The gait standard deviation, a single measure of kinematic variability

Gait & Posture 46 (2016) 194–200

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Gait & Posture journal homepage: www.elsevier.com/locate/gaitpost

The gait standard deviation, a single measure of kinematic variability Morgan Sangeux a,b,c,*, Elyse Passmore a,b,c, H. Kerr Graham a,b,d, Oren Tirosh a a

The Royal Children’s Hospital, Melbourne, Australia The Murdoch Childrens Research Institute, Australia c The University of Melbourne, School of Engineering, Australia d The University of Melbourne, Department of Paediatrics, Australia b

A R T I C L E I N F O

A B S T R A C T

Article history: Received 24 December 2015 Received in revised form 17 March 2016 Accepted 21 March 2016

Measurement of gait kinematic variability provides relevant clinical information in certain conditions affecting the neuromotor control of movement. In this article, we present a measure of overall gait kinematic variability, GaitSD, based on combination of waveforms’ standard deviation. The waveform standard deviation is the common numerator in established indices of variability such as Kadaba’s coefficient of multiple correlation or Winter’s waveform coefficient of variation. Gait data were collected on typically developing children aged 6–17 years. Large number of strides was captured for each child, average 45 (SD: 11) for kinematics and 19 (SD: 5) for kinetics. We used a bootstrap procedure to determine the precision of GaitSD as a function of the number of strides processed. We compared the within-subject, stride-to-stride, variability with the, between-subject, variability of the normative pattern. Finally, we investigated the correlation between age and gait kinematic, kinetic and spatio-temporal variability. In typically developing children, the relative precision of GaitSD was 10% as soon as 6 strides were captured. As a comparison, spatio-temporal parameters required 30 strides to reach the same relative precision. The ratio stride-to-stride divided by normative pattern variability was smaller in kinematic variables (the smallest for pelvic tilt, 28%) than in kinetic and spatio-temporal variables (the largest for normalised stride length, 95%). GaitSD had a strong, negative correlation with age. We show that gait consistency may stabilise only at, or after, skeletal maturity. ß 2016 Elsevier B.V. All rights reserved.

Keywords: Gait Variability Typically developing

1. Introduction Clinical gait analysis tends to focus on the shape of the kinematic and kinetic waveforms during a walking stride (e.g. [1]). However, variability of the gait pattern may provide additional, relevant, information about a condition or pre-post an intervention [2]. Mathematical tools to report the variability in kinematic, kinetic or electromyographic (EMG) data exist but there is no tool to summarise overall gait kinematic variability. The aim of this study was to propose and validate such a tool. Research regarding variability in gait analysis data began with the reliability of electromyographic waveforms [3]. Hershler and Milner introduced the variance ratio (VR) to estimate the

* Corresponding author at: The Hugh Williamson Gait Analysis Laboratory, The Royal Children’s Hospital, 50 Flemington Road, Parkville 3052, VIC, Australia. Tel.: +61 3 9345 6792. E-mail address: [email protected] (M. Sangeux). http://dx.doi.org/10.1016/j.gaitpost.2016.03.015 0966-6362/ß 2016 Elsevier B.V. All rights reserved.

repeatability of EMG waveforms over several gait cycles. In [4], Kadaba et al. used the variance ratio for EMG data but later [5] introduced the Coefficient of Multiple Correlation (CMC) to estimate the repeatability of kinematic and kinetic waveforms. In [5], Kadaba et al. did not use VR or CMC to measure variability of EMG data but the waveform coefficient of variation (W-CV) described by Winter [6]. Subsequent research regarding variability in gait waveforms utilised these indices. Dynamic stability is another field of human motion analysis interested in kinematic variability of gait. Researchers developed additional tools such as detrended fluctuation analysis, fractal dynamics or the Lyapounov exponent (e.g. [7,8]). Although related to variability, these tools do not measure variability per se but how well, or how fast, one adapts for the variability during movement. These tools require large number of strides and may not be easily used in the context of clinical gait analysis, where small number of strides, typically 10 or less, is captured during overground walking. The VR, CMC or W-CV indices are all dimensionless ratios. This allows the comparison of variability in data expressed in different

M. Sangeux et al. / Gait & Posture 46 (2016) 194–200

units or waveforms that vary over markedly different amplitudes. For example, Tirosh et al. used the VR to compare confidence in the mean waveforms from different treatment of EMG data and with respect to kinematic and kinetic data [9]. However, ratios cannot be combined to obtain a summary index across multiple variables. VR and CMC are two ways to express the same relationship in the data, and VR and W-CV ratios share the same numerator, the variance around the mean waveform. This variance can be combined across several variables to create a summary index of gait variability, which we will call GaitSD. Most research efforts about gait variability have focused on the reliability of the gait experiment and researchers have mostly been interested in between-session variability (between days, between assessors or both) [10]. The within-session (and intra-subject) variability has been calculated in some studies, but mainly to compare with the variability between-sessions. Intra-subject gait variability per se has been studied in normal adults or children [5,6,11] as well as in populations with various motor control problems: ataxia [12], stroke [13], spastic diplegia [14] or spastic hemiplegia [15], or skeletal problems such as scoliosis [16]. Most of the above studies utilised Kadaba’s CMC or Winter’s W-CV to measure variability of the kinematic and kinetic waveforms. However, the precision of the measurement of variability may depend on the number of strides captured and processed. Researchers used varying number of strides to calculate variability, a minimum of 2 strides was reported in [13], 3 in [5], 4 in [15], 5 in [16], 9 in [6], and 10 in [11,12,14]. What is the precision of the waveforms variability calculated from two strides, and from ten strides? We will address this question and provide reference data for the precision of CMC, W-CV and the newly introduced GaitSD. The definition of gait in the dictionary encompass two concepts. The first refers to the pattern of movement of the limbs that form the manner of walking. The second refers to different pace of forward progression adopted by horses and other animals (e.g. walk, trot, and gallop). In the scientific literature, search results about ‘‘gait variability’’ mostly refer to the second concept, and report the variability of spatio-temporal parameters such as walking speed, cadence and stride length. We will compare kinematic variability with the variability of spatio-temporal parameters. Sutherland et al. have shown that gait pattern may mature as early as age 4 [17]. However, little is known about the consistency of the pattern once it has matured. Does gait consistency continue to improve after the pattern has matured? We will try to answer this question and provide reference data about the kinematic, kinetic and spatio-temporal variability in typically developing children. 2. Material and methods

195

between the sum of the squares of the residuals (SSR) and the number of degrees of freedom of the residuals (DFR):

GVSD2 ¼

SSR ¼ DFR

PT PN j¼1

2 i¼1 ðX ij X¯j Þ

TðN1Þ

The total variance (TV) is the ratio between the total sum of squares (SST) and the total number of degrees of freedom (DFT):

TV ¼

SST ¼ DFT

PT PN

i¼1 ðX ij X¯j Þ

j¼1

2

TN1

with the overall mean X defined by: ¯ 1 XT XN X X¼ j¼1 i¼1 ij ¯ TN Hershler and Milner’s variance ratio VR is the ratio between the variance of the residuals and the total variance, that is 2

VR ¼ GVSD TV . In regression analysis, VR is called the fraction of variance unexplained but it is seldom used because the coefficient of determination is preferred. Kadaba’s CMC was defined as the square root of the coefficient of determination [5]:

CMC 2 ¼ R2adj ¼ 1

GVSD2 ¼ 1VR TV

Hence, VR and CMC measure the same thing, which is how representative of the variance in the data is the variance of the mean waveform across time. However, the variability of the data around the mean waveform is solely described by the variance of the residuals: GVSD2. GVSD2 is also found in Winter’s W-CV [6] where it is divided by the magnitude of the mean waveform. W-CV expresses how large the residuals around the mean waveform are with respect to the magnitude of the waveform. pffiffiffiffiffiffiffiffiffiffiffiffiffiffi GVSD2 W-CV ¼ PT 1 j¼1 jX¯j j T In the remaining of the text we will refer to GVSD2 as the gait variable variance and GVSD as the gait variable standard deviation, hence the acronym. We define GaitSD as the square root of the average variance over V kinematic variables. That is:

2.1. Gait kinematic variability: GaitSD In 1978, Hershler and Milner presented the analogy between the variance ratio and the analysis of variance [3]. If we consider N waveforms X defined over T time samples and a regression model of the data by the mean waveform:

X ij ¼ X j þ 2 ij ¯ with Xij a waveform from the stride i defined over j time samples, X j ¯ the mean of the N waveforms defined for each time instant j: P X j ¼ N1 N i¼1 X ij and 2ij the residuals. ¯ The variance of the residuals, which we will call GVSD2 for later use, is calculated from the mean square of the errors, the ratio

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u V u1 X GaitSD ¼ t GVSD2k V k¼1 GaitSD expresses the standard deviation of the residuals around V mean kinematic waveforms in degrees. We chose the same set of kinematic variables as other index of kinematic normalcy in clinical gait analysis [18,19]. GaitSD for one subject is composed of 15 kinematic variables: pelvic tilt, pelvic obliquity, pelvic rotation, left and right hip flexion, left and right hip abduction, left and right hip rotation, left and right knee flexion, left and right ankle dorsiflexion and left and right foot progression angles. GaitSD for one side is made of 9 kinematic variables: 3 from the pelvis and the 6 from the right or left lower limb.

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2.2. Data collection

3. Results

Gait analysis data were collected on 35 typically developing children as part of a previous study [9]. After a short period of adaptation, the subjects were asked to walk 20 trials at their preferred speed. Data from every full stride were processed. The markerset and modelling followed the Plug-in-Gait (PiG, Vicon, UK) procedure. Kinetic data were normalised by body mass (standard PiG output) and body mass and leg length (nondimensional normalisation, [20]). Spatio-temporal data were normalised by leg length (dimensionless, [21]). Variability of the kinetic data may be larger in part because of calibration of the force plates with the motion capture system. The calibration protocol described in [22] was followed before data collection.

Data was collected on 14 females and 21 males (Subject demographics, Table 1). There were no significant differences in age, height and mass between males and females. The average number of strides collected per subject was 45 (SD: 11) for kinematic and spatio-temporal data and 19 (SD: 5) for kinetic data. Fig. 1A presents the effect of m, the number of strides, on GaitSD. The number of strides m had little effect on mean GaitSD. This was expected given the 2000 random samples and calculation. Mean GaitSD for m = 5 was equal to mean GaitSD for m = 30, trueness was 100%. SD(GaitSD) decreased non-linearly with the number of strides m. The relative precision: SD(GaitSD)/GaitSD was 26% for m = 2, 11% for m = 5 and 4% for m = 30. Comprehensive results on the effect of the number of strides are tabulated in the supplementary materials (tab ‘‘Variability’’). These concern kinematics, kinetics and spatio-temporal variability. Results are provided for GVSD, W-CV and CMC for kinematic and kinetic variables and for the standard deviation for spatiotemporal variables. Variability of spatio-temporal variables required more strides to reach the same relative precision. For example, SD(Walking speedSD)/Walking speedSD for m = 30 was 12% compare to SD(GaitSD)/GaitSD of 11% for m = 5 strides. W-CV and GVSD had similar results and overall required m  15 strides to reach 10% relative precision. CMC required the least amount of strides, m = 2, to reach 10% relative precision for most variables except for foot progression (m = 8) and pelvic tilt (m = 30). Fig. 2 presents the regression fits and equations between the log transform of SD(GaitSD), m and GaitSD. The interaction between GaitSD  m was not significant. Regression lines had different

A 3. 2.5

Average GaitSD (°)

We constructed a normative kinematic and kinetic pattern from the most representative stride of the subjects. The total functional median distance depth, a multivariate measure of centrality, was calculated and the deepest (most representative) kinematic stride was retained [23]. The most representative strides were pooled and the normative pattern variability was calculated for the kinematic variables. For each subject, outlying strides were removed automatically following the procedure described in [23]. The functional median absolute deviation (MAD) was calculated to provide a robust estimate of the standard deviation. Strides that were found distant from the other strides by more than 3 times the MAD (Rscore  3, [23]) were considered outliers and removed from the pool of strides. We then calculated GaitSD, CMC and W-CV. We performed a bootstrap analysis to obtain the precision of the GaitSD depending on the number of strides processed. For each subject, m strides were selected at random, with replacement, from the pool of strides and the GaitSD was calculated. This procedure was repeated 2000 times and performed for m = 2–10, 15 and 30 strides. We hypothesised that the precision of GaitSD, measured by the standard deviation over 2000 repetitions, SD(GaitSD) varies with the number of strides m and GaitSD. We performed a stepwise regression analysis with SD(GaitSD) as the response and m, GaitSD and GaitSD  m as factors. Factors were eliminated forward and backward at a < 0.05. The variability of normalised spatio-temporal parameters was expressed as the standard deviation of the parameter over m strides. The same bootstrap procedure was followed to estimate the precision of the variability for: normalised walking speed, cadence and stride length. For the remaining of the analysis, we used the results for m = 30 strides as reference values. Pearson’s correlation coefficient was calculated to measure the correlation between individual kinematic GVSDs and GaitSD. We hypothesised that gait variability may change with age and sex and performed an ANOVA with GaitSD as the response, sex as a factor and age as a covariate. We calculated Pearson’s correlation coefficient for multiple correlation between the kinematic, kinetic and spatio-temporal variability and age. All processing was performed in Matlab (The Mathworks, USA) and statistical analysis performed in Minitab (Minitab Inc, USA).

2. 1.5 1. 0.5 0. 0

5

10

15

20

25

30

20

25

30

Number of strides m

B 0.06

Average Nwalking speedSD

2.3. Data analysis

0.05 0.04 0.03 0.02 0.01 0.

Table 1 Demographics, anthropometrics and gait kinematics variability for the population. Sex

Male

Female

All

Number Age (years) Height (m) Mass (kg) GaitSD (8)

21 11.1 1.5 39.6 2.4

14 11.8 1.5 41.0 2.1

35 11.4 1.5 40.1 2.3

(3.2) (0.2) (16.3) (0.5)

(3.0) (0.1) (10.4) (0.5)

(3.0) (0.2) (13.9) (0.5)

0

5

10

15

Number of strides m Fig. 1. For each subject, m = [2–10, 15, 30] strides were selected randomly (with replacement) from the pool of strides (mean: 45 SD: 11) and the various measures of variability were calculated from m strides. The process was repeated 2000 times to calculate the average (corresponding to trueness) and standard deviation (corresponding to precision). (A) Average and standard deviation for GaitSD and (B) average and standard deviation for the standard deviation of normalised walking speed (Nwalking speedSD).

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197

Fig. 2. Relationship between the logarithm of the standard deviation of GaitSD over 2000 random samples of m strides (Y, LOG(SD(GaitSD))) and GaitSD (X) and the number of strides m. One data point per subject and regression lines for m = [2–8, 10, 15, 30]. The purpose is to predict the precision of GaitSD given the value of GaitSD and the number of strides collected.

intersects for different m but the same slope coefficient with GaitSD. Correlation between individual kinematic GVSDs and GaitSD was strong, minimum of r = 0.81 for hip rotation and maximum of r = 0.96 for foot progression. The correlation was strong with kinematics in the sagittal plane: r = 0.92, 0.94, 0.90 and 0.88 for pelvic tilt, hip flexion, knee flexion and ankle dorsiflexion respectively. Fig. 3 shows that GaitSD decreased with age (p < 0.001, r = 0.71) but sex was not a significant factor. We hypothesised that GaitSD reaches an asymptotic value with age and a power regression line was fitted to the data (Fig. 2, R2 = 0.53). The results from multiple correlation between age and the kinematic, kinetic 4.

3.5

GaitSD (°)

3.

Female 2.5

Male All

Power (All)

2.

GaitSD = 9.65 Age-0.623 R² = 0.53 1.5

1. 4

6

8

10

12

14

16

18

and spatio-temporal variability are provided in Table 2. Most variables were correlated with age. However, variability of the kinematic variables was more highly correlated with age than kinetic (normalised by body mass) and spatio-temporal variables. Of note, sagittal plane hip and knee kinetic variability was not significantly correlated to age and sagittal plane ankle kinetic variability was just significant at a = 0.05. Kinetic variables after non-dimensional normalisation became highly correlated with age since leg length, a variable highly correlated with age, was factored in the scaling. Table 3 summarises the results for the stride-to-stride variability (S2S), normative pattern variability (NPV) and the ratio S2S/NPV. For kinematic variables, S2S was least for pelvic obliquity and tilt (1.08 and 1.38 respectively) and greatest for knee flexion and foot progression (both 3.08). For kinetic variables, S2S for hip extensor moment (0.110 N/m kg) was greater than for knee and ankle extensor moments (0.085 and 0.092 N/m kg respectively). Gait variables with large S2S/NPV ratio (80%) could be considered not subject specific because variability between strides was comparable to variability between subjects. Variables with large ratio were knee extensor moment (normalised by body mass), knee power, normalised walking speed and stride length. Interestingly, NPV for knee extensor moment after non-dimensional normalisation increased relative to S2S and the ratio decreased by 21%. In all other kinetic variables, non-dimensional normalisation slightly decreased the variability between subjects, between 0% and 8%. Variables with a small S2S/NPV ratio (40%) could be considered subject specific because variability between strides was small compared to variability between subjects. Variables with small ratios were pelvic tilt, hip flexion and hip rotation kinematics. Overall, the ratios were smaller for kinematic variables than for kinetic and spatio-temporal variables.

Age (years) Fig. 3. Relationship between GaitSD and Sex as factor and Age as covariate. Sex was not significant (p = 0.95). Age was significant (p < 0.001). Assuming GaitSD would reach an asymptotic value with age, a power regression line was fitted to the data (R2 = 0.53).

4. Discussion The aim of this study was to propose a tool to summarise gait kinematic variability. We explained the relationship between

M. Sangeux et al. / Gait & Posture 46 (2016) 194–200

198

Table 2 Correlation (Pearson’s r) between age and kinematic and kinetic (normalised by body mass and non-dimensional normalisation) variability (waveform standard deviation, GVSD) and spatio-temporal variability (standard deviation, SD). Only correlation with hip extensor, knee extensor and ankle abductor moments normalised by body mass were not significant (NS).

Kinematics GaitSD (8) GVSD

Kinetics GVSD

Kinetics GVSD

Variables

Pearson’s r

p

Pelvic tilt (8) Pelvic obliquity (8) Pelvic rotation (8) Hip flexion (8) Hip adduction (8) Hip rotation (8) Knee flexion (8) Ankle dorsiflexion (8) Foot progression (8)

0.71 0.59 0.61 0.69 0.65 0.60 0.50 0.66 0.59 0.71

<0.001 <0.001 <0.001 <0.001 <0.001 <0.001 0.002 <0.001 <0.001 <0.001

Normalised by body mass Hip extensor (N m/kg) Hip abductor (N m/kg) Knee extensor (N m/kg) Knee abductor (N m/kg) Ankle plantarflexor (N m/kg) Ankle abductor (N m/kg) Hip power (W/kg) Knee power (W/kg) Ankle power (W/kg)

0.28 0.52 0.26 0.42 0.34 0.06 0.37 0.42 0.34

NS: 0.1 0.002 NS: 0.1 0.01 0.05 NS: 0.7 0.03 0.01 0.05

Non-dimensional normalisation Hip extensor Hip abductor Knee extensor Knee abductor Ankle plantarflexor Ankle abductor Hip power Knee power Ankle power

0.66 0.69 0.66 0.67 0.69 0.42 0.58 0.63 0.62

<0.001 <0.001 <0.001 <0.001 <0.001 0.01 <0.001 <0.001 <0.001

0.51 0.51 0.56

0.002 0.002 <0.001

Spatio-temporal Normalised cadence (step) SD Normalised walking speed Normalised stride length

indices of variability found in the literature and identified the gait variable standard deviation (GVSD) as the common numerator between the indices and the true measure of variability. We introduced an overall measure of gait kinematic variability, GaitSD, as the pooled standard deviation of 15 kinematic variables commonly used in gait normalcy indices [18,19]. We investigated the statistical properties of all indices to report gait variability. Relative precision of GaitSD fell below 10% when it was calculated from 6 strides or more. Relative precision of kinematic and kinetic GVSDs required data from 10 to 15 strides to fall below 10% and relative precision of spatio-temporal variability required more than 30 strides to fall below 10%. These results highlight the increased precision of GaitSD, or the GVSDs, over the variability of spatio-temporal parameters when small number of strides is collected. A non-linear relationship between SD(GaitSD) and GaitSD may be extracted from Fig. 2. For m = 6 strides: SD(GaitSD) = 101.12+0.20GaitSD. Relative precision for GaitSD deteriorates when GaitSD increases. In our study, relative precision was 10% and 7% for m = 6 and 10 strides (GaitSD = 2.38) but increased to 14% and 10% if GaitSD doubles. This has implications when data from a particularly variable subject are compared to reference data from a cohort. When gait variability is the main outcome measure, it is advised that a minimum of 6 strides are collected for patients with low variability and 10 strides for patients with high variability.

Results on precision for W-CV behaved similarly to GVSD with the number of strides m, but that of CMC behaved differently (Supplementary material). This highlights an essential difference between the indices. W-CV is calculated as the variability of the waveforms divided by the magnitude of the mean waveform. Since the magnitude of the mean waveform varies markedly less with m than the waveforms, the precision of W-CV is almost the same as GVSD. CMC is the square root of the coefficient of determination and measures the proportion between the variance of the mean waveform across time and the total variance, from all waveforms and across time. As a result, the precision of CMC depends on the variance of the mean waveform across time. When the variance of the mean waveform across time is small (e.g. for pelvic tilt) data from 30 strides are required to reach 10% relative precision but when it is large (e.g. for knee flexion) data from 2 strides suffice to reach 1% relative precision. Within subject, pelvic obliquity and pelvic tilt had the smallest variability (GVSD of 1.0 and 1.38 respectively) and knee flexion and foot progression had the largest variability (3.1 and 3.28 respectively). However, pelvic tilt was considered the least

Table 3 Stride-to-stride variability (S2S, within subject), normative pattern variability (NPV, between-subjects) and the ratio S2S/NPV in % for kinematics, kinetics (normalised by body mass or non-dimensional normalisation) and spatio-temporal variables. Average over 35 typically developing children. For kinetic data after nondimensional normalisation, the difference in % with kinetics normalised by body mass is specified for S2S/NPV ratio. S2S/NPV ratios 40% and 80% are specified in bold font. The increasing effect of non-dimensional normalisation on NPV for knee extensor moment, leading to a decreased S2S/NPV ratio, is also highlighted in bold font. Stride to stride variability (S2S)

Normative pattern variability (NPV)

Ratio S2S/NPV (%)

Kinematics Pelvic tilt (8) Pelvic obliquity (8) Pelvic rotation (8) Hip flexion (8) Hip adduction (8) Hip rotation (8) Knee flexion (8) Ankle dorsiflexion (8) Foot progression (8)

1.3 1.0 2.6 2.2 1.5 1.8 3.2 2.2 3.2

4.7 1.7 3.3 6.4 2.3 4.9 5.4 3.8 4.9

28% 60% 79% 34% 66% 37% 59% 58% 65%

Kinetics, normalised by body mass Hip extensor moment (N m/kg) Hip abductor moment (N m/kg) Knee extensor moment (N m/kg) Knee abductor moment (N m/kg) Ankle plantarflexor moment (N m/kg) Ankle abductor moment (N m/kg) Hip power (W/kg) Knee power (W/kg) Ankle power (W/kg)

0.11 0.078 0.085 0.054 0.095 0.029 0.20 0.24 0.34

0.15 0.11 0.11 0.092 0.15 0.056 0.27 0.30 0.48

72% 70% 80% 59% 61% 52% 75% 81% 72%

Kinetics, non-dimensional normalisation Hip extensor moment Hip abductor moment Knee extensor moment Knee abductor moment Ankle plantarflexor moment Ankle abductor moment Hip power Knee power Ankle power

0.015 0.011 0.012 0.0075 0.013 0.0040 0.0075 0.0093 0.013

0.020 0.014 0.020 0.012 0.020 0.0076 0.010 0.011 0.018

74% + 2% 78% + 8% 60%  21% 62% + 3% 66% + 5% 52% + 0% 75% + 1% 81% + 0% 72% + 0%

Spatio-temporal Normalised cadence (step) Normalised walking speed Normalised stride length

0.028 0.038 0.091

0.045 0.047 0.096

62% 80% 95%

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repeatable by CMC (0.48) since the variance of pelvic tilt across time is small and pelvic obliquity was considered the least repeatable by W-CV (0.44) since the mean magnitude of pelvic obliquity is close to zero. On the contrary, knee flexion, in which variance across time is large and mean magnitude is also large, was considered second most repeatable by CMC (0.99) and W-CV (0.14). W-CV ranked the ankle plantarflexor moment as least repeatable (0.22) but both GVSD (0.095 N/m kg) and CMC (0.99) ranked the ankle plantarflexor moment as among the most repeatable (all data in supplementary material, tab ‘‘Reference data m = 3000 ). These results further highlight the difference between the indices and the difficulties in interpreting their results. In the end, only the gait variable standard deviation truly measures variability and we advocate its use to report kinematic or kinetic variability. In some applications, it may be impossible to collect data from the whole lower limb but only from one joint and in one plane. In this instance, foot progression angle or any of the sagittal plane kinematic variables would be appropriate as they had strong correlation with GaitSD (r > 0.88). We compared the stride-to-stride (S2S) variability and the variability of the normative pattern (NPV) in a population of typically developing children. We found that S2S/NPV was smaller for kinematic variables compared to kinetic and spatio-temporal variables. Three kinematic variables, pelvic tilt, hip flexion and hip rotation had the smallest ratios with 27, 33 and 37% respectively. It is however difficult to interpret these results fully because our design could not differentiate the between-session (of the same subject) and the between-subject effects. Results published in the literature shows that a large part of NPV may be attributable to test–retest of the gait experiment (between-session). A systematic review by McGinley et al. in 2009 compiled the test–retest results (in ‘‘Fig. 1: Summary of gait studies reporting 3DGA reliability as S.D. or S.E.’’, [10]). For hip rotation, our results for NPV resemble the average results from the review and it may be that a large part of the variability may be attributable to the test–retest error. Our results for pelvic tilt and hip flexion NPV are larger than that of the review so it may be that test–retest error is less prominent for these variables. On a side note, the studies included in the review often calculated the waveform’s standard deviation by averaging the standard deviation at each time instant. This is incorrect since standard deviations are not additive (but the variances are) and introduces a small bias in the results. We did not differentiate the variability of the mean values of the waveforms (offset variability) and the variability of the shapes of the waveforms (waveform pattern variability) and these two components are included in our measure of kinematics variability. It may be beneficial to analyse separately the offset and waveform pattern variability when the two components may be linked to different sources of variability. For example, between sessions of the same subject, the offset variability may be predominantly related to factors that are extrinsic to the subject, e.g. the repeatability of marker placement. However, the focus of our study was stride-to-stride variability and its comparison with variability of the normative pattern. Stride-tostride variability is obtained within-session where both components of variability relate to factors intrinsic to the subject. The normative pattern variability is obtained between subjects where offset variability relates to factors that are both intrinsic (a person’s choice of mean posture during walking, e.g. mean pelvic tilt) and extrinsic (repeatability/accuracy of marker placement, e.g. positioning of the anterior and posterior superior iliac spine markers on the pelvis). Because of the equivocal nature of offset variability between subjects, we decided to keep the analysis consistent and to report the overall kinematics variability.

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There was a strong, negative correlation between GaitSD and age (p < 0.001, r = 0.71). A power regression line was fitted to the data (GaitSD = 9.65Age0.623, R2 = 0.53) and predicts GaitSD of 3.28 at 6 years of age and almost half of that, GaitSD = 1.78, at 16 years of age. Even though the gait pattern may have matured as early as 4 years of age [17], it seems the consistency of the pattern continues to increase during childhood and adolescence. A limitation of our study is the narrow age range (6–17.5 years). This study aimed at providing reference data in a paediatric population and we found retrospectively that the age range was too young to determine when gait kinematic variability stops decreasing with age. One study, utilising a similar gait protocol and statistical analysis, provided data in five adults (range 21–33 years old) with no prior conditions affecting gait [24]. We calculated a GaitSD of 1.78 from the data reported in the manuscript (in ‘‘Table 2. inter-trial kinematic variation (SD) across intervals’’). Given that GaitSD at 16 years was 1.78 in our study, gait kinematic variability may not decrease after 16 years old and coincide with skeletal maturity. Interestingly, the correlation between GaitSD and age was the strongest (r = 0.71) compared to all the other kinematic, kinetic or spatio-temporal variables. Some kinetic variables were not significantly correlated with age when normalised by body mass but were after nondimensional normalisation. This is explained because nondimensional normalisation introduces leg length, a variable highly correlated with age, as a scaling factor. Correlation between kinetic variability and age were similar to that of kinematics after nondimensional normalisation. 5. Conclusion GaitSD summarises a subject’s kinematic variability during walking in a single number. The main clinical application of GaitSD may be to help understand different conditions affecting gait or the effect on kinematic variability pre-/post-intervention [2]. Our results showed that GaitSD reached a satisfying precision (SD(GaitSD)/GaitSD  10%) as soon as 6 strides were processed. Comparatively, single kinematic or kinetic variables required 10– 15 strides and spatio-temporal parameters required more than 30 strides. GaitSD was the most sensitive to changes with age and decreased until skeletal maturity. Acknowledgement This work has been funded through a Clinical Science theme grant from the Murdoch Childrens Research Institute. We would like to acknowledge Tandy Hasting-Ison, Jill Rodda and Pam Thomason senior physiotherapists at the Hugh Williamson Gait Analysis Laboratory, for their participation to data collection. Adrienne Fosang, senior physiotherapist at The Royal Children’s Hospital, coordinated patient recruitment and participated to data collection. She is gratefully acknowledged. Conflict of interest statement Each author certifies that he or she has no commercial associations that might pose a conflict of interest in connection with the article. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.gaitpost.2016. 03.015.

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