Variability in gait and sample sizes required for an accurate

Variability, or the spread or dispersion of a set of data, is typically measured by range, interquartile range, standard deviation and coefficient of variance (Vincent, 1999). Inter-individual variability, or between-group variability, is typically used as a measure of group homogeneity by reporting, for example, group mean and standard deviations. Intra-individual variability is also an important parameter that, in the case of gait kinematics, gives an indication of locomotor control and dynamic stability. Intra- individual variability is less often reported and has traditionally been examined by measuring fluctuations in certain gait parameters including temporal and spatial kinematics, kinetics and electromyography. Researchers have attempted to correlate variability of these parameters with control and consistency of the locomotor system, and with falling behaviour. This section discusses variability as:

2. findings of measures of intra-individual variability and their relationship with falls; and

3. sample size required for accurate representation of intra-individual variability and selection of appropriate measure of intra-individual variability.

The major focus of this research is on intra-individual as opposed to inter-individual variability and, therefore, variability will be referred to as either intra- or inter- individual variability throughout the thesis to easily distinguish between the two.

Increased intra-individual variability of various walking parameters has been equated with instability of the locomotor system and has been characterised as a predictor for falling by several researchers (e.g. Hausdorff et al., 2001; Hausdorff et al., 1997; Maki, 1997; Winter, 1991a; Gabell and Nayak, 1984; Guimaraes and Isaacs, 1980). Given the association with falling behaviour, several researchers have examined intra- individual variability of gait parameters in various elderly population groups and have attempted to explain any differences in relation to falls risk. Intra-individual variability in gait parameters, typically spatial and temporal parameters of step kinematics such as step length, step width, stride frequency and step/stride time, stance and swing phase time, has been reported for young subjects and elderly with and without a history of falls (Buzzi et al., 2003; Danion et al., 2003; Hausdorff et al., 2001; Guimaraes and Isaacs, 1980) or neuropathic elderly (Richardson et al., 2005; Dingwell and Cavanagh, 2001; Dingwell and Cusumano, 2000).

In general, the elderly compared with young have shown greater intra-individual variability in most, but not all, measured gait parameters (refer section 2.2.2.1). It has

been suggested that the increased intra-individual variability in elderly populations may be due to the reduced walking velocity and not to some pathological cause (e.g. Dingwell and Marin, 2006; Dingwell and Cavanagh, 2001; Dingwell and Cusumano, 2000) but not all researchers have found this (e.g. Owings and Grabiner, 2004b). These studies did, however, have some methodological differences. For example, Owings and Grabiner (2004b) examined treadmill walking during a subject-determined comfortable walking speed (normal) and then imposed a slower walking speed calculated as 90% of normal for slow walking speed. Dingwell and Cusumano (2000) collected overground walking data at a subject-determined comfortable walking speed and attempted to associate subjects’ different walking speeds with intra-individual variability for each individual. These studies show that important parameters such as walking speed need to be probed further in order to examine the influence on measures such as intra-individual variability.

Whilst many studies examining intra-individual variability in various temporal and spatial kinematic variables can be found, examination of the important variable of MTC is seldom reported. Given the need for a precise foot trajectory to prevent tripping, MTC could be described as a more sensitive measure than basic gait parameters such as step width, step length and time spent in stance and swing phase. As discussed in section 2.2.2.1, few studies have examined intra-individual variability of the foot trajectory (e.g. James, 1999; Winter, 1991a). The methodological approach utilised by James (1999) was, however, fundamentally different to that of the study by Winter (1991a). James (1999) analysed consecutive strides over a period of at least 30 minutes while Winter (1991a) analysed less than 10 non-consecutive strides. While neither study found significant age effects, the elderly in Winter’s study tended to have smaller

intra-individual variability compared with the young, while the elderly in James’ study tended to have greater intra-individual variability compared with young. There is a paucity of studies examining intra-individual variability in MTC and there is no clear definition of intra-individual variability in this sense. There is clearly a need for examination of intra-individual variability in the critical measure of foot clearance during continuous strides.

Intra-individual variability in locomotor patterns has traditionally been explained as solely noise in the system. For example, as Newell and Corcos (1993) assert, random intra-individual variability is inherent in all biological systems and the challenge for researchers is to “understand how order and regularity arise in the co-ordination and control of movement with noise (random fluctuations) as an inherent component to the system” (Newell and Corcos, 1993, p.4). However, the notion of ‘random’ variability proposed by Newell and Corcos has not been supported by all researchers. For example, Dingwell and Cusumano (2000) and Buzzi et al. (2003) proposed the stride- to-stride variability in human gait was not random but instead displayed a deterministic behaviour. They stated these fluctuations appeared to be chaotic and may be partly controlled by deterministic central nervous system processes. Further, Dingwell and Cusumano (2000) concluded that long-term and short-term patterns often underpin variability in long-term gait. Further research is warranted to determine the nature and origin of stride-to-stride variability.

An important consideration when examining intra-individual variability is the sample size studied. Most methods of examining intra-individual variability include a few strides that can be averaged to generate mean ensemble curves and in the process some

of the intra-individual variability in gait patterns is difficult to identify (Buzzi et al., 2003). As Owings and Grabiner (2004b) recognised, the accuracy of intra-individual variability estimates is proportional to the amount of data collected. Other studies have acknowledged this consideration and utilised larger sample sizes. For example, Hausdorff et al. (2001) used 6 minutes of overground walking data while Owings and Grabiner (2003) analysed 10 minutes of treadmill walking data. These studies identified greater intra-individual variability in some gait measures for elderly vs. young (Owings and Grabiner, 2004b; 2004a) and elderly fallers vs. non-fallers (Hausdorff et al., 2001). Owings and Grabiner (2004b) compared spatial and temporal step kinematics, namely step length, step width and step time, between healthy young and elderly subjects. While variability was greater in the elderly compared with young on all measures, significant difference was only found for step width variability (young = 2.1cm, n = 18 vs. elderly = 2.5, n = 12, p=.037). In the study by Hausdorff et al. (2001), increased variability in stride time and swing time was observed in elderly fallers compared with elderly non-fallers. Stride time variability for elderly fallers was 106ms (n = 20) vs. 49ms for elderly non-fallers (n = 32), p=.04.

Owings and Grabiner (2004b; 2004a; 2003) and Hausdorff et al. (2001) also deleted points defined as greater than 3.77 standard deviations from the mean (Owings and Grabiner, 2004b; 2004a; 2003) and 3 standard deviations from the median (Hausdorff et al., 2001) due to the fact they were deemed to be ‘extreme’. The study by Hausdorff et al. was the first to employ this method, while Owings and Grabiner based their method on Hausdorff et al.’s work. Hausdorff et al. (2001) excluded the first 10 seconds of data from each data set to minimise any start-up effect. Next, they deleted data points ±3 standard deviations from the median. The only other information

supplied stated each subject typically had several hundred strides. Owings and Grabiner (2004b; 2004a; 2003) sequentially sorted the data to obtain mean and standard deviation of the middle 90% of data. Next, any data from the original, unsorted series (i.e. 100% of the data) that was ±3.77 standard deviations, described as a conservative estimate of 3 standard deviations of 100% of the data points, was removed. Since each subject would vary in number of steps due to differing step frequency and the number of steps eliminated, each subject’s data set was truncated to match the subject with the least number of steps. No further information was provided on deleted data points and it is therefore not known how many and from which subjects data was deleted. It is also not known how many steps were analysed for each subject. The intra-individual variability measured may therefore have been underestimated and deleting extreme points may misrepresent the differences between the groups.

Owings and Grabiner (2003) examined the validity of measuring intra-individual variability in various gait parameters with respect to the size of the data set. They found that a minimum of 400 strides was required in order to obtain an accurate estimation of step kinematic intra-individual variability. It is important to know how many strides are necessary for accurate calculation of intra-individual variability data. One extreme point can substantially increase the intra-individual variability in the measured gait parameters, particularly in small data sets. A representation of the intra- individual variability in MTC descriptive statistics can be gained by observing the ‘stability’ of each statistic throughout the gait cycle. Stability of MTC descriptive statistics is derived by plotting each statistic with the addition of each new MTC data point. Recent research using this method has shown that MTC outliers can

substantially influence the stability of MTC descriptive statistics (Best et al., 2000; James, 1999).

Figure 2.13 gives a representation of the stability of four MTC descriptive statistics, namely mean (M), standard deviation (SD), skew (S) and kurtosis (K), for 1382 strides for one healthy young female during a 30 minute treadmill walking period (James, 1999). Descriptive statistics of the data removed series for this subject for M, SD, S and K were 1.00cm, 0.23cm, 0.17 and –0.12, respectively. It can be seen in Figure 2.13 that a large MTC occurred at stride 132 and was in fact the largest MTC for the entire data set at 2.15cm, an increase of 1.14cm or 114% from the mean MTC of 1.00cm.

Figure 2.13: Stability of descriptive statistics.

Data shown for one young female subject during 1382 consecutive strides of a 30 minute treadmill walking period. Descriptive statistics for each data set: a) data removed series (1382 strides) - M = 1.00cm, SD = 0.23cm, S = 0.17, K = -0.12; and b) raw series (1385 strides) - M = 1.00cm, SD = 0.24cm,

S = 0.30, K = 0.43.

The largest MTC in the data series (2.15cm) occurred at stride number 132. Adapted from James (1999)

SD stability 0.16 0.18 0.2 0.22 0.24 0.26 1 10 1 20 1 30 1 40 1 50 1 60 1 70 1 80 1 90 1 10 01 11 01 12 01 13 01 Stride number MT C S D / c m raw data removed Stride 132 Mean stability 1 1.05 1.1 1.15 1.2 1 10 1 20 1 30 1 40 1 50 1 60 1 70 1 80 1 90 1 100 1 110 1 120 1 130 1 Stride number M T C Me an / c m raw data removed Stride 132 Kurtosis stability -1 0 1 2 3 1 10 1 20 1 30 1 40 1 50 1 60 1 70 1 80 1 90 1 100 1 110 1 120 1 130 1 Stride number MT C K u rt o s is raw data removed Stride 132 Skew stability -0.2 0.2 0.6 1 1 10 1 20 1 30 1 40 1 50 1 60 1 70 1 80 1 90 1 10 01 11 01 12 01 13 01 Stride number MT C S k ew raw data removed Stride 132

The large MTC at stride 132 caused increases of M, SD, S and K of 0.7% (1.13 to 1.14cm), 8.5% (0.21 to 0.23cm), 1677% (0.04 to 0.65) and 1315% (0.18 to 2.52), respectively. These figures demonstrate that one extreme outlier substantially influences descriptive statistics and therefore support the need for large data sets in order to obtain data representative for the individual.

Best et al. (2000) used this method of examining stability of descriptive statistics for 2,766 strides for one healthy young male adult during a 60-minute treadmill walking period. An unusual block of 12 strides was identified and, on closer examination, it was discovered that this unusual block contained 12 of the 20 most extreme data points in the MTC distribution. It was thought that the subject might have been distracted during this short period. The influence of this ‘distracted’ block, which occurred at stride 824, approximately one third of the way through the walking trial, resulted in increases in SD of 12% and K of 50%. Additionally, a single MTC value of 2.55cm which occurred at stride 1751, slightly over half way through the walking trial, caused increases in M, SD, S and K of 0.1%, 1%, 10% and 20% respectively. It took K a further 250 strides to re-stabilise after this single extreme outlier.

The walking trial analysed by Best et al. (2000) did not intentionally include distractions, however, it is expected that an individual would be distracted at times for many different reasons and the data shows that such distractions have a substantial impact on MTC. The extreme MTC achieved during the distracted period might indicate areas of increased tripping risk or could be the individual’s response to prevent a tripping incident during distracted attention. These findings indicate a need for closer examination of the effect of distracted walking. Consistent with the data presented in

Figure 2.13, the study by Best et al. (1999) highlights the important finding that extreme MTC have a substantial influence on descriptive statistics and support the need for larger data sets for accurate analysis.

Studies examining only a few strides (e.g. Winter, 1991a) make the assumption that these trials form a normal distribution and represent typical gait characteristics. Table 2.2 and Figure 2.14 present the range of MTC descriptive statistics obtained by dividing a 60-minute walking period (3,318 strides) for one healthy young adult female into different time intervals (James, 1999). For example, M, SD, S, and K were calculated for 332 x 10 strides, 120 x 30s intervals of the 60 minutes of data, 60 x 1 minute intervals, 30 x 2 minute intervals, 12 x 5 minute intervals, 6 x 10 minute intervals, 4 x 15 minute intervals, 3 x 20 minute intervals, 2 x 30 minute intervals and 1 x 60 minute interval. The median values of each set of intervals are shown together with the range of values for each set of time/stride interval. Minimum and maximum values are shown in Figure 2.14.

Table 2.2: Range of MTC descriptive statistics in different time intervals, n = 3318 continuous strides for one healthy young female (adapted from James, 1999).

Note: descriptive statistics given are for MTC in various time/stride intervals (sd = standard deviation, s = skew, k = kurtosis). Median (med) and range of the 5 descriptive statistics are shown. Actual value for mean, median, sd, s and k are shown for 1 x 60 minute interval.

med range med range med range med range med range

332x10strides 10 0.96 1.09 0.94 1.12 0.21 0.38 0.25 5.50 0.03 10.72 120 x 30s 28 0.95 0.86 0.94 0.87 0.23 0.25 0.39 3.26 0.09 10.38 60 x 1min 55 0.94 0.73 0.95 0.73 0.24 0.16 0.36 2.01 0.21 6.56 30 x 2min 111 0.95 0.57 0.94 0.58 0.25 0.09 0.40 1.36 0.30 4.66 12x5min 277 0.94 0.39 0.93 0.43 0.27 0.08 0.34 0.72 0.33 2.75 6x10min 553 0.94 0.28 0.93 0.29 0.27 0.04 0.41 0.66 0.34 2.12 4x15min 830 0.93 0.30 0.92 0.32 0.27 0.03 0.35 0.52 0.35 1.63 3x20min 1106 0.94 0.23 0.93 0.23 0.27 0.01 0.42 0.15 0.30 0.81 2x30min 1659 0.96 0.20 0.95 0.20 0.27 0.02 0.39 0.25 0.39 0.66 1x60min 3318 sd s k 0.96 0.94 0.29 0.36 0.15 Time/stride interval Strides/ interval mean median

Descriptive statistics for the entire 60-minute walking period were M = 0.96cm, SD = 0.29cm, S = 0.36 and K = 0.15. As Table 2.2 shows, the range of values is substantially greater in the smaller intervals, particularly the 10-stride intervals. In general, range of values becomes progressively smaller as the time intervals increase to include greater number of strides (i.e. inter-interval intra-individual variability in descriptive statistics decreases). The two largest time intervals, i.e. 3 x 20 minute intervals and 2 x 30 minute intervals, have the smallest range of values for each of the five descriptive statistics.

Figure 2.14 and Table 2.2 clearly shows the greater range of each descriptive statistic is within the smallest time intervals, i.e. 232 x 10 strides and 120 x 30 seconds.

Figure 2.14: Comparison of median, minimum and maximum descriptive statistic values (mean, SD, skew and kurtosis) for various time/stride intervals (adapted from James, 1999).

MTC Me an 0.4 0.6 0.8 1.0 1.2 1.4 1.6 T im e / s t ride int e rv a l MTC SD 0.0 0.1 0.2 0.3 0.4 0.5 T im e / s t ride int e rv a l MTC Skew -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 T im e / s t ride int e rv a l MTC Kurtos is -3.0 0.0 3.0 6.0 9.0 T im e / s t ride int e rv a l

Conversely, it can be seen that the range of each descriptive statistic decreases as the time interval increases. This demonstrates that inter-interval variability in descriptive statistics is substantially greater in small sample sizes. Small sample sizes, therefore, may not be sufficient to achieve stable values and may not be representative of the gait for the individual.

The greater range of values within each time/stride interval in the smaller set of intervals, i.e. the 332 x 10 strides and 120 x 30 second intervals, also has implications for the normality of each data set within the set of intervals. For example, S during the 332 x 10 stride intervals range from –2.84 to 2.66 and K ranges from –2.06 to 8.66 while ranges for M and SD are 0.48 to 1.48cm and 0.08 to 0.46cm, respectively. Given that S and K for a normal distribution are zero, small data sets with greater inter- interval variability have the potential for greater deviation from the assumptions of a normal distribution. In data sets that are not normally distributed, measures of standard deviation, and indeed mean as a measure of central tendency, may not be appropriate.