To what extent does not wearing shoes affect the local dynamic stability of the gait? Effect size and intra-session repeatability

Local dynamic stability (LDS) quantifies how a system responds to small perturbations. Several experimental and clinical findings have highlighted the association between gait LDS and fall risk. Walking without shoes is known to slightly modify gait parameters. Barefoot walking (BW) may cause unusual sensory feedback to individuals accustomed to shod walking (SW), and this may impact on LDS. The objective of this study was therefore to compare the LDS of SW and BW in healthy individuals and to analyze the intrasession repeatability. Forty participants traversed a 70 m indoor corridor wearing normal shoes in one trial and walking barefoot in a second trial. Trunk accelerations were recorded with a 3D-accelerometer attached to the lower back. The LDS was computed using the finite-time maximal Lyapunov exponent method. Absolute agreement between the forward and backward paths was estimated with the intraclass correlation coefficient (ICC). BW did not significantly modify the LDS as compared to SW (average standardized effect size: +0.12). The intrasession repeatability was high in SW (ICC: 0.73-0.79) and slightly higher in BW (ICC: 0.82-0.88). Therefore, it seems that BW can be used to evaluate LDS without introducing bias as compared to SW, and with a sufficient reliability.


Introduction
Although bipedal locomotion is inherently unstable, 1 human beings exhibit a high resilience to external perturbations that could lead to falls. 2,3 However, many pathologies and conditions may alter the capabilities to efficiently manage obstacles and perturbations while walking. In particular, fall-related injuries are a major health issue in elderly people. 4,5 Analytical methods derived from nonlinear analysis of dynamical systems have been proposed to analyze gait stability and related fallrisks. 6 Following the nomenclature on stability theory and nonlinear dynamical systems, the largest perturbation that individuals can tolerate without falling is referred to as "global stability" 7 or "stability margins." 8 Inside those margins, motor control continuously adjusts gait parameters to compensate for infinitesimal perturbations induced, for instance, by neuromuscular noise. This is referred to as "local dynamic stability" (LDS). 9 LDS can be characterized using the maximal Lyapunov exponent, which is a parameter that assesses how infinitesimal perturbations grow over time (butterfly effect) or, in other words, how fast the system diverges. 6,[9][10][11] Rosenstein et al. 12 proposed a practical method for calculating maximal Lyapunov exponents from small data sets. With this method, local divergence exponents (λ) are computed from the slopes of divergence curves, which quantify how fast the neighboring trajectories of a reconstructed state space diverge from nearest neighbor points ("initial perturbation"). Strictly speaking, due to the nonlinearity of the divergence curves, multiple slopes could be defined. Hence, no true single maximal Lyapunov exponent exists. Different slopes (divergence exponents) quantify local divergence (and hence local stability) at different time scales. Classically, two different time scales have been proposed to assess gait LDS: long-term divergence λ l 6 (long-term LDS) and short-term divergence λ s 7 (short-term LDS). The former is based on the time interval between four and 10 strides after the initial perturbation, and the latter is based over a time interval corresponding to one stride (or one step) after the initial perturbation. Authors first proposed to use longterm LDS to assess gait stability. 6 However, recent developments in the field have shown that short-term LDS is a more responsive index. 13,14 Consequently, throughout the article, the generic LDS term is used to refer to short-term LDS, and the time scale reference is used when necessary.
Although there are no theoretical reasons why local stability should predict global stability, 13

Procedure
The participants were instructed to walk straight ahead at a self-selected comfortable walking speed along a 70 m hallway and then to do a U-turn and return.
The walking surface was hard, corresponding to standard hospital flooring. The participants wore their own shoes. They were instructed to wear shoes in which they felt quite comfortable, but no high heels were allowed. They performed one trial (2 ×70 m) with shoes and one trial barefoot. The sequence between the BW and the SW trials was randomized. Trunk accelerations were recorded with a tri-axial accelerometer (Physilog system, BioAGM, Switzerland), which was attached to the lower back (over the spine, L3-L4 level) with a belt and connected to a lightweight data logger (

Data Analysis
Following graphical inspection, U-turns were discarded from the raw acceleration signals, and two segments of steady gait were selected (one for the forward path and one for the backward path). The step frequency (SF, Hz) was computed by fast Fourier transform of the raw acceleration signals. Eighty steps were then selected. To improve the normalization of the data, the 3D acceleration signals pertaining to the 80 steps were time-normalized at 8,000 samples (resampling), ensuring a constant sample length.
The method for quantifying the LDS from divergence exponents has been described in detail in numerous articles. 6,7,11,35 Interested readers will find a thorough theoretical background in the review by Dingwell. 9 Here, only the parameters necessary to reproduce the results are summarized. The state space was reconstructed according to Takens' theorem, as classically applied in gait dynamics studies. 6  Rosenstein's algorithm. 12 Time was normalized by the average stride time (1/SF) in each trial, taking into account the resampling. The divergence exponents were computed over a time scale corresponding to one step (0.5 stride, short-term LDS λ s ) 7, 13 and over the fourth to the 10th strides (long-term LDS λ l ). 6

Statistics
To analyze the footwear effect, the results of both the forward and backward paths were averaged together. Notched boxplots (median and quartiles) were used to describe the data ( Fig. 1 and 2). The means and standard deviations (SD) are also presented, including the average change and the corresponding SD (BW minus SW).
The coefficient of variation (CV = SD/mean × 100) was used to assess interindividual variability.
The standardized effect size (ES) is reported using Hedges's g, 36 which is a variant method of Cohen's d for inferential measures. The difference between SW and BW was the contrast measure, and the standardizer was the pooled standard deviation (s p ). Ninety-five percent CIs were estimated by bootstrapping. 34 The results are presented ( Fig. 3) with the arbitrary limits for small (0.2), medium (0.5) and large (0.8) ES. 37 To minimize type I error risk induced by the numerous comparisons, the analysis was completed with a multivariate comparison test (Hotelling's T-squared).
The null hypothesis (H0) was that the mean differences (BW-SW) along the three axes were equal to zero.
To explore whether BW induced a kinematics change that would explain the change in LDS, the correlation (Perason's r) between the change in cadence (SF) and the change in LDS (BW minus SW) was assessed, with corresponding 95% CIs (asymptotic estimates).
Both the intraclass correlation coeffecient (ICC) and the standard error of measurement (SEM) were used to characterize the intrasession reliability. 38 Both forward and backward paths were separately analyzed as two intrasubject repetitions.
The approach was that proposed by McGraw and Wong, 39 based on the classical work of Shrout and Fleiss. 40 The ICC(1) model was used, which assesses the degree of absolute agreement among measurements made on randomly selected objects (one-way model). 39 This approach was justified by the fact that the study was focused on the intrasession reliability evaluated in two trials, which were consecutively measured with no changes in the measurement method. The agreement between the two repetitions was separately analyzed under the two conditions (SW and BW). The 95% CIs on the ICC were computed using traditional F statistics. 39 SEM is the group-level estimation of the intrasubject average variability (expected trial-to-trial noise in the data). It was computed with the following equation: , where S T is the grand SD, (i.e., the SD of the pooled data of the two repetitions), and R is the ICC result. To facilitate the comparison among the parameters, the CV (i.e., SEM/grand mean × 100) was also computed.
Finally, the Spearman-Brown prophecy formula was used to predict the number of strides necessary to achieve high reliability (i.e. R A * = 0.90), taking into account that a normalized number of 40 strides was tested for repeatability. The formula was as follows: where N is the estimated number of trials needed to achieve the expected level of repeatability R A * given the observed repeatability R A . The number of trials N was converted to the corresponding number of strides. An example of this approach applied to the field of gait analysis can be found in the work of Hollman and others. 41

Results
Regarding the cadence results, there was a small increase in BW condition compared to the SW condition ( Fig. 1). In other words, individuals tended to walk at a higher step rate when they walked barefoot. The absolute effect corresponded to 2.4 steps·min -1 (+2%). Compared to the average SD among the individuals (0.13), the SD of the difference (0.04) was small, indicating a high homogeneity of the response to BW among the participants (relative agreement between BW and SW, Pearson's r = 0.92). The interindividual variability (CV) was 4.3% (SW) and 5.7% (BW).
Regarding the ES results (Fig. 3) Regarding reliability results, a high repeatability was present under both conditions regarding cadence (SF). On the contrary, poor repeatability was observed for long-term LDS, with the ICC ranging from 0.22 to 0.63. There were also high within-subject errors (SEM CV: 23-29%). The repeatability was higher for shortterm LDS, with the ICC ranging from 0.74 to 0.87 and the SEM CV ranging from 6% to 8%. BW walking induced a more consistent LDS compared to SW.
The analysis of the ICC results using the Spearman-Brown prophecy formula (Table 1) confirmed the low reliability of long-term LDS (211-1276 strides to reach 90% reliability) and the sufficient reliability of short-term LDS (54-126 strides).

Discussion
In 40 15 reported an effect on LDS of -11%. In another study, treadmill walking significantly increased LDS compared to overground walking by +9%. 11 The effect of aging, defined as the relative difference between young and older adults, on LDS has been found to be about -70% 49 or -40%. 50 Although omnibus testing (T 2 , p = 0.11) revealed no significant modification of long-term LDS, it should be taken into account that the low observed repeatability ( Fig. 4) greatly increased the risk of type II statistical error, which is also highlighted by the large CIs (Fig. 3). Therefore, the existence of an effect at the population level cannot be excluded. Namely, the average ES (Fig. 3, -0.36), the significant changes along the ML and AP directions, and the average relative change (Fig. 2, -11%), which was in the range of the reported change in the literature appear to suggest that individuals exhibit more local stability in BW. Many studies have shown that longterm LDS is poorly related to actual fall risk. 13,14,17 However, other authors have suggested that enhanced long-term LDS could be related to compensatory mechanisms under destabilizing situations 15 or to a more cautious gait. 24 Those factors may constitute a valid explanation for the results in the present study.
Unlike long-term LDS, short-term LDS exhibited high repeatability. Thus, a lower risk of type II error is expected. Short-term LDS exhibited no significant change in multivariate testing, but the results obtained were close to the 5% significance level (T 2 , p = 0.06). This is mainly due to the small (ES: 0.32, relative change: +5%) but significant destabilizing effect (higher λ s ) that was observed along the vertical axis. As compared to the results of the other studies (see above), 11,15,44 the effect is probably of limited relevance. Conversely, no change in the ML and the AP directions (average ES: 0.03) were observed. The spread of the CIs excludes with high confidence that a substantial effect exists at the population level.
As step duration is a highly controlled parameter, which exhibits low strideto-stride variability even during long duration walking, 47  They reported that at least 3 min were necessary to reach good repeatability (ICC >0.75) for short-term LDS, whereas long-term LDS ICC leveled-off around 0.6.
They observed that in 1 min walking, the ICC was around 0.45 for short-term LDS and around 0.30 for long-term LDS. Recently, using a similar method to that employed in the present study, van Schooten et al. 33 analyzed both intra-and intersession repeatability during long-duration outdoor walking (500 m). Only shortterm LDS was analyzed. They found that the ICC was around 0.8 for intrasession repeatability. Although the current study used shorter walking tests, we found higher repeatability than Kang's study and similar results as in van Schooten's study. On the other hand, we confirmed that long-term LDS exhibited large intraindividual variability (CV SEM 23-30%), which severely compromises its use at the individual level.
From the recent literature, it has become increasingly clear that short-term rather than long-term LDS is the most appropriate parameter to assess global stability and fall risk. 16,17,52 Furthermore, the importance of lateral LDS has been emphasized. 16 The present study showed that this parameter was not modified in BW. Therefore, healthy individuals seem able to maintain optimal dynamic stability, even when faced with the unusual situation of walking without shoes, despite the fact that they probably adopt a slightly more cautious gait. 26 Furthermore, high intrasession repeatability was observed in the present study. Consequently, short duration walking tests might be appropriate to assess gait stability, even to measure differences between conditions at the individual level. Fifty-four strides could be sufficient to reach 90% reliability (       The Spearman-Brown prophecy formula was used with the ICC presented in Figure   4 as an input. SW, shod walking; BW, barefoot walking; SF, step frequency; LDS, local dynamic stability; ML, medio-lateral; V, vertical; AP, antero-posterior.