ORIGINAL ARTICLE

Statistical Considerations When Assessing Short Latency Stretch Reflexes in the Human Soleus Muscle Asger Roer Pedersen, Peter William Stubbs, and Jørgen Feldbæk Nielsen Aarhus University The aim was to investigate trial-by-trial response characteristics in the short-latency stretch reflex (SSR). Fourteen dorsiflexion stretches were applied to the ankle joint with a precontracted soleus muscle on 2 days. The magnitude and variability of trial-by-trial responses of the SSR were assessed. The SSR was log-normally distributed and variance heterogeneous between subjects. For some subjects, the magnitude and variance differed between days and stretches. As velocity increased, variance heterogeneity tended to decrease and response magnitude increased. The current study demonstrates the need to assess trial-by-trial response characteristics and not averaged curves. Moreover, it provides an analysis of SSR characteristics accounting for log-normally distributed and variance heterogeneous trial-by-trial responses. Keywords: Human, stretch reflex, short latency reflex, variability, statistics, distribution.

When assessing reflex responses during sitting, standing and walking following an external perturbation/stimulation, the electromyography (EMG) responses of a number of trials are often averaged and the response magnitude is calculated and expressed in various ways (see Pedersen, Stubbs, & Nielsen (2013) for references). Once the responses have been quantified, conclusions are based on the magnitude or changes in the magnitude of these values. Assessing the magnitude of responses using averaged responses was challenged by Pedersen et al. (2013). When assessing short-latency crossed spinal inhibitory responses, Pedersen et al. (2013) noted that commonly used methods of averaging trials ignored statistical assumptions by concealing the characteristics of trial-by-trial responses. Pedersen et al. (2013) observed that trial-by-trial root mean square responses (when expressed as a percentage of the baseline) were log-normally distributed and variance heterogeneous between subjects. This has important consequences for the statistical analysis of data, which conventionally assumes normally distributed and The authors are with the Hammel Neurorehabilitation Hospital and Research Center, Aarhus University, Hammel, Denmark. Address author correspondence to Asger Roer Pedersen at [email protected] 253

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variance homogeneous data. The log-normal distribution is characterized by two (population) parameters, μ and σ2, and defined by its relation to the normal distribution. A random variable Y follows the log-normal distribution with parameters μ and σ2 if the natural logarithm of Y, log(Y), follows the normal distribution with mean μ and variance σ2. The variance parameter σ2 is called the variance on log-scale. The center of the log-normal distribution is given by the population median, exp(μ), whereas the population mean is given by exp(μ + 0.5σ2). This implies that the mean of a log-normal sample overestimates the distribution center, exp(μ), because it estimates exp(μ + 0.5σ2), which is larger than exp(μ) and moreover depends on the variance on log-scale, σ2. Hence, standard statistical analyses of EMG data may produce erroneous results because the response magnitude is overestimated and patterns of variance heterogeneity may appear as patterns of response magnitude. With a log-normal sample, the distribution center, exp(μ), can be estimated by the geometric mean (the exponential of the arithmetic mean of log-transformed values), and based on that, Pedersen et al. (2013) suggested statistical methods for analyzing data that respects the log-normal and variance heterogeneous trial-bytrial distributions. Particularly, Pedersen et al. (2013) demonstrated that standard statistical analysis of the LogG statistic (the logarithm of the geometric mean of trials) could be applied. Pedersen et al. (2013) noted that ignoring the log-normal trial-by trial distribution resulted in an understatement of the response (less prominent inhibition). Moreover, between subject variance heterogeneity erroneously appeared as changes in the response magnitude. As acknowledged by the authors, the conclusions made were applicable for inhibitory responses and facilitatory responses may show different trial-by-trial response characteristics. The short-latency stretch reflex (SSR) is a commonly investigated facilitatory response that can be elicited in a number of muscles over a number of joints during a number of tasks following a rapid stretch of the muscle (see Pierrot-Deseilligny & Burke, 2005; Zehr & Stein, 1999 for overview). At the ankle joint this can be elicited during sitting (Toft, Sinkjær, & Andreassen, 1989; Toft, Sinkjær, Andreassen, & Larsen, 1991) and walking (Andersen & Sinkjaer, 1999; Grey, Ladouceur, Andersen, Nielsen, & Sinkjaer, 2001; Sinkjaer, Andersen & Larsen, 1996a; Yang, Stein, & James, 1991) in the soleus (SOL) (Sinkjaer et al., 1996a; Grey et al., 2001; Toft et al., 1991; Yang et al., 1991) and tibialis anterior (Toft et al., 1989) following dorsiflexion and plantarflexion stretches, respectively. At the ankle, stretch reflexes have been investigated in patient groups with stroke (Nielsen, Andersen, Barbeau, & Sinkjaer, 1998; Lamontagne, Stephenson, & Fung, 2007), multiple sclerosis (Grey, Klinge, Crone, Lorentzen, Biering-Sorensen, Ravnborg, & Nielsen, 2008; Nielsen, Andersen, & Sinkjaer, 2000; Nielsen, Bech, Gadeberg, & Sinkjaer, 2004; Sinkjaer, Andersen, & Nielsen, 1996b; Toft, Sinkjær, Andreassen, & Hansen, 1993) and spinal cord injury (Grey et al., 2008) with impairments such as spasticity [see Nielsen, Crone, and Hultborn (2007) for overview]. Pedersen et al. (2013) demonstrated that standard statistical analysis of the averaged SSR response may produce erroneous results because the trial-by-trial distribution and variance heterogeneity are not accounted for. Therefore it is important to investigate trial-by-trial response magnitudes of SSR (facilitatory) responses to ascertain if the same violations that occurred with the short latency crossed inhibitory response could have occurred with this response. MC Vol. 19, No. 4, 2015

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The first aim of the current study was to determine the population distribution of the SOL SSR trial-by-trial responses for a number of different stretch velocity/ amplitude combinations. The second aim of the study was to determine if the magnitude and trial-bytrial variability were maintained over 2 days for the different stretches. Previous studies have noted that the magnitude of responses is maintained between days for stretches to the biceps brachii (Segal & Wolf, 1993) and tendon tap reflexes to the biceps brachii, triceps brachii, quadriceps femoris and triceps surae (Stam & van Crevel, 1989). Although it is maintained if administered over repeated sessions, the magnitude of the stretch reflex has a tendency to decrease (Segal & Wolf, 1993; Stam & van Crevel, 1989). The trial-by-trial variability of the stretch reflex is also significantly different between sessions within the same day and between days for the biceps brachii muscle (Segal & Wolf, 1993). Stam and van Crevel (1989) noted that the trial-by-trial distribution of magnitudes was right-skewed (nonsymmetric distribution with mean larger than median) for tendon jerk reflexes, although it was difficult to ascertain if the subsequent analysis accounted for this distribution. While these studies examined the magnitude and trial-by-trial variation of responses between days, the trial-by-trial distribution did not appear to be considered. If the trial-by-trial distribution and variance heterogeneity of responses are considered, and the LogG analysis is applied (Pedersen et al., 2013), it is possible that the magnitude and trial-by-trial variability will show different characteristics on day 1 and day 2. The third aim was to determine if there were differences in the trial-by-trial variability of responses between subjects and within subjects over different stretches. Investigating tendon jerk reflexes, Stam and van Crevel (1989) noted ‘large variations of reflex amplitudes between subjects, and within subject’s from tap to tap after correction for stimulus strength’. In addition, using electrical stimulation, Pedersen et al. (2013) found significant differences in trial-by-trial variances between subjects. Therefore it is hypothesized that trial-by-trial variability between subjects will be significantly different. However, the trial-by-trial variability between stretches within subjects may be similar. Although Stam and van Crevel (1989) noted that within subject trial-by-trial variability was significantly different between subjects, they still analyzed data by standard statistical methods that assume normal trial-by-trial distribution and variance homogeneity. Pedersen et al. (2013), assessing the short latency crossed inhibitory response and using electrical stimulation of different intensities, noted that within subjects, regardless of the stimulation intensity, trial-by-trial response variability was conserved between stimulation intensity. Therefore it is possible that the within subject trial-by-trial variability over the different stretches will be conserved.

Methods Subjects Fourteen subjects [five males (36%); mean age 43.4 ± 14.2 (SD) years] participated in the study. All subjects were neurologically healthy and provided written informed consent. Approval was given by the Scientific Ethics Committee of Mid-Jutland (approval number: 20110194) and conformed to the standards of the Declaration of Helsinki. MC Vol. 19, No. 4, 2015

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Apparatus, Instrumentation and Experimental Setup Surface electrodes (20 mm Blue Sensor Ag/AgCl, AMBU A/S) recorded the EMG activity of the right SOL muscle.The electrodes were placed in accordance with the recommendations of Cram, Kasman, and Holtz (1998). All data were sampled at a frequency of 4 kHz. The EMG signals were amplified, band pass filtered at 10 Hz–1 kHz, full wave rectified and smoothed with a 1st order, 40 Hz low pass (Butterworth) filter. In all experiments, subjects were seated, the hip and knee were positioned at an angle of 100° and the ankle was positioned at 110°. Both feet were strapped firmly to two separate footplates aligned parallel to each other. The footplate was attached to and driven by a motor [95DSE9230 Dutymax motor (2.6 kW) and DB Digitax AC-servo amplifier (3 × 380 V, IP20, 8.5 A/17A) and a plenary transmission (Harmonic Drive HPGP36 I = 12:1)], which provides a peak torque of 331 Nm of the output shaft of the gear. Therefore the motor is capable of rotating the footplate at a range of amplitudes and velocities. The angular position of the footplate was measured using a goniometer. As the foot was firmly attached to the footplate the rotation of the ankle was deemed to be the same as the rotation of the footplate.

Protocol The subjects were tested on 2 days spaced 1–14 days apart (mean: 4 days). The same protocol was performed on the 2 testing days. On both days, subjects were asked to perform a maximum voluntary contraction (MVC) of the SOL of the dominant leg (one left and 13 right sided dominant). For the MVC, subjects were instructed to push down and perform a maximal isometric contraction of their SOL so that the resulting force was due to the contraction of the SOL. The subjects were asked to avoid contraction of proximal leg muscles and cocontraction of the tibialis anterior. The EMG activity of the SOL was displayed on a screen in front of the subjects. For testing, 14 dorsiflexion stretches were imposed to the ankle joint. The average velocity of the stretches was approximately 20 deg/s at an amplitude of 4.5 deg, 8 deg and 12 deg, 60 deg/s at an amplitude of 4.5 deg, 8 deg and 12 deg, 100 deg/s at an amplitude of 4.5 deg, 8 deg and 12 deg, 140deg/s at an amplitude of 8 deg and 12 deg, 180 deg/s at an amplitude of 8 and 12 deg, and 200 deg/s at an amplitude of 12 deg. Stretches were imposed every 4–6 s for 30 stretches per amplitude/ velocity combination (for a total of 450 stretches). Two subjects performed only 24 stretches per amplitude/velocity combination. During the experiment, subjects contracted the SOL from 5–15% of the MVC indicated by arrows on a screen in front of the subject. The instructions to the subjects were to maintain the required SOL EMG at all times during testing. Subjects rested as required (0–3 × during testing) and were able to pause the experiment at any stage if they reported fatigue.

Measurements Recorded The onset and time of peak were recorded. The onset of the SSR was defined as the time of the first major deflection following the onset of the stretch. The SSR peak was defined as the first response peak in the rectified and filtered average EMG trace 25 ms following the onset of the stretch. For some subjects, a 1st order MC Vol. 19, No. 4, 2015

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100 Hz low pass (Butterworth) filter was used to better identify the first response peak. The magnitude of the reflex was defined as the area under the curve in a 20 ms period around the response peak. The baseline was defined as the area under the curve for the 20 ms period, 10 ms before the onset of the stretch.

Statistical Analysis A paired t test compared the onset time and time of peak for day 1 vs. day 2. One way ANOVA compared the onset time and time of peak for each investigated amplitude/velocity combination. Downloaded by Boston University on 09/25/16, Volume 19, Article Number 4

Determining Trial-by-Trial Distributions and the Appropriate Summary of Trials.

Following Pedersen et al. (2013), the trial-by-trial distributions we examined through the following steps: (i) Determining the trial-by-trial distribution of absolute baseline and response magnitudes by investigation of quantile plots and histograms for each of the 308 trial samples. Testing for normality and log-normality separately for each sample by means of the Shapiro-Wilk test (test for log-normality was done by applying the Shapiro-Wilk test to log-transformed values). (ii) Based on the appropriate summary of trials derived from (i), i.e., summarizing trials by the arithmetic/geometric mean in case of normal/log-normal trial-by-trial distributions, a Bland-Altman plot of the summaries was used to determine whether the stretch intervention induces an additive or multiplicative effect on the baseline value (additive: normalize each absolute response magnitude by subtracting the baseline summary of the sample; multiplicative: normalize each absolute response magnitude by expressing it as a fraction of the baseline summary of the sample). Log-normal trial-by-trial distributions and a multiplicative effect of the stretch intervention would justify the LogG analysis proposed by Pedersen et al. (2013). In that case, the normalized response magnitudes (relative response magnitudes) were summarized for each sample by the LogG statistic (the logarithm of the geometric mean of the relative response magnitudes) and the sample estimate of the variance on log-scale (sample variance of log-transformed relative response magnitudes) to perform statistical analysis of, respectively, the relative response magnitude and the trial-by-trial variance. Analysis of Relative Response Magnitudes. As the applicable velocities differed

between amplitudes, data were analyzed separately for each amplitude for the effects of velocity, day and subject. The LogG statistic was analyzed by two different linear mixed models characterized by different handling of subject effects: (i) One considered fixed subject effects, thus providing results specifically for the involved subjects, whereas (ii) A second analysis assumed random subject effects, thus providing results for the mean across subjects within the population represented by the specific subjects. In both analyses, the starting point was a statistical model with subject specific linear relationships between velocity as the independent variable and the LogG statistic as the dependent variable, where both intercept and slope were allowed to vary between days and subjects. This corresponded to log-linear relationships between velocity and the relative response magnitude. The models were then simplified by removing statistically nonsignificant terms (significance level 5%), and the final model was used for post hoc testing. MC Vol. 19, No. 4, 2015

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Analysis of Variances on Log-Scale. The trial-by-trial variance estimates on

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log-scale followed stochastically independent chi-square distributions (HoffmannJørgensen, 1994) and could therefore be analyzed for the effects of velocity, day and subjects, separately for each amplitude, by means of a generalized linear model with logarithmic link function (McCullagh & Nelder, 1983). This models

Figure 1 — Example traces for one subject (subject K) for a 12-degree, 140 deg/s stretch on day 1 (31 trials) and day 2 (30 trials). The subject was tested 14 days apart. For each plot, the horizontal axis indicates time since stretch onset (ms), i.e., time zero indicates stretch onset. A: Rectified and smoothed (40Hz low pass Butterworth filter) SOL EMG for individual trials (gray lines) and the average of all trials (black line). The time of the peak of the response is 46.25 ms for day 1 and 44.75 ms for day 2. The vertical dashed lines indicate the analysis windows. B and C: Histograms of the response (B) and baseline magnitudes (C) for the traces shown in A. MC Vol. 19, No. 4, 2015

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Figure 2 — Histograms of the p-values from Shapiro-Wilk tests of normality and log-normality of the 308 samples of absolute baseline and response magnitudes. Gray bars mark the p-values for which normality or log-normality is rejected on a 5% significance level.

log-linear relationships between velocity as the independent variable and the trialby-trial variances with subject and day specific model parameters. The models were simplified by removing statistically nonsignificantly terms, and the final model was used for post hoc testing.

Results As consistent SSRs were not observed in all subjects for the slowest stretch velocity (20 deg/s), these stretches were removed from all subsequent analysis. For the remaining stretches there were no statistically significant differences of the onset time or time of peak between day 1 (onset time: 42.1 ± 5.8 ms; time of peak: 54.7 ± 7.0 ms) and day 2 (onset time: 41.7 ± 5.5 ms; time of peak: 54.4 ± 7.2 ms) or between the different velocity/amplitude combinations (onset range: 40.6–43.5 ms; time of peak range: 53.3–55.2 ms). There was, however, a tendency toward a faster response peak latency as the stretch velocity increased. Figure 1 is an example of the EMG traces (Figure 1A) and resulting histograms [showing the distribution of the absolute response and baseline magnitudes] for one subject following a 12-degree, 140 deg/s stretch. The trial-by-trial distributions MC Vol. 19, No. 4, 2015

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for this subject/stretch combination were well approximated by log-normal distributions. Figure 2 shows that the absolute baseline and response trial-by-trial distributions could be approximated by log-normal distributions and displays the result of testing each sample for normality and log-normality by the Shapiro-Wilk test (p-value less than 0.05 rejects normality). Overall, the Shapiro-Wilk p-values are much larger for the test applied to log-transformed absolute baseline and response magnitudes relative to the values obtained for the untransformed magnitudes. Moreover, for normal samples, the Shapiro-Wilk test produces a false significance with probability 5% (approximately), which is similar to what was observed for the test applied to log-transformed magnitudes (13% and 8%; Figure 2), whereas a much larger rejection rate was observed for the untransformed magnitudes (27% and 47%; Figure 2). Hence, absolute baseline and response magnitudes could be summarized within samples by the geometric mean, and Figure 3 shows a Bland-Altman plot of the summaries, which shows that the stretch intervention has a multiplicative effect on the baseline magnitudes (additive effect: points scattered parallel to the identity line; multiplicative effect: points scattered around a nonunity sloped line intersecting the origin). Therefore, the LogG analysis (see Pedersen et al., 2013) was justified, and relative response magnitude trials were computed by dividing each absolute response magnitude trial value by the corresponding within-sample geometric mean of the absolute baseline magnitudes. Following from the above, these could also be approximated by log-normal distributions.

Figure 3 — Bland-Altman plot of the geometric mean of absolute baseline and response magnitudes. MC Vol. 19, No. 4, 2015

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Analysis of the Variance of the Response

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The trial-by-trial variances of the relative response magnitudes on log-scale largely differed between subjects, between stretch velocities within subjects, and for some subjects between days. At amplitude 4.5 deg, the variances on log-scale did, however, not depend on velocity with statistical significance. Differences in Between Subject Variances Figure 4(A–C) demonstrates significant between subject variance heterogeneity for the 4.5 deg, 60 and 100 deg/s stretches (independent of velocity; Figure 4A), 8 deg, 100 deg/s stretch (Figure 4B) and 12 deg 100 deg/s stretch (Figure 4C) for all subjects for day 1 and day 2. Figure 4(A–C) shows that 257/546 (47%) tests are significant (i.e., significantly different between subject variance). As the expected number of false positive significant differences is 5% of 546, i.e., 27–28, there is clear evidence of different variances between some subjects. This pattern is observed for all stretch velocities/amplitudes.

Figure 4 — Differences in variances on log-scale between subjects for (A) amplitude 4.5 deg stretch (joint model for the 60 deg/s and 100 deg/s stretches), (B) 8 deg, 100 deg/s stretch, and (C) 12-degree, 100 deg/s stretch comparing all subjects (A–N) for day 1 and day 2. The diagonal line delineates day 1 (below diagonal line) and day 2 (above diagonal line). The gray and black squares represent significant differences in variance between subjects significance levels 5% and 1%, respectively. Blank areas represent no significant difference in the variance between subjects at significance level 5%. Figure 4(A–C) shows that 257/546 (47%) tests are significant (i.e., significantly different between subject variance). As the expected number of false positive significant differences is 5% of 546, i.e., 27–28, there is clear evidence of different variances between some subjects. MC Vol. 19, No. 4, 2015

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The Effect of Day on Variance on Log-Scale Figure 5 shows differences in the variances on log-scale between day 1 and day 2 within subjects for the 4.5 deg, 8 deg, 100 deg/s stretch, and the 12 deg, 100 deg/s stretch. In Figure 5, 12/42 (29%) tests are statistically significant, which exceeds the expected number of false significances of 2–3 thus providing evidence of different variances between days for some subjects and stretches. Table 1 shows the number of subjects with within stretch variances that were significantly different between day 1 and day 2.

Table 1 Table Showing the Percentage of Subjects and Number of Subjects (N = 14) That Showed Significantly Different Variances on Day 1 and Day 2 Amplitude

Velocity

Percentage of Subjects

4.5

All

29% (4)

8

All

36% (5)

12

60

14% (2)

12

100

21% (3)

12

140

36% (5)

12

180

43% (6)

12

200

36% (5)

The Effect of Velocity on Variance Figure 6(A–D) represents models for the

variance on log-scale as a function of velocity for the 8- and 12-degree stretches on day 1 and 2 for each subject. For the 8-degree stretch, the dependence on velocity was not statistically different over the 2 days, and a joint model for day 1 and day 2 showed that 8/14 subjects had variances dependent on velocity (as velocity increased the variance decreased) and 6/14 subjects had variances that did not significantly increase or decrease with an increase in velocity. For the 12-degree stretch, the dependence on velocity was significantly different between days. For both days, 12/28 tests had variances dependent on velocity and 16/28 tests had variances that did not significantly increase or decrease with an increase in velocity. For the 8and 12-degree stretches, no subjects showed an increase in variance with increased velocity. Variance did not depend significantly on velocity for the 4.5 deg stretch. The probable reason was that only two velocities were assessed and therefore no velocity dependence could be observed.

Analysis of the Geometric Mean Magnitude of the Response at the Individual Level. After removing statistically

insignificant model terms, similar models were observed for 4.5 deg, 8 deg and 12 deg. Figure 7 models the geometric mean as a function of velocity for each subject at amplitude 4.5, 8 and 12 deg on day 1 and day 2. The models showed a log-linear relationship between the geometric mean and velocity. There was a subject × day interaction meaning that the intercepts depended on both subject and day (p = .0151, MC Vol. 19, No. 4, 2015

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Figure 5 — Within subject variance ratios and 95%-confidence intervals between day 1 and day 2 for all tested subjects (A–N). The open circle (with solid lines) represents the 4.5 deg stretch (model independent of velocity), filled squares (with dashed lines) represents the 8 deg, 100 deg/s stretch and filled circles (with dotted lines) represents the 12 deg, 100 deg/s stretch. The horizontal line represents a variance ratio of 1 (i.e., equal variance) between day 1 and day 2.

Figure 6 — Model plots showing variance on a log scale as a function of velocity for the 8-degree (A and B) and 12-degree stretch (C and D) for day 1 (A and C) and day 2 (B and D) for each subject. Each line represents one subject. The solid line represents subjects that demonstrate a significant decrease in variance with an increase in velocity and the dashed line represents subjects that demonstrate no significant increase or decrease in variance with an increase in velocity. MC Vol. 19, No. 4, 2015

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Figure 7 — Model plots showing geometric mean (relative response magnitude) as a function of velocity for the 4.5, 8 and 12 deg stretches at all tested velocities on day 1 and day 2. Solid lines represent a significant increase in the relative response magnitude as velocity increases. Dashed lines represent no significant increase in response magnitude as velocity increases. For the 8 deg and 12 deg stretches, all subjects showed a significant increase in the relative response magnitude as velocity increased. Nonsignificance effects were only observed for the 4.5 degree stretch.

Figure 8 — Relative response magnitudes and 95%-confidence intervals comparing day 1 and day 2 for all tested subjects (A–N). The open circle (with solid lines) represents the 4.5 deg stretch (model independent of velocity), filled squares (with dashed lines) represents the 8 deg, 100 deg/s stretch, and filled circles (with dotted lines) represents the 12 deg, 100 deg/s stretch. The horizontal line represents equal relative response magnitudes between day 1 and day 2. 264

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p < .0001, p < .0001 at amplitudes 4.5, 8 and 12 deg, respectively). There were also subject specific slopes, where the relationship between velocity and magnitude differed between subjects (p = .0044, p < .0001, p < .0001 at amplitudes 4.5, 8 and 12 deg, respectively). Figure 8 demonstrates the relative response magnitude between day 1 and day 2 for each subjects for the 4.5 deg [independent of velocity], 8 deg, 100 deg/s and 12 deg 100 deg/s stretch. In Figure 7, 17 tests are statistically significant (i.e., significantly different between day). As the expected number of false positive significant differences is 2.1, there is evidence of different response magnitudes between days for some subjects and some stretches. When response magnitude is modeled for all subjects and includes all velocities for 4.5, 8 and 12 deg stretches, 14% (2/14), 50% (7/14) and 57% (8/14) have significantly different response magnitudes between day 1 and day 2. Magnitude of the Response at the Group Level. The model describing the magnitude of the response as the group model is the same model as the individual model except that individual differences in terms of intercepts and slopes of the linear relationships between velocity and LogG are modeled as random individual divergences from common values at the group level. Hence, individual differences become part of the random variation of LogG measurements around linear relationships at the group level. Thereby, the velocity dependency of the relative response magnitude at the 2 days can be studied at the group level. Within this model, intercepts and slopes between the 2 days were not significantly different for any amplitude. There was a dependence of velocity at all amplitudes (p < .0001 at all amplitudes). Hence, at the group level, there is a significant dependency of response magnitude on velocity but no significant differences between days. The random variation of values around this model is primarily due to overall differences between subjects and differences between days within subjects and less due to subject differences with regards to the dependency on velocity.

Discussion The current study shows that the facilitatory SSR has log-normal trial-by-trial distributions of absolute baseline, absolute response and relative response magnitudes. There was variance heterogeneity between subjects and differences between days (for some subjects) and stretch velocities. As velocity increased, the variance on a log-scale significantly decreased or neither increased/decreased. The right-skewed log-normal distribution and widespread variance heterogeneity violates standard statistical assumptions and therefore, an analysis by which an average of untransformed trials is taken is incorrect. The LogG analysis [proposed by Pedersen et al. (2013)] accounts for the log-normal and variance heterogeneous trials and is therefore more appropriate. The relative magnitude of the SSR was assessed using the LogG analysis. There were significant differences in the magnitude of the reflex between some subjects and for some subjects between days. For all amplitudes, there was a significant effect of velocity such that as the stretch velocity increased the magnitude of the response increased. At the group level, there was a significant effect of velocity that was not significantly different between days. MC Vol. 19, No. 4, 2015

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Consequences of Violating Statistical Assumptions The current study has shown that standard statistical assumptions are violated using the standard method of averaging trial-by-trial responses. Averaging responses ignores the log-normal trial-by-trial distribution and ignores patterns of variance heterogeneity. The consequences of ignoring these statistical assumptions was demonstrated and discussed by Pedersen et al. (2013). By using data simulations Pedersen et al. (2013) noted that patterns of variance heterogeneity manifested as changes in the response magnitude which can produce false significant/nonsignificant results. This may have implications when assessing SSRs (and tendon reflexes) from patients and drawing conclusions based on these. At times, changes in response magnitudes are small and therefore, although significant/nonsignificant differences may be observed, these may only reflect differences in the patterns of response variation rather than the magnitude itself. Subsequently, these may alter the results in some studies that investigate stretch reflexes and changes in stretch reflexes following treatment regimes or drug administration.

Differences in Variance and Magnitude Between Day 1 and Day 2 The current study shows widespread variance heterogeneity on log-scale. Within subject, variability was less pronounced than between subject variability. There were also differences in variance heterogeneity for some subjects between days although the majority of subjects had variances that were not significantly different. This possibly shows that patterns of response variance and the state of nervous system are conserved for some subjects over days. The magnitude of the response was significantly different for some subjects and nonsignificant for others over day. This was an unexpected finding as previous studies investigating the biceps brachii SSR (Segal & Wolf, 1993) and tendon tap reflexes (Stam & van Crevel, 1989) indicated that reflexes over a short time have nonsignificantly different magnitudes. However, after 2.5 years the agreement of measurements reduced (Stam & van Crevel, 1989). In that study the short time between tests was 1–2 hr in which the electrodes were not removed and therefore it is likely that more agreement between average magnitudes will be observed. Segal and Wolf (1993) reported no day-to-day differences in the SSR magnitude, which is different to the current study (but noted a reduction in the magnitude as trials progressed). In that study, the distribution of the trial-bytrial responses were not reported and it was for a different target muscle (biceps brachii) which may have contributed to the differences. Interestingly, when the data in the current study were analyzed with subject treated as a fixed and not random factor (magnitude on a group and not individual level), there were no significant differences in the magnitude of the response between day 1 and day 2 which is in agreement with the results of Segal and Wolf (1993) and Stam and van Crevel (1989). The conditions during testing were kept as uniform as possible (for example; subject position and preparation, comfort level, time of day tested and similar baseline EMG levels). Despite this, the subjects were tested 1–14 days apart meaning the testing conditions may not have been exactly the same on the 2 days. These MC Vol. 19, No. 4, 2015

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differences might be the cause of some of the variance and magnitude differences between days other than day-to-day fluctuations of the nervous system.

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The Effect of Velocity For the 8-degree and 12-degree stretches, as the stretch velocity increased subjects either had a decrease or no change in variance on log-scale. The magnitude of the SSR is considered velocity dependent and the results of the current study confirms these previous findings (for example, Gollhofer, Schopp, Rapp, & Stroinik, 1998; Gottlieb, Agarwal & Jaeger, 1981). The current study also demonstrates that the trial-by-trial variance on log-scale of the reflex also depends on stretch velocity. As SSRs are quantified by an averaged curve, the variance on log-scale of the trialby-trial responses in previous studies could be concealed by the averaged curve and therefore not reported. Pedersen et al. (2013) investigated the short latency crossed spinal inhibitory response following electrical stimulation of the ipsilateral tibial nerve at motor threshold, 35% of the maximal peak-to-peak M-wave (M-max), and at 85% M-max of the ipsilateral SOL EMG, and investigated inhibitory responses in the EMG of the precontracted contralateral SOL muscle. In that study there were differences in variance on log-scale between subjects but the same variance within subjects when stimulated at MT, 35% M-max and 85% M-max. The current study supplement these findings. There were significant differences in between subject variances for many subjects over all investigated stretches demonstrating that variance is subject dependent. However, within subject variances were the same in Pedersen et al. (2013) over different stimulation intensities but differed with stretch velocity in the current study. This is could be due to the nature of the stimulus or the different type of investigated response (ipsilateral facilitatory response compared with a contralateral inhibitory response). Burke, Gandevia, and McKeon (1983) reported that the Achilles tendon jerk consisted of more dispersed afferents volleys when compared with sub threshold electrical stimulation evoking an H-reflex. This means that following a stretch of the muscle, afferent volleys are temporally dispersed (more asynchronous) as compared with an electrically evoked stimulus (more synchronous). This increase in synchronicity could result in a less variable electrically evoked than mechanically evoked reflex response. Other studies have also noted differences between mechanically and electrically evoked responses. Morita, Petersen, Christensen, Sinkjaer, and Nielsen (1998) reported that presynaptic inhibition evoked by a tap to the biceps femoral tendon modulated the presynaptic inhibition of electrical evoked responses (H-reflex), SSRs, and tendon tap reflexes differently. In that study there was a more prominent inhibition to the H-reflex compared with the tendon tap reflex and from the H-reflex compared with the SSR (which had no inhibition). In Morita et al. (1998), the SSR was evoked by a 5 deg at 125 deg/s stretch and therefore is within the amplitude/ velocity range of the current study. The authors of that study cited that temporal dispersion and the different waveform in mechanical and stretch evoked response were the likely source of the differences between the two afferent stimuli. For this reason, it is plausible that the difference in temporal dispersion of afferent volleys between electrically evoked and mechanically evoked afferent volleys is large enough that it is the cause of the difference in within subject variance between Pedersen et al. (2013) and the current study. MC Vol. 19, No. 4, 2015

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Variance tended to decrease as the velocity increased. As stretch velocity increases the impulses/second from the afferent nerve endings in the muscle increase (Matthews, 1963). As with the reduction in variance comparing electrically and mechanically evoked response, the increased synchronicity with increased stretch velocities could result in a reduction in the response variability at the spinal cord. The synchronicity results in temporal summation and an increased likelihood for the excitatory post synaptic potential to reach threshold (Muir & Porter, 1973) and depolarize the motor neuron. A greater synchronicity could cause more uniform input and as such a reduction in the variability of the response in some subjects with an increase in velocity.

Implications From the current study it is possible to speculate about the functional implications of the results. As SSRs are log-normally distributed, previous studies documenting SSRs probably overestimated the size of the SSR as they have taken the mean of untransformed SSR variance heterogeneous magnitudes (see introduction for references). This may or may not have implications to the conclusions in these studies. Pedersen et al. (2013) noted that untransformed variance heterogeneous results underestimated the size of the inhibition in the measured response although conclusions (in terms of significant p-values) were not changed. In studies that report marginal significance, untransformed variance heterogeneous results may alter the significance levels such that the conclusions may not hold true. Differences in the trial-by-trial variance in responses are important to assess. For example, a patient may have ‘normal reflexes’ based on the average of 30 reflex trials. However, when these trials are assessed individually, 26 trials may be within the normal range of reflex magnitudes and four maybe very different (small and/or large). On average, the responses may appear normal but when assessing the trial-by-trial variance it can be observed that the patient has a very abnormal composition of these averaged responses. If this patient is walking and over time 30 reflexes occur, 26 of these times the patient will continue walking with no problem, however on 4 occasions these reflexes are not normal (too large or small) and on these occasions there may be a higher likelihood for the patient to fall. For this reason assessing the variance of trial-by-trial responses in each patient could be beneficial and provide additional information about the patient.

Conclusion The current study shows that the facilitatory SSR in the SOL following a rapid dorsiflexion at a range of velocities and magnitudes follow log-normal trial-by-trial distributions with widespread between subject differences of the trial-by-trial variance on log-scale. Some subjects also showed different variances on logscale between days. Given these observations, the common method of averaging responses conceals the characteristics of the trial-by-trial and could show erroneous results. The LogG analysis is more appropriate for these data as it allows for log-normal trial-by-trial distributions and variance heterogeneity. As velocity increased, the trial-by-trial variance on log-scale tended to decrease with MC Vol. 19, No. 4, 2015

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subjects showing either a decrease in variance or no significant increase/decrease in variance. The response magnitude significantly increased with an increasing velocity. The characteristics with an increase in velocity did, however, depend on subject, and for some subjects also on day and amplitude. However, when the responses were treated on a group level, there were no differences between day or stretch amplitude. The current study demonstrates the characteristics of the SSR over different velocities and amplitudes. It highlights the importance of assessing trial-by-trial responses as important information can be concealed if an averaged response is used.

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Acknowledgments This study was supported by Hammel Neurorehabilitation and Research Center. The authors would like to thank Anne Sofie Baunsgaard for her assistance.

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