent}(\overline{t},\varphi)\) values for the overall trajectory are obtained for the elderly subjects. The average \(ID_{agent}(x,\varphi)\) profiles have been calculated for the categories \"Young\" and \"Old\", at \"Slow\" and \"Fast\" speed levels (Figure 7). At each \(x\) value, a higher speed leads to higher \(ID_{agent}(x,\varphi)\) average values for both young and old subjects; higher speed values also lead to greater stride length. A mixed-design analysis of variance (ANOVA) is conducted to assess the effects of age and speed on the Index of Difficulty of each trajectory \(ID_{agent}(\overline{t},\varphi)\). The age group is defined as the \"between-subjects\" factor and the speed condition is defined as the \"within-subjects\" factor. Figure 6: Index of Difficulty \(ID_{agent}(x,\varphi=0.96)\) calculated along the x coordinate of the swing foot trajectory in “Young” (red lines, \(N=20\)) and “Old” (blue lines, \(N=14\)) subjects, for walking trials at “Slow” (**a**) and “Fast” (**b**) speed. Figure 7: Average \(ID_{agent}(x,\varphi=0.96)\) observed for “Young” (continuous lines) and “Old” (dotted lines) at “Slow” (red lines) and “Fast” (black lines) speed conditions. Prior to the ANOVA, both between-subjects and within-subjects assumptions are checked. All subjects are independent from each other. The Shapiro-Wilk test is used to determine whether each subset of data (according to the categorization from Table 1) is approximately normal; the older subjects within the slow condition did not meet this assumption (\(p=0.018\)). Aside from this subset, the remainder of the dataset is approximately normally distributed. The ANOVA is typically robust to violations of normality, and therefore the analysis was pursued even though one subset of the data did not meet the assumption. The variances are tested using the Levene's test, where it is confirmed that the variances among both age groups for each speed condition are approximately similar (Slow vs. Age: \(p=0.789>0.05\); Fast vs. Age: \(p=0.701>0.05\)). The within-subjects assumptions are also met: each subject participated in both the slow and fast speed conditions, the total number of subjects (\(N=34\)) is large enough to assume normality, and sphericity can be assumed since there are only two conditions.