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The probability of false positives in zero-dimensional analyses of one-dimensional kinematic, force and EMG trajectories

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A false positive is the mistake of inferring an effect when none exists, and although α controls the false positive (Type I error) rate in classical hypothesis testing, a given α value is accurate only if the underlying model of randomness appropriately reflects experimentally observed variance. Hypotheses pertaining to one-dimensional (1D) (e.g. time-varying) biomechanical trajectories are most often tested using a traditional zero-dimensional (0D) Gaussian model of randomness, but variance in these datasets variance is clearly 1D. The purpose of this study was to determine the likelihood that analyzing smooth 1D data with a 0D model of variance will produce false positives. We first used random field theory (RFT) to predict the probability of false positives in 0D analyses. We then validated RFT predictions via numerical simulations of smooth Gaussian 1D trajectories. Results showed that, across a range of public kinematic, force and EMG datasets, the median false positive rate was 0.382 and not the assumed α=0.05, even for a simple two-sample t test involving N=10 trajectories per group. The median false positive rates for experiments involving three-component vector trajectories was p=0.764. This rate increased to p=0.945 for two three-component vector trajectories, and to p=0.999 for six three-component vectors. This implies that experiments involving vector trajectories have a high probability of yielding 0D statistical significance when there is, in fact, no 1D effect. Either (a) explicit a priori identification of 0D metrics or (b) adoption of 1D methods can more tightly control α.
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The probability of false positives in zero-dimensional analyses of
one-dimensional kinematic, force and EMG trajectories
Todd C. Pataky 1, Jos Vanrenterghem2, and Mark A. Robinson2
1Department of Bioengineering, Shinshu University, Japan
2Research Institute for Sport and Exercise Sciences, Liverpool John Moores University, UK
March 18, 2016
Abstract
A false positive is the mistake of inferring an eect when none exists, and although controls the
false positive (Type I error) rate in classical hypothesis testing, a given value is accurate only if the
underlying model of randomness appropriately reflects experimentally observed variance. Hypotheses
pertaining to one-dimensional (1D) (e.g. time-varying) biomechanical trajectories are most often tested
using a traditional zero-dimensional (0D) Gaussian model of randomness, but variance in these datasets
variance is clearly 1D. The purpose of this study was to determine the likelihood that analyzing smooth
1D data with a 0D model of variance will produce false positives. We first used random field theory
(RFT) to predict the probability of false positives in 0D analyses. We then validated RFT predictions
via numerical simulations of smooth Gaussian 1D trajectories. Results showed that, across a range of
public kinematic, force and EMG datasets, the median false positive rate was 0.382 and not the assumed
=0.05, even for a simple two-sample ttest involving N=10 trajectories per group. The median false
positive rates for experiments involving three-component vector trajectories was p=0.764. This rate
increased to p=0.945 for two three-component vector trajectories, and to p=0.999 for six three-component
vectors. This implies that experiments involving vector trajectories have a high probability of yielding
0D statistical significance when there is, in fact, no 1D eect. Either (a) explicit aprioriidentification
of 0D metrics or (b) adoption of 1D methods can more tightly control .
Corresponding Author:
Todd Pataky, Ph.D., Institute for Fiber Engineering, Department of Bioengineering, Shinshu University
Tokida 3-15-1, Ueda, Nagano, Japan 386-8567
tpataky@shinshu-u.ac.jp, T.+81-268-21-5609, F.+81-268–21-5318
Keywords: statistical parametric mapping; random field theory; time series analysis; kinematics; ground
reaction force; three-dimensional analysis
1
1 Introduction
In classical hypothesis testing pvalues represent the probability that a random process would produce
an eect larger than the observed one. There are unfortunately many ways in which pvalue computations
can go astray to yield ‘false positives’ (Knudson, 2005; Kundson and Lindsey, 2014): the mistake of inferring
an experimental eect when none exists in reality. This paper deals with one specific pitfall in pvalue
computations which has not previously been quantified and is relevant to many branches of Biomechanics:
zero-dimensional (0D) pvalues for one-dimensional (1D) (e.g. time varying) data.
Imagine a simple experiment which yields five scalar 1D force trajectories for each of two groups (Fig.1b).
Classical hypothesis testing can be conducted using either a “0D” or a “1D” approach (Pataky et al., 2015).
Zero-dimensional analysis: One could analyze the data using a 0D summary metric like local maxima
(Fig.1a). In this case a one-tailed two-sample ttest of the maxima yields t=2.357, p=0.023 and one would
reject the null hypothesis at =0.05. More completely, the 0D residuals (Fig.1c) represent the variance
about the group means, and one judges the eect size x(Fig.1a) against this variance. If the null hypothesis
(x=0) were true, random 0D data with the same variance would produce a distribution of tvalues over
an infinite number of identical experiments (Fig.1e) and only 2.3% of those values would be greater than
the observed t=2.357. The null hypothesis is rejected because the observed tvalue exceeds the threshold
t
0D corresponding to (Fig.1e,g). This t
0D value can be rapidly computed in nearly all statistical software
packages, or it can be computed by iteratively simulating thousands of ttests on randomly generated samples
of 0D Gaussian data.
One-dimensional analysis: One could alternatively analyze the data using 1D methods (Lenhoet al.,
1999; Pataky et al., 2015) (Fig.1, right panels). Analogous to the 0D procedure, the 1D residuals (Fig.1d)
embody the variance about mean trajectories, and the null hypothesis is the null dierence trajectory:
x(q) = 0, where qis time or the 1D measurement domain. If the null hypothesis were true, random 1D
data with the same variance and same smoothness would produce a distribution of ttrajectory maxima over
an infinite number of identical experiments (Fig.1f), and in this case random 1D data would produce the
observed 0D eect of t=2.357 with a probability of approximately 55%, which is well above . In other
words, random 1D data produce particular tvalues with generally much greater probability than do random
0D data. The null hypothesis is not rejected because the observed maximum tvalue, across the whole
trajectory, does not exceed the threshold t
1D corresponding to (Fig.1f,h). The t
1D value can be readily
calculated using random field theory (RFT) procedures (Adler and Taylor, 2007; Friston et al., 2007), or it
can be computed by iteratively simulating thousands of ttests on randomly generated samples of smooth
2
1D Gaussian data (Fig.2) (Pataky, 2016).
The 0D and 1D procedures have yielded opposite hypothesis testing results, so which is correct? The
answer is easy: both are correct but both cannot be simultaneously correct within a predefined level. If,
prior to conducting an experiment, one explicitly identified that particular 0D metric as the sole metric of
empirical interest then the empirical question is inherently 0D and the 0D result is correct. If, however,
one measured 1D data and did not specify that particular 0D metric prior to the experiment then the
empirical question is inherently 1D and the 1D result is correct. Failing to specify a 0D metric prior to a 1D
experiment and then adopting 0D methods has been termed ‘regional focus bias’ (Pataky et al., 2013) and
is a potential source of false positives. False positive prevalence for 1D biomechanics datasets has previously
been estimated for 0D procedures (Knudson, 2005) but not, to our knowledge, in the context of 0D vs. 1D
procedures.
The purpose of this study was to quantify the false positive rates that could be expected in real 1D
biomechanical datasets when employing 0D statistical inference. To that end we analyzed nine public
datasets (Table 2) which represent a variety of experimental tasks (walking, running, cutting, cycling) and
data modalities (forces, kinematics, EMG). Based on the data’s temporal smoothness we estimated the
likelihood of producing false positives using a simplified experimental design: a two sample ttest with N=10
for each group. Analyses for more complex designs like ANOVA are addressed in the Discussion (§4.3). The
key theoretical concept we shall attempt to convey is that two parameters — the mean (µ) and standard
deviation () — completely describe 0D Gaussian behavior, and that only one additional parameter — 1D
smoothness (FWHM) (Fig.2) (Appendix A, B)— is needed to completely describe 1D Gaussian behavior.
To clarify our “0D” and “1D” terminology we shall also employ the term: nDmD”, where nand m
are the dimensionalities of the measurement domain and dependent variable, respectfully (Table 1). In
nDmD datasets the physical nature of the variables changes across the mcomponents but not across the nD
measurement domain. Biomechanics studies often measure 1DmD data but use 0D1D models of randomness
to define critical statistical thresholds, and this paper quantifies false positive rates associated with that
approach. Throughout this paper “0D” and “1D” represent to “0D1D” and “1DmD”, respectively.
We emphasize that this paper focusses on just a single statistical issue: the probability of false positives
in a single 0D1D two-sample t test conducted on 1DmD data. We use only the two-sample t test because (a)
this simple test suciently demonstrates the magnitude of the false positives problem and (b) the problem
is exacerbated in more complex designs like ANOVA. We acknowledge that many other issues must be
considered when conducting statistical analyses including: small sample sizes, non-sphericity, normality,
3
outliers, etc. Just as it is useful to consider each of these issues individually, we feel it is equally useful to
consider 0D vs 1D analysis individually because this issue is relevant to all 1D data analyses but has not
been explicitly addressed in the literature.
2 Methods
All analyses were implemented in Python 2.7 (van Rossum, 2014) using Canopy 1.4 (Enthought Inc.,
Austin, USA) and the open-source software package ‘rft1d’ (Pataky, 2016) (www.spm1d.org/rft1d). For
readers unfamiliar with Python, MATLAB source code (The MathWorks, Natick, USA) replicating the
study’s main analyses and results is provided as Supplementary Material (Appendix C).
2.1 Experimental data smoothness estimations
Since 1D smoothness determines the height that random 1D trajectories reach (Fig.2), we first estimated
the smoothness of 1D biomechanical trajectories from nine public datasets (Table 2). Each dataset contained
1D scalar and/or vector time series spanning a normalized interval of 0 – 100%. All data were analyzed in
their publicly available form and no extra signal processing was conducted. Detailed descriptions of the data
are available in the original papers and in the public datasets themselves.
After computing residual trajectories (Fig.1d) as the dierence between group/subject means (as appro-
priate for each dataset), we then calculated the residuals’ smoothness using a robust FWHM estimation
procedure (Kiebel et al., 1999) (Appendix D). The ‘FWHM’ is the full-width at half-maximum of a Gaus-
sian kernel (Appendix A) which, when convolved with completely uncorrelated Gaussian data (Appendix
B) yields random 1D trajectories with the same smoothness as the observed 1D residuals. Eectively the
estimated FWHM is simply the mean temporal gradient normalized by residual magnitude. This FWHM
estimation procedure has been validated elsewhere for 1D data (Pataky, 2016) and a similar validation is
available in the MATLAB Supplementary Material (Appendix C). Subsequent analyses consider the full
range of estimated FWHM values.
2.2 Theoretical false positive rates
2.2.1 False positive rate for random 0D1D data
The probability that random 0D1D data, or equivalently: 0D univariate Gaussian data, would produce
atvalue which exceeds an arbitrary height uis given by the survival function:
4
P0D(t>u)=Z1
u0
@
(+ 1)/2
p⌫⇡ (/2) 1+x2
!(+1)/21
Adx (1)
where is the degrees of freedom and is the gamma function. Note that, given a height u,P0D(t>u)is
dependent on only sample size as manifested in the parameter .
Classical hypothesis testing on 0D1D data is conducted by setting Eqn.1 to :
P0D(t>t
0D)=(2)
and then solving for the critical threshold t
0D . If the experimentally observed tvalue exceeds t
0D then the
null hypothesis is rejected. To be clear, false positives occur in 0D analysis of 0D data when random 0D
data exceed t
0D , and this occurs at a rate of by definition.
2.2.2 False positive rate for random 1D1D data
The probability that random 1D1D data, or equivalently 1D univariate Gaussian trajectories, produce t
trajectories which reach arbitrary heights uis (Worsley et al., 2004; Friston et al., 2007):
P1D(tmax >u)=1exp 2
4P0D(t>u)S
W
p4 log 2
2 1+u2
!(1)/23
5(3)
where tmax is the trajectory maximum, Sis the trajectory length (constant for all trajectories in one dataset,
usually S=100) and Wis the FWHM representing trajectory smoothness. Note that, relative to the 0D case
(Eqn.1), just one additional parameter (S/W ) is needed to describe the probabilistic behavior of tmax.
The probability that 1D1D data will reach the 0D threshold for significance (t
0D )isthussimply:
False positive rate {0D1D analysis, 1D1D data}=P1D(tmax >t
0D) (4)
To our knowledge this false positive rate has not been previously reported. We thus calculated Eqn.4 as a
function of for three values: 0.01, 0.05 and 0.10, and also over the range of the experimentally estimated
FWHM values.
Last, we computed the false positive rate for the case of completely uncorrelated data (Fig.2a) using the
Bonferroni correction:
5
PBonftmax >t
0D=1(1 )100 (5)
Note that Eqn.5 is accurate only for the case of 100 independent tests and is therefore inaccurate when
1D data are smooth. We nevertheless use this Bonferroni result as a reference, to demonstrated that 1D
results converge to this result as 1D trajectories become increasingly rough.
2.2.3 False positive rate for random 1DmD data
We last considered three separate cases of 1DmD data: one, two and six vector trajectories where each
vector has three components. Equivalently, these are 1DmD multivariate Gaussian trajectories with: m=3,
m=6 and m=18, respectfully. For example, m=6 could represent an experiment involving two joints’ three
rotations. For each case we assumed independently varying vector components, thereby yielding:
False positive rate {0D1D analysis, 1DmD data}=11P1D(tmax >t
0D)m(6)
Note that Eqns.4 and 6 are equivalent when m=1.
2.3 Theoretical validations
We validated all theoretical results using 0D and 1D Gaussian random data generators as implemented
in SciPy (Jones et al., 2001) and rft1d (Pataky, 2016), respectively (c.f. Appendix E). Specifically, we
generated 100,000 random datasets then conducted one ttest for each dataset, thereby yielding 100,000 t
values or 100,000 ttrajectories. To validate theoretical predictions (Eqns.1, 3) we calculated the percentage
of the 0D tvalues and 1D tmax values to exceed arbitrary thresholds u.
For 0D data we repeated these simulations for two sample sizes (=4, =48), and for the 1D case
we additionally repeated simulations for ten dierent FWHM values to span the range of experimentally
observed smoothness values. Last, we repeated all 1D simulations for the aforementioned multivariate cases
of one-, two- and six three-component vector trajectories which could represent the analysis of a single joint
in three dimensions, the synergistic function of the hip and knee and all the three main joints of the lower
limb, respectively.
6
3 Results
3.1 Experimental data smoothness
Residual 1D trajectories from all datasets, two of which are depicted in Fig.3, were qualitatively consistent
with simulated smooth Gaussian 1D trajectories (Fig.2). Quantitative consistency between 1D residuals and
Gaussian 1D trajectories has been demonstrated elsewhere (Pataky et al., 2015).
Residual smoothness estimates yielded minimum, median and maximum FWHM values of: 6.2%, 16.5%
and 67.0%, respectively, across all datasets (Table 3). Kinematic residuals were smoothest on average,
followed by force and then EMG residuals. Half of all trajectory variables analyzed lay within a range of
FWHM=[11.9%, 29.5%] and ninety percent of all trajectory variables lay within a range of FWHM=[9.4%,
36.5%] (Appendix F). Most of the experimental trajectories investigated therefore lay in the smoothness
range depicted in Fig.2b through Fig.2e.
3.2 Theoretical false positive rates and validations
The 0D and 1D survival functions (Eqns.1 and 3, respectively) are depicted in Fig.4 for two dierent
sample sizes (=4 and =48). Considering first the 0D survival functions, it is clear that 50% of tests on
random 0D data yield tvalues larger than zero, implying that 50% are less than zero. Next, the critical
threshold is sample-size dependent: t
0D =1.67 and 2.13 for =4 and 48, respectively (=0.05). For these
two sample sizes Fig.4 shows that the maximum tvalue produced by random 1D1D Gaussian trajectories
will reach these critical heights (t
0D ) with much greater probability. For the median observed smoothness
of FWHM=16.5%, that probability is p=0.431 and 0.376 for =4 and 48, respectively. Simulating random
Gaussian 0D and 1D data (depicted as dots in Fig.4) validated all results.
For two-sample t tests with N=10 in each group (=2N2 = 18), false positive rates were greater
than for all smoothness values (Fig.5), and this rate approached the Bonferroni rate (depicted as stars in
Fig.5) as the FWHM approached zero. The 1D false positive rate converged to the 0D only for FWHM=1
because an infinitely smooth 1D trajectory is equivalent to a 0D scalar.
Last, Fig.6 depicts results for two-sample experiments involving 1DmD data (i.e. m-component vector
trajectories). For 1D3D data, Fig.6 suggests that the probability of a false positive is approximately p=0.90
when FWHM=10%. For 1DmD data where m6, Fig.6 suggests that false positives are nearly certain
irrespective of trajectory smoothness. For the smoothest observed data (FWHM=67.0), the probability of a
false positive was estimated to be p=0.145 and p=0.940 for for 1D1D and 1D18D data, respectfully (Table
7
4). This implies that methods which control 1D false positive rates should also control false positive rates
associated with multivariate data when multivariate data are measured and analyzed.
4 Discussion
4.1 Main implications
The convention of =0.05 implies that one accepts a 5% false positive rate when conducting classical
hypothesis testing. The main result of this study was that smooth, random 1D tra jectories generally produce
false positives in 0D analyses with a probability much higher than . Even for the best case — maximum
smoothness (FWHM=67.0) and one scalar trajectory — false positive rates were nearly three times greater
than (p=0.145, Table 4). For the median smoothness observed across all datasets (FWHM=16.5), the
false positive rates for three-component vector trajectories was greater than p=0.76. In the worst case
— maximum roughness (FWHM=6.2) and two or more three-component vectors — the false positive rate
exceeded p=0.999. Since Biomechanics studies often measure and analyze multiple three-component vector
trajectories (e.g. 3D joint angles and moments at the hip, knee and ankle during gait), these results imply
that false positives are nearly certain when conducting 0D analyses of typical 1D datasets. Inflated false
positive rates in 0D analyses of 1D data have previously been suggested (Lenhoet al., 1999; Pataky et al.,
2015; Robinson et al., 2015) but to our knowledge have not been previously quantified.
We stress that these results in no way invalidate published 0D analyses of 1D data most obviously because
large eects are generally discoverable irrespective of the analysis procedure; clearly a somewhat stronger
signal in Fig.1h would cross both 0D and 1D thresholds for significance. Second, as a general limitation of
classical hypothesis testing: significance does not imply practical meaning and vice versa. Last, all results
are valuable regardless of their magnitude if they lead to subsequent independent scrutiny and verification
through repeated experimentation.
4.2 Context of ideal hypothesis testing
An ideal hypothesis-driven experiment (Fig.7)a involves formulating a testable null hypothesis, from
which the independent variables (IVs), dependent variables (DVs) and often the experiment itself directly
emerge. When dealing with complex systems it can be dicult to follow that procedure because it may
be dicult to formulate specific hypotheses prior to conducting the experiment. In this case, exploratory
analyses (Fig.7)b provide an extended framework for valid hypothesis testing. In particular, the investigator
8
can freely choose arbitrary measurements, processing procedures, IVs, DVs and tests provided the reported
results are confirmed by applying the identical procedures to an independent dataset. In other words,
exploratory analyses are useful for narrowing the empirical scope of the study and to generate specific null
hypotheses to be tested on independent data.
Biomechanics studies often adopt a hybrid of the aforementioned hypothesis testing and exploratory
procedures (Fig.7)c. The result is a unfortunately a procedure which contains critical scientific flaws. The
fundamental problem is that hypothesis tests are 0D1D (see Table 1), but this is incommensurate with the
dimensionality of the aprioriDV which is 1DmD. In other words, models of 0D1D randomness cannot
describe the probabilistic behavior of 1DmD data (Fig.1), so statistical tests cannot pertain to the collected
1DmD data. The relative absence of 1DmD procedures in the Biomechanics literature is, in our opinion,
a major oversight. Access to 1DmD theory and procedures would help future studies converge to ideal
hypothesis testing and exploratory procedures.
To summarize, the recipe below will ensure that ideal classical hypothesis testing is conducted when
analyzing 1DmD data (Table 1):
1. Did I formulate a specific null hypothesis regarding a specific 0D1D dependent variable prior to con-
ducting the experiment?
2. If yes to #1, then I must use 0D1D hypothesis testing methods and I mustn’t report 1DmD data
except qualitatively because 1DmD data are irrelevant to the hypothesis.
3. If no to #1, then I must use 1DmD methods and mustn’t report 0D1D data except qualitatively
because 0D1D data are irrelevant to the hypothesis.
4.3 Limitations
The false positive results (Table 4) pertain directly only to two-sample experiments involving ten tra-
jectories in each group, and these values will change for dierent sample sizes and dierent designs. We
nevertheless found qualitatively identical trends after repeating these analyses for a large variety of designs
(one-sample, regression, one-way and two-way ANOVA and MANOVA). In particular, false positive rates
increased with the number of measured trajectories in all designs, and numerical values were only slightly dif-
ferent from those in Table 4. For example, one-way ANOVA with three groups and ten trajectories per group
yielded a false positive rate of p=0.433, which is just slightly higher than the two-sample case (p=0.382).
We report only two-sample results for brevity.
9
A second consideration is that we conducted univariate analysis of multivariate (vector) trajectories using
two-sample ttests even though we should have conducted multivariate analysis using Hotelling’s T2test (Cao
and Worsley, 1999; Pataky et al., 2013). We report univariate results to be consistent with the Biomechanics
literature’s high prevelance of univariate analyses of multivariate data (Knudson, 2005).
A third apparent limitation is that one is not obliged to analyze maxima (Fig.1d). For example, the
maximum or ‘peak’ does not necessarily lie within aprioritemporal windows of empirical interest in sports
maneuvers (Besier et al., 2001; Besier et al., 2003) so other metrics may be chosen to represent those move-
ment phases. False positive rates for non-maxima would be lower than those reported here. We’d nevertheless
argue that choosing non-maxima is scientifically unjustified for the following reasons: if one has an apriori
hypothesis regarding a 0D variable, then it is irrelevant whether that variable is a maximum, minimum, or
intermediate value because only that 0D variable must be analyzed. The issue of maxima vs. non-maxima
therefore pertains only to 1D analyses. For 1D analysis the statistical goal is to quantify the probability
that random 1D trajectories would produce an eect as large as the observed eect, and this by definition
pertains to the maximum signal.
A fourth, and real limitation is the broad issue of classical vs. Bayesian inference. This paper considered
only classical inference. Bayesian inference aords a much broader class of statistical inferences and is
generally regarded to supersede classical inference as a more objective means of scientific inquiry (Kruschke,
2013). Since the Biomechanics literature almost exclusively adopts classical inference (Knudson, 2005) we
leave Bayesian inference for future work.
A fifth limitation is that the present analyses assumed isotropically smooth residual trajectories (Fig.3).
This may not be a good assumption for some datasets which have mixed-frequency signals. As an example,
ground reaction forces during running typically have high-frequency impact signals followed by compara-
tively low-frequency propulsive forces (Cavanagh and Lafortune, 1980). Fortunately, there are two factors
mitigating potential problems associated with anisotropic smoothness. First, it is possible to correct for
anisotropic smoothness simply by estimating the trajectory length which yields constant smoothness (Wors-
ley et al., 1999). Second, the unbiased smoothness estimation approach adopted herein (Kiebel et al., 1999)
assures that FWHM estimates are robust to minor anisotropy. Regardless, research is needed to estimate
the prevalence and seriousness of smoothness anisotropy in biomechanical trajectories.
A general limitation of 1D methods is sensitivity, which is the ability of a statistical test to detect a
true eect. Statistical thresholds are naturally higher for 1DmD analyses than for 0DmD analyses (Fig.1)
so sensitivity is generally lower in 1D procedures. However, in our experience 1D procedures usually have
10
sucient sensitivity because eect sizes are usually large in biomechanics datasets, even for small datasets
(Pataky et al., 2015). In fact it could be argued that sensitivity is much lower in 0D vs. 1D analyses because
0D analyses fail to consider the entire dataset and therefore cannot detect all signals. Regardless, in order
to ensure adequate sensitivity one should generally conduct power analysis before proceeding with a full
experiment. Procedures for 0D1D power analysis are well known and procedures for nDmD power analysis
are described elsewhere (Friston et al., 2007). Additionally, a simple way to boost sensitivity when dealing
with 1DmD data is to first conduct 1D analyses, then identify a 0D variable of interest and last conduct a
separate experiment on independent subjects using 0D analysis of the identified variable (Table ??).
A second general limitation of 1D methods is complexity. 1D procedures are naturally more complex
than 0D procedures and are usually more dicult both to learn and to deploy. Exacerbating the complexity
problem is a lack of suitable reference material. While many textbooks detail the dierences between
0D1D and 0DmD procedures (Rencher and Chistensen, 2012), none of which we are aware details 1DmD
procedures. Some books detail subsets of 1D1D procedures (Ramsay and Silverman, 2005; Zhang, 2013), and
others thoroughly summarize 3D3D and 4D1D procedures (Friston et al., 2007) procedures, but these contain
many concepts which are unneeded in 1DmD data analysis. Regardless of reference material availability,
we feel that complexity is an unavoidable necessity: 1DmD methods are the least complex methods for
controlling false positives in 1DmD datasets. A side benefit of learning 1DmD methods is that all aspects
of simpler analyses become clearer; from an nDmD perspective typical 0D1D procedures are simply the
special case: n=0, m=1, and all concepts relevant to 0D1D analysis (e.g. ANOVA, normality, outliers,
non-sphericity, etc) also apply directly to nDmD data analysis.
Perhaps the most important limitation of 1D techniques is software availability. No commercial statistical
software package of which we aware implements 1D procedures. The most prominent open-source software
packages to implement nD procedures, including packages like SPM8 (Friston et al., 2007), are all tailored
for n=3 or n=4, making them somewhat bulky and overly complex for 1D datasets. A few open-source
packages tailored specifically for 1D analysis exist including: FDA (Ramsay and Silverman, 2005), spm1d
(Pataky, 2012) and rft1d (Pataky, 2016), but all are still at relatively early stages of development. Clear in-
terfaces to 1D procedures in commercial packages would remove a major barrier to 1D procedure accessibility
in Biomechanics.
11
4.4 Summary
Conducting scalar 0D analyses of 1D data without clear apriorispecification of the 0D scalar produces
false positives at relatively high rates (p>0.38 and p>0.76 for 1D scalar trajectories and 1D three-component
vector trajectories, respectively). The most robust protection against false positive results is good experi-
mental design focussing on the testing of a small number of specific hypotheses whose variables are clearly
defined apriori. Since 0D analysis of ambiguous 0D variables fails to consider 1D randomness, 0D techniques
cannot control false positive rates () in experiments whose hypotheses pertain to 1D data. To solve the
problem one should either (i) explicitly identify 0D variables prior to conducting an experiment or (ii) adopt
1D procedures.
Acknowledgments
This work was supported by Wakate A Grant 15H05360 from the Japan Society for the Promotion of
Science. We also wish to thank Cyril J. Donnelly for helpful discussions and continued support.
Conflict of Interest
The authors report no conflict of interest, financial or otherwise.
References
Adler, R. J. and Taylor, J. E. 2007. Random Fields and Geometry, Springer-Verlag, New York.
Besier, T. F., Lloyd, D. G., Chochrane, J. L., and Ackland, T. R. 2001. External loading of the knee joint during running and
cutting maneuvers, Medicine & Science in Sports & Exercise 33(7), 1168–1175.
Besier, T. F., Lloyd, D. G., and Ackland, T. R. 2003. Muscle activation strategies at the knee during running and cutting
maneuvers, Medicine & Science in Sports & Exercise 35(1), 119–127.
Besier, T. F., Fredericson, M., Gold, G. E., Beaupre, G. S., and Delp, S. L. 2009. Knee muscle forces during walking and running
in patellofemoral pain patients and pain-free controls, Journal of Biomechanics 42(7), 898–905, data: https://simtk.org/
home/muscleforces.
Cavanagh, P. R. and Lafortune, M. A. 1980. Ground reaction forces in distance running, Journal of Biomechanics 13(5),
397–406.
Cao, J. and Worsley, K. J. 1999. The detection of local shape changes via the geometry of Hotelling’s T2 fields, Annals of
Statistics 27(3), 925–942.
12
Dorn, T. T., Schache, A. G., and Pandy, M. G. 2012. Muscular strategy shift in human running: dependence of running speed
on hip and ankle muscle performance., Journal of Experimental Biology 215, 1944–1956, data: https://simtk.org/home/
runningspeeds.
Duhamel, A., Bourriez, J., Devos, P., Krystkowiak, P., Destee, A., Derambure, P., and Defebvre, L. 2004. Statistical tools for
clinical gait analysis, Gait and Posture 20(2), 204–212.
Friston, K. J., Worsley, K. J., Frackowiak, R. S. J., Mazziotta, J. C., and Evans, A. C. 1994. Assessing the significance of focal
activations using their spatial extent., Hum Brain Mapp 1, 210–220.
Friston, K. J., Holmes, A., Poline, J. B., Price, C. J., and Frith, C. D. 1996. Detecting activations in PET and fMRI: levels of
inference and power., NeuroImage 4(3), 223–235.
Friston, K. J. 1997. Testing for anatomically specified regional eects., Hum Brain Mapp 5, 133–136.
Friston, K. J., Ashburner, J. T., Kiebel, S. J., Nichols, T. E., and Penny, W. D. 2007. Statistical Parametric Mapping: The
Analysis of Functional Brain Images, Elsevier/Academic Press, Amsterdam.
Jones, E., Oliphant, T., and Peterson, P. 2001. SciPy: Open Source Scientific Tools for Python,http://www.scipy.org.
Kautz, S. A., Feltner, M. E., Coyle, E. F., and Baylor, A. M. 1991. The pedalin technique of elite endurance cyclists: changes
with increasing workload at constant cadence, International Journal of Sport Biomechanics 7, 29–53.
Kiebel, S. J., Poline, J., Friston, K. J., Holmes, A. P., and Worsley, K. J. 1999. Robust smoothness estimation in statistical
parametric maps using standardized residuals from the general linear model, NeuroImage 10(6), 756–766.
Knudson, D. V. 2005. Statistical and reporting errors in applied biomechanics research, Proceedings of the 23rd International
Conference on Biomechanics in Sports, 811–814.
Kundson, D. V. and Lindsey, C. 2014. Type I and Type II errors in correlations of various sample sizes, Comprehensive
Psychology 3(1), Article 1.
Kruschke, J. K. 2013. Bayesian estimation supersedes the t test, Journal of Experimental Psychology: General 142(2), 573–603.
Lenho, M. W., Santer, T. J., Otis, J. C., Peterson, M. G., Williams, B. J., and Backus, S. I. 1999. Bootstrap prediction and
confidence bands: a superior statistical method for analysis of gait data, Gait and Posture 9, 10–17.
Neptune, R. R., Wright, I. C., and van den Bogert, A. J. 1999. Muscle coordination and function during cutting movements,
Medicine & Science in Sports & Exercise 31(2), 294–302, data: http://isbweb.org/data/rrn/.
Nichols, T. E. and Holmes, A. P. 2002. Nonparametric permutation tests for functional neuroimaging a primer with examples,
Human Brain Mapping 15(1), 1–25.
Pataky, T. C . 2012. One-dimensional statistical parametric mapping in Python, Computer Methods in Biomechanics and
Biomedical Engineering 15(3), 295–301.
Pataky, T. C., Robinson, M. A., and Vanrenterghem, J. 2013. Vector field statistical analysis of kinematic and force trajectories.,
Journal of Biomechanics 46(14), 2394–2401.
Pataky, T. C. 2016. RFT1D: Smooth one-dimensional random field upcrossing probabilities in Python, Journal of Statistical
Software, in press.
13
Pataky, T. C., Vanrenterghem, J., and Robinson, M. A. 2015. Zero- vs. one-dimensional, parametric vs. non-parametric, and
confidence interval vs. hypothesis testing procedures in one-dimensional biomechanical trajectory analysis., Journal of Biome-
chanics 48(7), 1277–1285.
Ramsay, J. O. and Silverman, B. W. 2005. Functional Data Analysis, Springer, New York.
Rencher, A. C. and Chistensen, W. F. 2012. Methods of Multivariate Analysis, Wiley, New Jersey.
Robinson, M. A., Vanrenterghem, J., and Pataky, T. C. 2015. Statistical parametric mapping for alpha-based statistical analyses
of multi-muscle EMG time-series, Journal of Electromyography and Kinesiology 25(1), 14–19.
Schwartz, M. H., Trost, J. P., and Wervey, R. A. 2004. Measurement and management of errors in quantitative gait data, Gait
and Posture 20(2), 196–203.
van Rossum, G. 2014. The Python Library Reference Release 2.7.8,https://docs.python.org/2/library/.
Worsley, K. J., Andermann, M., Koulis, T., MacDonald, D., and Evans, A. C. 1999. Detecting changes in nonisotropic images,
Human Brain Mapping 8, 98–101.
Worsley, K. J., Taylor, J. E., Tomaiuolo, F., and Lerch, J. 2004. Unified univariate and multivariate random field theory,
NeuroImage 23, S189–S195.
Zhang, J. T. 2013. Analysis of variance for functional data, CRC Press, London.
14
Table 1: Dataset dimensionality terminology. An “nDmD” dataset contains an m-dimensional
vector or tensor measured over an n-dimensional spatiotemporal domain.
Example
nm ndomain mdomain
01 ground impact instant knee flexion
03instant of max propulsion ground reaction force
11 gait cycle time knee flexion
12 gait cycle time left and right knee flexion
13 stance phase ground reaction force
21 foot contact surface pressure
31 femur von Mises stress
36 femur strain tensor
Table 2: Dataset overview. Nis the total of number of trajectories. Note: the Kautz et al.
(1991) dataset contains 26 trials but six are duplicates. Variables: COP = center of pressure,
EMG = electromyography, GRF = ground reaction force.
Source NTasks Variables Link
Besier et al. (2009) 43 Walking,
Running
GRF, Muscle
forces
simtk.org/home/muscleforces
Caravaggi et al. (2010) 30 Walking plantar arch de-
formation
spm1d.org/Downloads.html
Dorn et al. (2012) 8 Running GRF simtk.org/home/runningspeeds
Fregley et al. (2012) 19 Walking GRF, COP,
Knee implant
forces
simtk.org/home/kneeloads
Kautz et al. (1991) 20Cycling pedal dynamics isbweb.org/data/kautz/
Murley et al. (2014) 5 Walking EMG ncbi.nlm.nih.gov/pubmed/24618372
Neptune et al. (1999) 20 Cutting Kinematics,
EMG
isbweb.org/data/rrn/index.html
Pataky et al. (2008) 600 Walking GRF spm1d.org/Downloads.html
Schwartz et al. (2008) 161 Walking Kinematics ncbi.nlm.nih.gov/pubmed/18565753
Table 3: Smoothness estimation results, summary across all datasets. FWHM values represent
1D residual smoothness (Appendix F) and higher FWHM values reflect smoother residuals. The
FWHM represents Data unit: % of the 1D trajectory length. Refer to Fig.2 for a qualitative
representation of similar smoothness values.
Data modality Min Median Max Mean ±SD
Kinematics 10.8 33.1 67.0 34.7 ±14.1
Forces 6.2 14.3 32.4 16.1 ±7.1
EMG 7.2 11.8 15.6 12.1 ±2.5
Overall 6.2 16.5 67.0 21.9 ±13.6
Table 4: False positive results for two-sample t tests with N=10 per group; summary of results
from Figs.5&6. The three FWHM columns represent the minimum, median, and maximum
value observed across all datasets (Table ??). Vector data are assumed to contain three inde-
pendent components each.
Type of 1D data Reference FWHM=6.2 FWHM=16.5 FWHM=67.0
One scalar Fig.5 p= 0.699 p= 0.382 p= 0.145
One vector Fig.6 p= 0.973 p= 0.764 p= 0.374
Two vectors Fig.6 p= 0.999 p= 0.945 p= 0.609
Six vectors Fig.6 p= 1.000 p= 0.999 p= 0.940
Figure 1: Overview of 0D vs. 1D statistical analyses. (a,b) 0D data extracted from 1D dataset (unit:
decanewton). (c,d) Residuals with 1D residual maxima highlighted. (e,f) Theoretical and simulated probability
density functions; 1D densities correspond to the 1D maximum (RFT= random field theory). Simulations
validate the theoretical predictions and involved 100,000 random residuals (Gaussian densities) and 100,000
two-sample t tests using Gaussian residuals (t densities). (g,h) Hypothesis testing results; the observed t value
exceeds the α-defined t* value for 0D analysis but not 1D analysis so the null hypothesis is rejected for 0D but
not 1D analysis.
Figure 2: Simulated 1D Gaussian random fields. (a) Uncorrelated Gaussian data. (b-e) Data from panel (a)
but smoothed with Gaussian kernels of varying FWHM (see Appendix A). Circles depict trajectory maxima
whose FWHM-dependent probabilistic behavior Random Field Theory (RFT) describes (Adler and Taylor
2007). (f) Infinitely smooth 1D random fields are equivalent to 0D random scalars; RFT results converge to
0D results as FWHM approaches .
Figure 3: Example residuals from two public datasets. (a,b) Original data. (c,d) Residuals; here the
differences between individual trajectories and the mean trajectory. The estimated smoothness values for (c)
and (d) were FWHM=19.6% and FWHM=18.9%, respectively.
Figure 4: Survival functions for the t statistic for different degrees of freedom (ν=4 and ν=48) when the
underlying data are (i) 0D Gaussian scalars (Eqn.1) or (ii) smooth 1D Gaussian trajectories with the median
observed smoothness of FWHM=16.5% (Eqn.3). Solid lines depict theoretical probabilities and dots depict
validation results. The broken horizontal line depicts the classical hypothesis testing threshold for significance,
and broken vertical lines depict the critical t values for the two 0D cases. The intersection of those vertical lines
with the 1D curves gives the true false positive rate (approximately 37% and 47%, respectively) if one regards
the residuals as 1D rather than 0D (Fig.1c,d).
Figure 5: Probability of false positives for two-sample t tests (N=10) if using a 0D randomness model and if
analyzing the maximum 1D t value. Star symbols (FWHM=0) depict the case of completely uncorrelated
trajectories (Eqn.4).
Figure 6: Probability of false positives for two-sample t tests (N=10) for three-component vectors with
independently varying components.
a priori nDmD null
hypothesis
Collect nDmD data
(hypothesis driven)
Conduct nDmD test
(a) Ideal hypothesis testing (b) Ideal exploratory analysis (c) Biomechanics’ hybrid approach
Process data
Collect nDmD data
Formulate post hoc
nDmD null hypothesis
Collect nDmD data
Formulate ad hoc 0D null
hypothesis
Visualize results
Conduct 0D test
(Same dataset)
Visualize results
Ideal hypothesis testing
(independent dataset)
Process data
(hypothesis driven)
Process data
Figure 7: General procedures for hypothesis testing and exploratory analysis. In Biomechanics the
dimensionalities of the test and the data are incommensurate.
Appendix A. Full width at half maximum (FWHM)
The FWHM parameter can be used to describe the smoothness of experimentally observed
1D residuals (Fig.1d, main manuscript). Most generally, the FWHM describes the shape of a
Gaussian kernel (Fig.A1), which is typically defined as:
g(x)= 1
p2e(xµ)2
22(A.1)
Here µand are its mean and standard deviation, respectively. Gaussian kernels can
alternatively be expressed in terms of the FWHM (Fig.A2) through the following identity:
FWHM = 2p2 log 2 2.4(A.2)
The FWHM is somewhat more-intuitive than for describing 1D field smoothness because
it is linked directly to kernel height: the kernel loses half of its maximum height over a dis-
tance of 0.5 FHWM units (Fig.A2). More specifically, the FWHM represents the width of a
Gaussian kernel which, when convolved with uncorrelated Gaussian data (Appendix B), yields
1D Gaussian trajectories with the same smoothness as the observed 1D residuals.
Random field theory (RFT) (Adler & Taylor, 2007; Friston et al. 2007) regards experi-
mentally observed 1D residuals as 1D random fields, and uses the estimated FWHM value to
describe the probabilistic behavior of an infinite number of identically smooth fields. Thus,
once one knows or estimates the FWHM, one can use RFT to calculate the maximum 1D
dierences / eect that random 1D fields would produce in arbitrary experiments.
While could be used in place of the FWHM to describe field smoothness, the main
manuscript uses the FWHM parameter preferentially over because: (i) is typically used
to represent population standard deviation, and (ii) the literature convention is to use FWHM
(Friston et al. 2007).
Figure A1: Gaussian kernels.
Figure A2: Breadth parameters for Gaussian kernels: and FWHM.
Appendix B. Convolution and random 1D Gaussian fields
This Appendix describes one procedure for generating smooth 1D Gaussian fields. Consider
the functions f(q) and g(q0) which are defined on one-dimensional (1D) domains qand q0,
respectively (Fig.B1a,b). Convolution is a procedure which slides g(q0)overf(q) (Fig.B1c) to
yield an ‘overlapping area’ function h(q) (Fig.B1d). Convolving f(q) with a Gaussian kernel
— also called “Gaussian filtering” — yields a similar but smoother result (Fig.B2). In the
context of the main paper the functions f(q), g(q0) and h(q) represent the experimental data,
a smoothing kernel, and the smoothed data, respectively.
Figure B1: Convolution of two square waves. (a) Stationary function. (b) Moving function.
(c) Depiction of g(q0) moving across f(q). (d) Convolution result: colored circles depict the
overlapping area between f(q) and g(q0) when the right edge of g(q0) reaches the position q.
Figure B2: Convolution of a square wave with a Gaussian pulse.
When the data f(q) are more like an experimental time series but consist of completely un-
correlated random Gaussian values (Fig.B3a) then convolving with a Gaussian kernel (Fig.B3b,c)
yields a smooth Gaussian random field (Adler & Taylor, 2007) (Fig.B3d). The broader the
smoothing kernel, the smoother the resulting random field (Fig.B4). Kernel breadth is param-
eterized by its full-width-at-half-maximum (FWHM) (Appendix A) and the FWHM parameter
is central to 1D probability results (Friston et al. 2007).
Figure B3: Convolution of uncorrelated Gaussian data (a) with a Gaussian kernel (b–c) yields
a Gaussian random field (d).
Figure B4: Identical to Fig.B3, but with a broader kernel (b) which yields a smoother random
field (d).
Appendix C. MATLAB functions and scripts
The attached MATLAB functions and scripts (Table C1) replicate the paper’s main re-
sults. Note that these MATLAB files are meant primarily for exploring concepts of 1D anal-
yses through Random Field Theory (RFT) (Adler and Taylor, 2007). We encourage readers
interested in actually conducting 1D analyses to access our free-and-open-source software at
www.spm1d.org.Thespm1d software package contains 1D functionality for standard tests in-
cluding t tests, regression and ANOVA, with multivariate and repeated-measures functionality
currently in-development.
Table C1: Overview of attached MATLAB files.
Category File name Summary
Functions (1D)
estimate fwhm.m Unbiased estimates of 1D field smoothness
randn1d.m Random 1D Gaussian field generator
rft Tsf.m RFT survival function for the t statistic
Functions (0D)
spm invBcdf.m Inverse cumulative distribution function
spm invTcdf.m Inverse t cumulative distribution function
spm Tcdf.m Cumulative distribution function for the t statistic
Data Schwartz2008.mat 1Public dataset from Schwartz et al. (2008)
Scripts (FWHM)
sA0 fwhm verbose.m Verbose smoothness estimation
sA1 fwhm.m Smoothness estimation using estimate fwhm
sA2 fwhm validate.m Validation of estimate fwhm using known FWHM
Scripts (random)
sB0 randn1d verbose.m Verbose random field generation
sB1 randn1d.m Random field generation using randn1d
sB2 randn1d meaning.m The meaning of ‘Gaussian 1D random field’
Scripts (results)
sC0 falsepos.m Verbose estimate of false positive rate.
sC1 validate onesample.m Validation of rft Tsf.m for one-sample tests
sC2 validate twosample.m Validation of rft Tsf.m for two-sample tests
The first three functions embody the core functionality of RFT-based 1D t tests. The
second three functions embody 0D functionality as implemented in the open-source software
package SPM8 (www.fil.ion.ucl.ac.uk/spm); MATLAB users who have access to the Statis-
tics Toolbox may which to use the equivalent 0D three functions: betaincinv”, “tinv” and
tcdf”, respectively. Neither MATLAB nor any other commercial software package of which
1Redistributed with author’s permission.
we are aware implements the first three (1D) functions.
The attached dataset from Schwartz et al. (2008) is redistributed with permission of the
author; if you use these data in your own work please cite the original Schwartz et al. (2008)
article. This dataset is attached to demonstrate how we computed the paper’s smoothness
results (Appendix F). All other datasets can be accessed using the links provided in Table 1
(main manuscript).
Last, the scripts are grouped in three sets of files labeled “sA”, “sB” and “sC”which
correspond to smoothness, 1D randomness and false positive results, respectively. All scripts
and functions are commented for clarity. If any section of any of these scripts and functions is
unclear, please contact us for support through our site at www.spm1d.org.
Appendix D. Estimating 1D residual smoothness
This Appendix summarizes the smoothness estimation procedures of Kiebel et al. (1999).
While that paper describes the procedure for 3D data, the procedure is conceptually identical
for 1D data. The 1D domain could be time, space or any other continuous variable, but for
writing convenience we shall consider only 1D temporal trajectories.
The ultimate goal is to estimate the temporal smoothness of experimentally observed 1D
residuals (Fig.1d, main manuscript) using a single scalar parameter: the FWHM (Appendix A).
That single FWHM value represents the breadth of a Gaussian kernel which, when convolved
with uncorrelated Gaussian time series (Appendix B) would yield random trajectories which
have the same smoothness as the observed residuals. The procedure is depicted in Fig.D1 and
is described in detail below.
Figure D1: Overview of the FWHM estimation procedure of Kiebel et al. (1999). The residual
data (a) are from the cycling normal pedal force of Kautz et al. (1991). SS = sum-of-squares.
Imagine that there are Jresidual trajectories which are each sampled at Qdiscrete time
points and that the jth residual trajectory is denoted “rj(q)”. The first step is to compute the
temporal gradient at each point qfor each of the Jtrajectories (Fig.D1b):
r0
j(q)drj(q)
dq (D.1)
The gradients can be estimated most easily using the dierences between adjacent samples
(i.e. rj(q+ 1) rj(q)), but could also be done using alternative procedures. Practically,
dierences in gradient estimation procedures will likely have negligible eects on the ultimately
estimated FWHM value. Regardless of the procedure, gradient estimation yields a total of J
gradient trajectories.
Next, the true gradient magnitude at point qis estimated as the sum-of-squares of the
observed gradients (Fig.D1c):
SS[r0](q)=
J
X
j=1 r0
j(q)2
(D.2)
In order to normalize across datasets and experiments the sum-of-squared residual values
is also needed (Fig.D1d):
SS[r](q)=
J
X
j=1 rj(q)2
(D.3)
The estimated gradients are then normalized by the residual magnitudes (Fig.D1e):
(q)=SS[](q)
SS[r](q)(D.4)
Last, the FWHM trajectory (Fig.D1f) is given as:
FWHM(q)=s4 log 2
(q)(D.5)
where an unbiased estimate of the true FWHM is simply the mean of the FWHM trajectory:
ˆ
FWHM = 1
Q
Q
X
q=1
FWHM(q) (D.6)
Note that estimated FWHM values are generally dierent at each point qin the 1D field
(Fig.D1f). Smoothness which is non-constant across the field is termed ‘anisotropic’. In order
to deal with this issue let’s consider ‘apparent’ vs. ‘real’ anisotropy. If the anisotropy is merely
apparent, then Eqn.D.6 is valid. To understand why, consider a 0D random variable xwhich
comes from a population with constant variance. Although the population variance is constant,
random samples of xwill yield dierent sample variances. Nevertheless, sample variance is an
unbiased estimate of the true population variance. Similarly, even when the true FWHM is
constant, randomly sampled 1D data will yield sample FWHM estimates which vary not only
from sample to sample but also from point to point in the 1D field. Thus the mean FWHM
value is an unbiased estimate of the true population FWHM when the anisotropy is merely
apparent. Note also this FWHM estimation procedure has been validated for 1D data elsewhere
(Pataky, 2015).
‘True anisotropy’ is a theoretically less trivial problem which exists when dierent field
regions actually do have dierent population-level smoothnesses. As an example, consider
impacts in running ground reaction forces: the initial impact phase is generally associated
with higher signal frequencies than the midstance and push-ophases, so in this situation
the true population FHWM is likely dierent in the dierent phases. There is fortunately an
easy solution to the problem (Worsley et al. 1999). If there are Qpoints in the field, one
simply computes the field length Q0for which smoothness is isotropic. Since the procedures
are validated in Worsley et al. (1999), and since this anisotropy correction has no eect on the
main paper’s conclusions regarding 0D vs. 1D false positives, we leave the issue of anisotropic
smoothness for future projects where assuming isotropic smoothness may have less trivial eects
on analyses’ results.
Appendix E. Generating smooth 1D Gaussian trajectories
This Appendix describes how to generate smooth 1D Gaussian trajectories in MATLAB
(The MathWorks, Natick, USA) by summarizing the mathematical theory and code of Penny
(2008). Interested readers are encouraged to consult Penny (2008) for a variety of additional
background details regarding Random Field Theory and its applications. The target audience
for this Appendix is anyone with (a) familiarity with linear algebra basics and MATLAB
programming, and (b) a desire to generate their own random 1D trajectories for validating
and/or exploring Random Field Theory predictions.
Penny W (2008) Mathematics for Brain Imaging (Course Notes), Chapter 3: “Random Field Theory”.
Retrieved on 12 Feb 2015 from http://www.fil.ion.ucl.ac.uk/~wpenny/mbi/
Smooth Gaussian trajectories can be generated using the convolution procedure described
in Appendix B but in most programming languages it is considerably easier to use a single
matrix multiplication to achieve the same result:
y=Cz (E.1)
where, if Qis the number of trajectory nodes:
yis the (Q1) smooth 1D Gaussian trajectory
Cis the (QQ) convolution matrix which embodies the correlation between neighboring
points in the 1D trajectory
zis a (Q1) trajectory of uncorrelated Gaussian values (Fig.B3a)
In MATLAB zcan be generated as follows:
>> Q = 101;
>> z = randn(Q,1);
The convolution matrix Crequires a bit more work. First note that a Gaussian covariance
function with unit power is defined as:
r(d)=expd2
2s2(E.2)
where:
dis the distance between the current node and a reference node in the trajectory. If
Q=101 then d=0.01 for adjacent nodes, and d=0.01nfor two nodes which are nnodes
apart.
sis the trajectory smoothness which linearly scales with the FWHM (Appendix A) as:
FWHM =s(Q1)p4 log 2 (E.3)
The convolution matrix Cembodies this covariance (Eqn.E.2) simultaneously for all points
in the trajectory. To assemble Cin MATLAB first define the number of trajectory nodes and
the smoothness:
>> Q = 101; % number of trajectory nodes
>> FWHM = 10; % smoothness (FWHM units)
>> s = FWHM / ( (Q-1) * sqrt( 4*log(2) ) ) %smoothness (s=0.060 for FWHM=10)
and then build a distance matrix Das follows:
>> dq = 1 / (Q -1); % inter-node distance
>> x = dq * (1:Q); % trajectory position
>> X = repmat(x, Q, 1); % temporary holder of trajectory positions
>> D = X - repmat(x’, 1, Q); % matrix of distances from diagonal nodes
The (i,j)th element of the resulting matrix represents the distance between nodes iand j:
>> D(1:5, 1:5)
ans =
0 0.0100 0.0200 0.0300 0.0400
-0.0100 0 0.0100 0.0200 0.0300
-0.0200 -0.0100 0 0.0100 0.0200
-0.0300 -0.0200 -0.0100 0 0.0100
-0.0400 -0.0300 -0.0200 -0.0100 0
Next apply Eqn.E.2 to yield the covariance matrix A:
>> A = exp( -0.5 * D.^2 / s^2);
This covariance matrix can be visualized using the ‘pcolor’ command (Fig.E1):
>> pcolor(A)
>> colorbar
Figure E1: Visualization of the covariance matrix Afor Q=101 and FWHM=10.0. The value
of the (i,j)th matrix element represents the correlation values at 1D trajectory nodes iand
j. When i=jthe correlation is defined as 1.0, and since the data are smooth adjacent nodes’
values are positively correlated. The farther apart the nodes the weaker their correlation.
Last, to generate data which have the covariance structure depicted above in Fig.E1, the
convolution matrix C(Fig.E2) is assembled by first calculating the covariance matrix’s eigen-
solution:
>> [V,U] = eig(A);
where Vand Uare A’s eigenvectors and eigenvalues, respectively, and then projecting the
positive eigenvalues as follows:
>> U = diag( U );
>> U(U<0)=0;
>> C = V * diag( sqrt( U ) ) * V’ ;
Figure E2: Visualization of the convolution matrix (Eqn.E.2) for Q=101 and FWHM=10.0.
This convolution matrix Cneeds to be computed only once for each smoothness value. It
can then be used to generate single smooth Gaussian trajectories (Fig.E3) as follows :
>> y0 = C * randn(Q,1);
>> y1 = C * randn(Q,1);
>> y2 = C * randn(Q,1);
>> plot( y0 )
>> plot( y1 )
>> plot( y2 )
Figure E3: Three example 1D Gaussian trajectories.
Alternatively many random trajectories (Fig.E4) can be generated like this:
>> J = 50; % number of trajectories
>> Z = randn(Q, J); % uncorrelated Gaussian data
>> Y = C * Z; % (Q x J) array containing J separate random trajectories
>> plot( Y )
Figure E4: Fifty example 1D Gaussian trajectories.
For the analyses in the main manuscript we generated J=10 trajectories for each of two
groups, computed the two-sample ttrajectory using the usual definition of the two-sample
tstatistic, then repeated 100,000 times for each smoothness (FWHM) value. This yielded
100,000 ttrajectories for each FWHM value. Last, we validated Eqns.3&4 (main manuscript)
by extracting the maximum tvalue (tmax ) for each trajectory and then counting the number
of tmax values which exceeded particular heights u.
Appendix F. FWHM estimation results
This Appendix lists the smoothness estimation results for all nine datasets (Table 1, main
manuscript). First the 1D residuals for each dataset were calculated by subtracting the mean 1D
trajectory as depicted in Fig.3, main manuscript. Next 1D residual smoothness was estimated
as the FWHM (Appendix A) using the procedures of Kiebel et al. (1999) as summarized in
Appendix D. Results are organized below into kinematic, force and EMG variables in Tables
F1, F2 & F3, respectively.
Table F1: Smoothness estimates for the kinematics datasets. Smoothness was quantified using
the FWHM (Appendix A)
Category Source Task Variable FWHM (%)
Joint rotations
Besier et al. (2009) Walking
Hip flexion 51.5
Knee flexion 32.6
Angle dorsiflexion 30.8
Besier et al. (2009) Running
Hip flexion 64.4
Knee flexion 33.6
Angle dorsiflexion 33.5
Neptune et al. (1999) Cutting
Ankle supination/pronation 23.9
Ankle dorsi/plantar flexion 36.4
Knee extension/flexion 36.7
Knee adduction/abduction 21.4
Knee internal/external rotation 46.1
Hip extension/flexion 40.9
Hip adduction/abduction 33.6
Hip internal/external rotation 10.8
Schwartz et al. (2008) Walking
Pelvic Up/Dn 14.2
Pelvis Ant/Pst 33.5
Pelvic Int/Ext 15.6
Hip Flx/Ext 19.6
Hip Add/Abd 13.9
Hip Int/Ext 17.8
Knee Flx/Ext 10.5
Ankle Dor/Pla 9.2
Foot Int/Ext 15.1
Center of pressure Fregley et al. (2012) Walking Anterior/posterior 24.4
Medial/lateral 65.2
Other Kautz et al. (1991) Cycling Pedal angle 33.1
Caravaggi et al. (2010) Walking Plantar arch angle 18.8
Table F2: Smoothness estimates for the dynamics datasets. Results from Pataky et al. (2008)
are based on unsmoothed data.
Category Source Task Variable FWHM (%)
Ground reaction force
Dorn et al. (2012) Running
Anterior / Posterior 8.8
Medial / Lateral 9.5
Vertical 11.1
Fregley et al. (2012) Walking
Anterior / Posterior 8.4
Medial / Lateral 8.2
Vertical 6.2
Neptune et al. (1999) Cutting Vertical 11.9
Pataky et al. (2008) Walking Vertical 6.2
Muscle forces
Besier et al. (2009) Walking
Semimembranosus 16.4
Semitendinosus 15.1
Biceps femoris (long head) 16.7
Biceps femoris (short head) 13.7
Rectus femoris 9.3
Vastus medialis 12.5
Vastus intermedius 12.9
Vastus lateralis 12.9
Medial gastrocnemius 15.1
Lateral gastrocnemius 13.4
Besier et al. (2009) Running
Semimembranosus 29.1
Semitendinosus 29.3
Biceps femoris (long head) 32.4
Biceps femoris (short head) 27.8
Rectus femoris 17.9
Vastus medialis 20.5
Vastus intermedius 21.6
Vastus lateralis 21.8
Medial gastrocnemius 25.0
Lateral gastrocnemius 27.2
Joint implant forces Fregley et al. (2012) Walking
Knee: posterior-medial 14.7
Knee: anterior-medial 11.7
Knee: anterior-lateral 12.3
Knee: posterior-lateral 11.9
Other Kautz et al. (1991) Cycling
Pedal normal force 19.8
Pedal tangental force 15.3
Crank torque 12.7
Table F3: Smoothness estimates for the EMG datasets. Results from Murley et al. (2014)
are based on average time series and FWHM values were estimated relative to the average
cross-task trajectory.
Source Task Variable FWHM (%)
Murley et al. (2014) Walking
Tibialis posterior 10.3
Peroneus longus 10.8
Tibialis anterior 7.2
Medial gastrocnemius 7.9
Neptune et al. (1999) Cutting
Vastus lateralis 13.0
Rectus femoris 15.5
Biceps femoris 15.6
Medial hamstring 14.3
Tibialis anterior 14.2
Medial gastrocnemius 11.8
Gluteus maximus 13.8
Gluteus medius 11.8
Adductor magnus 11.6
Vastus medialis 12.8
Peroneus longus 10.7
... Whilst mean and peak kinetic and kinematic variables have been extensively reported, a more sophisticated and detailed analysis of the force-time data may provide additional insight into where the differences occur between loading conditions, and how practitioners can appropriately implement these exercises. It is recommended that when testing non-directed hypotheses involving biomechanical vector fields, researchers should implement statistical parametric mapping analysis (SPM) as it is generally biased to test one-dimensional data (1D) using zero-dimensional methods, and SPM may reduce such bias (Pataky et al., 2013(Pataky et al., , 2015(Pataky et al., , 2016. Researchers have utilized time-normalized curve analysis (sometimes termed waveform or temporal phase analysis) to assess force-, velocity-, power-and displacement-time data during weightlifting derivatives (Kipp et al., 2021;Suchomel & Sole, 2017a, 2017b and jumps (Cormie et al., 2008(Cormie et al., , 2009McMahon, Murphy et al., 2017;. ...
... Curve analysis indicated that the jump shrug exhibited greater ground reaction force from ~46% to 50% of the movement and lower vertical velocities and power from ~72% to 76% and ~70% to 76% of the movement, when compared to the hang power clean. However, these differences were not observed with the SPM analysis, highlighting that the differences observed in the curve analysis may be related to an increase in type one error (Pataky et al., 2016). Statistical parametric mapping has been previously used to compare performances in jumping (Hughes et al., 2021) and weightlifting derivatives (Kipp et al., 2021), and may be a more appropriate analysis of time-series data compared to a temporal phase analysis (Kipp et al., 2021). ...
Article
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The effect of load on time-series data has yet to be investigated during weightlifting derivatives. This study compared the effect of load on the force-time and velocity-time curves during the countermovement shrug (CMS). Twenty-nine males performed the CMS at relative loads of 40%, 60%, 80%, 100%, 120%, and 140% one repetition maximum (1RM) power clean (PC). A force plate measured the vertical ground reaction force (VGRF), which was used to calculate the barbell-lifter system velocity. Time-series data were normalized to 100% of the movement duration and assessed via statistical parametric mapping (SPM). SPM analysis showed greater negative velocity at heavier loads early in the unweighting phase (12-38% of the movement), and greater positive velocity at lower loads during the last 16% of the movement. Relative loads of 40% 1RM PC maximised propulsion velocity, whilst 140% 1RM maximized force. At higher loads, the braking and propulsive phases commence at an earlier percentage of the time-normalized movement, and the total absolute durations increase with load. It may be more appropriate to prescribe the CMS during a maximal strength mesocycle given the ability to use supramaximal loads. Future research should assess training at different loads on the effects of performance.
... The averaged ankle and knee kinematic waveforms, consisting of 101 points were imported to the SPM1d software (version 0.4, available for download at http://www.spm1 d.org/, accessed on 15 August 2022) in Matlab (The MathWorks, Natick, MA, USA, version 2020b). Clusters were formed if the critical threshold, based on random field theory, was crossed [50,82,83] (alpha = 0.05). In case of significant clusters, the p-value, details on the location and the duration (percentages of the entire gait cycle) were reported. ...
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Children with spastic cerebral palsy (SCP) are often treated with intramuscular Botulinum Neurotoxin type-A (BoNT-A). Recent studies demonstrated BoNT-A-induced muscle atrophy and variable effects on gait pathology. This group-matched controlled study in children with SCP compared changes in muscle morphology 8–10 weeks post-BoNT-A treatment (n = 25, median age 6.4 years, GMFCS level I/II/III (14/9/2)) to morphological changes of an untreated control group (n = 20, median age 7.6 years, GMFCS level I/II/III (14/5/1)). Additionally, the effects on gait and spasticity were assessed in all treated children and a subgroup (n = 14), respectively. BoNT-A treatment was applied following an established integrated approach. Gastrocnemius and semitendinosus volume and echogenicity intensity were assessed by 3D-freehand ultrasound, spasticity was quantified through electromyography during passive muscle stretches at different velocities. Ankle and knee kinematics were evaluated by 3D-gait analysis. Medial gastrocnemius (p = 0.018, −5.2%) and semitendinosus muscle volume (p = 0.030, −16.2%) reduced post-BoNT-A, but not in the untreated control group, while echogenicity intensity did not change. Spasticity reduced and ankle gait kinematics significantly improved, combined with limited effects on knee kinematics. This study demonstrated that BoNT-A reduces spasticity and partly improves pathological gait but reduces muscle volume 8–10 weeks post-injections. Close post-BoNT-A follow-up and well-considered treatment selection is advised before BoNT-A application in SCP.
... peak values) and 'one-dimensional' (1D; e.g. time-normalised kinematic waveform) variables [6]. Analyses of these common kinematic variables in both their 0D and 1D forms may provide valuable insight into the number of gait cycles required in biomechanical research. ...
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A common approach in biomechanical analysis of running technique is to average data from several gait cycles to compute a 'representative mean.' However, the impact of the quantity and selection of gait cycles on biomechanical measures is not well understood. We examined the effects of gait cycle selection on kinematic data by: (i) comparing representative means calculated from varying numbers of gait cycles to 'global' means from the entire capture period; and (ii) comparing representative means from varying numbers of gait cycles sampled from different parts of the capture period. We used a public dataset ( n = 28) of lower limb kinematics captured during a 30-second period of treadmill running at three speeds (2.5m·s ⁻¹ , 3.5m·s ⁻¹ and 4.5m·s ⁻¹ ). 'Ground truth' values were determined by averaging data across all collected strides and compared to representative means calculated from random samples (1,000 samples) of n (range = 5-30) consecutive gait cycles. We also compared representative means calculated from n (range = 5-15) consecutive gait cycles randomly sampled (1,000 samples) from within the same data capture period. The mean, variance and range of the absolute error of the representative mean compared to the 'ground truth' mean progressively reduced across all speeds as the number of gait cycles used increased. Similar magnitudes of 'error' were observed between the 2.5m·s ⁻¹ and 3.5m·s ⁻¹ speeds at comparable gait cycle numbers - where the maximum errors were < 1.5 degrees even with a small number of gait cycles (i.e. 5-10). At the 4.5m·s ⁻¹ speed, maximum errors typically exceeded 2-4 degrees when a lower number of gait cycles were used. Subsequently, a higher number of gait cycles (i.e. 25-30) was required to achieve low errors (i.e. 1-2 degrees) at the 4.5m·s ⁻¹ speed. The mean, variance and range of absolute error of representative means calculated from different parts of the capture period was consistent irrespective of the number of gait cycles used. The error between representative means was low (i.e. < 1.5 degrees) and consistent across the different number of gait cycles at the 2.5m·s ⁻¹ and 3.5m·s ⁻¹ speeds, and consistent but larger (i.e. up to 2-4 degrees) at the 4.5m·s ⁻¹ speed. Our findings suggest that selecting as many gait cycles as possible from a treadmill running bout will minimise potential 'error.' Analysing a small sample (i.e. 5-10 cycles) will typically result in minimal 'error' (i.e. << 2 degrees), particularly at lower speeds (i.e. 2.5m·s ⁻¹ and 3.5m·s ⁻¹ ). Researchers and clinicians should consider the balance between practicalities of collecting and analysing a smaller number of gait cycles against the potential 'error' when determining their methodological approach. Irrespective of the number of gait cycles used, we recommend that the potential 'error' introduced by the choice of gait cycle number be considered when interpreting the magnitude of effects in treadmill-based running studies.
... Pataky et al. (2015) stated that using one value may be biased. In fact it can sometimes show statistical significance when there is no 1-D effect (Pataky et al., 2016). Alternatively, the aforementioned, SPM method (Pataky, 2010) is suggested to provide a better insight concerning the movement pattern for the entire time series (Hughes et al., 2020). ...
Article
The aim of the current study was to compare the arm-stroke kinematics during maximal and sub-maximal breaststroke swimming using both discrete and continuous data analysis. Nine male breaststrokers swam 2 x 25 m with maximal and sub-maximal intensity and their full body 3-D kinematics were obtained using eight video cameras. The arm-stroke was divided into five phases: recovery, glide, out-sweep, in & down-sweep and in & up-sweep. The statistical treatment of selected discrete variables was conducted using t-test, while the analysis of their equivalent time series, when applicable, was conducted using Statistical Parametric Mapping. Sub-maximal trial, compared to maximal, presented lower swimming velocity, greater stroke length and less stroke rate. Moreover, the absolute and relative duration of the glide phase was longer, while the relative duration of all the other phases was shorter. The resultant hand velocity during the arm recovery was slower, as well as the hand velocity time series in the transverse and longitudinal axis which were slower from ∼45% to ∼60% and from ∼5% to ∼15% of the stroke cycle, respectively. Both discrete and continuous data analysis revealed that the main discriminating factor between the two conditions concerns to the adjustment of the glide and the recovery phase and consequently the continuation of the propulsive movements.
... The entire waveform of each joint angle and joint moment were contrasted between the six gait conditions based on one-dimensional statistical parametric mapping analysis (SPM(F)) [25]. After verification of the normality of the distribution, the temporal smoothness of SPM (F) based on its average temporal gradient was estimated. ...
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Background Knee braces and lateral wedge foot orthoses are two treatment options recommended for medial knee osteoarthritis, but the combination of both of them could further improve their effectiveness. Research Question The aim was to evaluate whether the combination of lateral wedge foot orthoses with two types of knee brace enhances the biomechanical effects and pain relief during the stance phase of gait while maintaining comfort. Methods Ten patients with medial knee osteoarthritis were fitted with a standard valgus brace, an unloader brace with valgus and external rotation functions, and 7° lateral wedge foot orthoses. The pain relief, comfort, kinematics and kinetics of the lower limb were measured during walking without orthotics, with the combined and with the isolated treatments. Results The valgus and external rotation brace significantly reduced the knee adduction moment and allowed more knee flexion both in isolation and in combination to foot orthoses compared to the valgus brace or without treatment. Pain relief was not significant with the different orthotic treatment modalities. The valgus brace and combined treatment with either brace significantly increased the discomfort level, whereas the valgus and external rotation brace or foot orthoses in isolation did not induce significant discomfort. Significance Amongst the tested orthotic treatment modalities, the valgus and external rotation brace obtained better biomechanical outcomes while maintaining comfort. The combined treatment with foot orthoses enhanced the effectiveness of the valgus brace, however foot orthoses may be unnecessary with the valgus and external rotation brace.
... On closer inspection, Hardee et al. (41) evaluated lifting technique by comparing each data point within the barbell trajectory between the first and lastrepetition using a t-test. Given that barbell trajectory data is continuous, this statistical method is not appropriate and is likely to result in false positives(81). As such, more research on the effect of cluster sets on weightlifting technique analyzed with an appropriate statistical method is warranted. ...
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... Statistical parametric mapping (SPM) is a multidimensional data analysis technique that allows the presentation of statistical outputs in the original time series, providing an understanding of temporal regions where significant differences may occur (Pataky, 2012). It has proven to be applicable to a variety of biomechanical datasets including joint angle waveforms (Pataky, 2012;Pataky et al., 2013;Pataky et al., 2016;Donnelly et al., 2017;Pincheira et al., 2019;Papi et al., 2020;Moisan et al., 2021). Analyzing the entire gait waveform directly can enhance the understanding of the strategies adopted for the full gait cycle (Donnelly et al., 2017). ...
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Precise identification of deficient intersegmental coordination patterns and functional limitations is conducive to the evaluation of surgical outcomes after total knee arthroplasty (TKA) and the design of optimal personalized rehabilitation protocols. However, it is still not clear how and when intersegmental coordination patterns change during walking, and what functional limitations are in patients with TKA. This study was designed to investigate lower limb intersegmental coordination patterns in patients with knee osteoarthritis before and after TKA and identify how intersegmental coordination of patients is altered during walking before and after TKA. It was hypothesized that 6-month after TKA, intersegmental coordination patterns of patients are improved compared with that before TKA, but still do not recover to the level of healthy subjects. Gait analysis was performed on 36 patients before and 6-month after TKA and on 34 healthy subjects. Continuous relative phase (CRP) derived from the angle-velocity phase portrait was used to measure the coordination between interacting segments throughout the gait cycle. Thigh-shank CRP and shank-foot CRP were calculated for each subject. Statistical parametric mapping (SPM), a one-dimensional analysis of the entire gait cycle curve, was performed directly to determine which periods of the gait cycle were different in patients and healthy subjects. Six-month after TKA, thigh-shank CRP was significantly higher during 5–12% of the gait cycle ( p = 0.041) and lower during 44–95% of the gait cycle ( p < 0.001) compared with healthy subjects, and was significantly higher during 62–91% of the gait cycle ( p = 0.002) compared with pre-operation. Shank-foot CRP was significantly lower during 0–28% of the gait cycle ( p < 0.001) and higher during 58–94% of the gait cycle ( p < 0.001) compared with healthy subjects, and was significantly lower during 3–18% of the gait cycle ( p = 0.005) compared with pre-operation. This study found that patients exhibited altered intersegmental coordination during the loading response and swing phase both before and after TKA. Six-month after TKA, the thigh-shank coordination was partially improved compared with pre-operation, but still did not recover to the level of healthy subjects, while there was no improvement in the shank-foot coordination pattern after TKA compared with pre-operation. CRP combined with SPM methods can provide insights into the evaluation of surgical outcomes and the design of rehabilitation strategy.
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Overloading of tendon tissue may result in overuse tendon injuries in runners. One possible cause of overloading could be the occurrence of biomechanical inter-limb differences during running. However, scarce information exists concerning the simultaneous analysis of inter-limb differences in external and internal loading-related variables in habitual runners. In this study ground reaction force, joint kinematics, triceps surae and tibialis anterior activations, and medial gastrocnemius muscle-tendon junction displacement were assessed bilaterally during treadmill running at 2.7 m.s⁻¹ and 4.2 m.s⁻¹. Statistical parametric t-tests and effect sizes were calculated to identify eventual inter-limb differences across the stance phase and stride cycle. Hip flexion angle was 9° greater (p = 0.03, ES = 0.30) in the non-preferred limb during the flight phase at 4.2 m.s⁻¹. Hip extension velocity was 45 deg.s⁻¹ greater (p = 0.04, ES = 0.41) during ground contact and 25 deg.s⁻¹ greater (p = 0.02, ES = 0.41) in the non-preferred limb immediately after toe-off at 4.2 m.s⁻¹. Hip extension velocity was also 40 deg.s⁻¹ greater (p = 0.01, ES = 0.46) in the non-preferred limb prior to touch-down at 4.2 m.s⁻¹. Brief inter-limb differences in joint kinematics were not accompanied by inter-limb differences in variables associated to internal loading, suggesting they are unlikely to be underlying factors leading to tendon overloading in healthy non-injured runners.
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Ankle osteoarthritis is a chronic debilitating disease marked by cartilage breakdown, pain and significant biomechanical impairment of the entire lower limb. Total ankle replacement (TAR) has been encouraged during the last decade as it has the potential to maintain the existing pre-operative ankle range of motion and to protect the more distally located joints of the foot. Three-dimensional gait analysis using a multi-segment foot model can provide an objective analysis of TAR for the treatment of end-stage ankle osteoarthritis. Thirty-six patients suffering from post-traumatic end-stage ankle osteoarthritis were evaluated before and after TAR. A four-segment kinematic foot model was used to calculate intrinsic foot joint kinematics during gait. Spatio-temporal parameters were also assessed. Kinematic results were compared to a control group of asymptomatic subjects. Differences in waveform patterns were mainly limited to dorsi-/plantarflexion inter-segment angles. At loading response, the Shank-Calcaneus plantarflexion angles as well as the Calcaneus-Midfoot dorsiflexion angle increased slightly in post-operative condition. During propulsion, an increase in Hallux-Metatarsus dorsiflexion angle was observed. Pain improved after surgery as supported by increased spatio-temporal parameters. While multi-segment foot and ankle kinematics were improved, they remained impaired compared to control values. This study confirms that TAR maintains the residual pre-operative range of motion after surgery from midstance to propulsion. Furthermore, the results suggest that the kinematic behavior of the foot joints distal to the affected ankle joint also improves post-operatively. The outcome of this study further emphasizes the clinical relevance of multi-segment foot modeling when assessing the outcome of TAR.
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This study documented the effect of sample sizes commonly seen in exercise science research on type I and type II errors in statistical tests of numerous correlations. Data on tennis string testing were used to examine zero-order and partial correlations between six variables for the population (N = 198) and three randomly drawn sub-samples of 99, 50, and 25. Sample size and statistical analysis procedure affected the rates of statistical errors. Reducing sample size increased type II errors 7% to 21% using correlation analysis. Partial correlation analysis of smaller samples increased type II errors 29% to 85%. Correlation studies of small sample sizes are likely vulnerable to type I or type II statistical errors and should be interpreted with caution.
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In an age where the amount of data collected from brain imaging is increasing constantly, it is of critical importance to analyse those data within an accepted framework to ensure proper integration and comparison of the information collected. This book describes the ideas and procedures that underlie the analysis of signals produced by the brain. The aim is to understand how the brain works, in terms of its functional architecture and dynamics. This book provides the background and methodology for the analysis of all types of brain imaging data, from functional magnetic resonance imaging to magnetoencephalography. Critically,Statistical Parametric Mappingprovides a widely accepted conceptual framework which allows treatment of all these different modalities. This rests on an understanding of the brain's functional anatomy and the way that measured signals are caused experimentally. The book takes the reader from the basic concepts underlying the analysis of neuroimaging data to cutting edge approaches that would be difficult to find in any other source. Critically, the material is presented in an incremental way so that the reader can understand the precedents for each new development. This book will be particularly useful to neuroscientists engaged in any form of brain mapping; who have to contend with the real-world problems of data analysis and understanding the techniques they are using. It is primarily a scientific treatment and a didactic introduction to the analysis of brain imaging data. It can be used as both a textbook for students and scientists starting to use the techniques, as well as a reference for practicing neuroscientists. The book also serves as a companion to the software packages that have been developed for brain imaging data analysis. * An essential reference and companion for users of the SPM software * Provides a complete description of the concepts and procedures entailed by the analysis of brain images * Offers full didactic treatment of the basic mathematics behind the analysis of brain imaging data * Stands as a compendium of all the advances in neuroimaging data analysis over the past decade * Adopts an easy to understand and incremental approach that takes the reader from basic statistics to state of the art approaches such as Variational Bayes * Structured treatment of data analysis issues that links different modalities and models * Includes a series of appendices and tutorial-style chapters that makes even the most sophisticated approaches accessible.