![Vector field schematic: a two-component vector varying in time. Depicted are mean ground reaction force (GRF) vectors F = [Fx Fy] T from one subject during running (Dorn et al., 2012), where +x and +y represent the anterior and vertical directions, respectively. These vectors, when projected on the (Time, Fx) and (Time, Fy) planes, produce common GRF plots (see Fig. 4a,b); here vertical dotted lines depict standard deviation 'clouds'. When F is projected on the (Fx, Fy) plane these standard deviations are revealed to arise from covariance ellipses, where ellipse orientation indicates the direction of maximum covariance between Fx and Fy (see Supplementary Material-Appendix B).](https://www.researchgate.net/profile/Todd-Pataky/publication/255738742/figure/fig8/AS:668428574683148@1536377199565/Vector-field-schematic-a-two-component-vector-varying-in-time-Depicted-are-mean-ground_Q320.jpg)
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Vector field statistical analysis of kinematic and force trajectories
Todd C. Pataky1, Mark A. Robinson2, and Jos Vanrenterghem2
1Department of Bioengineering, Shinshu University, Japan
2Research Institute for Sport and Exercise Sciences, Liverpool John Moores University, UK
Abstract
When investigating the dynamics of three-dimensional multi-body biomechanical systems it is often diffi-
cult to derive spatiotemporally directed predictions regarding experimentally induced e↵ects. A paradigm
of ‘non-directed’ hypothesis testing has emerged in the literature as a result. Non-directed analyses typ-
ically consist of ad hoc scalar extraction, an approach which substantially simplifies the original, highly
multivariate datasets (many time points, many vector components). This paper describes a commen-
surately multivariate method as an alternative to scalar extraction. The method, called ‘statistical
parametric mapping’ (SPM), uses random field theory to objectively identify field regions which co-vary
significantly with the experimental design. We compared SPM to scalar extraction by re-analyzing three
publicly available datasets: 3D knee kinematics, a ten-muscle force system, and 3D ground reaction
forces. Scalar extraction was found to bias the analyses of all three datasets by failing to consider suf-
ficient portions of the dataset, and/or by failing to consider covariance amongst vector components.
SPM overcame both problems by conducting hypothesis testing at the (massively multivariate) vector
trajectory level, with random field corrections simultaneously accounting for temporal correlation and
vector covariance. While SPM has been widely demonstrated to be e↵ective for analyzing 3D scalar
fields, the current results are the first to demonstrate its e↵ectiveness for 1D vector field analysis. It was
concluded that SPM o↵ers a generalized, statistically comprehensive solution to scalar extraction’s over-
simplification of vector trajectories, thereby making it useful for objectively guiding analyses of complex
biomechanical systems.
Keywords: biomechanics, random field theory, Statistical Parametric Mapping, multivariate statistics
1
Glossary
Category Symbol Other Description
Counts
Index:
I i Vector components
J j Responses (i.e. experimental recordings)
K k Predictor variables
NExtracted scalars (e.g. maximum force)
Q q Field measurement nodes (e.g. 100 points in time)
Responses
Mean, variance:
yiy,s2Scalar response (with st.dev.)
yi(q)y(q), s2(q) Scalar field response (with st.dev. field)
y(q)y(q), W(q) Vector field response (with covariance field)
Test statistics
field:
tSPM{t}⌘t(q)Student’s tstatistic
FSPM{F}⌘F(q) Variance ratio (e.g. from ANOVA)
T2SPM{T2}⌘T2(q)Hotelling’s T2statistic (vector equivalent of t)
RCanonical correlation coefficient
Probability ↵Type I error rate
pProbability value
Acronyms
CCA Canonical correlation analysis
EMG Electromyography
GRF Ground reaction force
PFP Patellofemoral pain
2
1 Introduction
Measurements of motion and the forces underlying that motion are fundamental to biomechanical ex-
perimentation. These measurements are often manifested as one-dimensional (1D) scalar trajectories yi(q),
where irepresents a particular physical body, joint, axis or direction, and where qrepresents 1D time or
space. Experiments typically involve repeated measurements of yi(q) followed by registration (i.e. homolo-
gously optimal temporal or spatial normalization) to a domain of 0–100% (Sadeghi et al., 2003). This paper
pertains to analysis of registered data yi(q).
Given that many potential sources of bias exist in yi(q) analysis (Rayner, 1985; James and Bates, 1997;
Mullineaux et al., 2001; Knudson, 2009), a non-trivial challenge is to employ statistical methods that are
consistent with one’s null hypothesis. Consider first ‘directed’ null hypotheses: those which claim response
equivalence in particular vector components i, and in particular points qor windows [q0,q1]:
Example ‘directed’ null hypothesis: Controls and Patients exhibit identical maximum knee flexion
during walking between 20% and 30% stance.
To test this hypothesis only maximum knee flexion should be assessed, and only in the specified time
window. Testing other time windows, joints, or joint axes in a post hoc sense would constitute bias. This is
because increasing the number of statistical tests increases our risk of incorrectly rejecting the null hypothesis
(see Supplementary Material – Appendix A). In other words, it is biased to expand the scope of one’s null
hypothesis after seeing the data. We refer to this type of bias as ‘post hoc regional focus bias’.
Next consider ‘non-directed’ null hypotheses: hypotheses which broadly claim kinematic or dynamic
response equivalence:
Example ‘non-directed’ null hypothesis: Controls and Patients exhibit identical hip and knee kine-
matics during stance phase.
To address this hypothesis both hip and knee joint rotations should be assessed, about all three orthogonal
spatial axes, and from 0% to 100% stance (i.e. the entire dataset yi(q)). It would be biased to assess only
maximum hip flexion, for example, in a post hoc sense but for the opposite reason: it is biased to reduce the
scope of one’s null hypothesis after seeing the data.
Non-directed hypotheses expose a second potential source of bias: covariance among the Ivector compo-
nents. Scalar analyses ignore covariance and are therefore coordinate-system dependent (see Supplementary
3
Material – Appendix B). This is important because a particular coordinate system — even one defined
anatomically and local to a moving segment — may not reflect underlying mechanical function (Kutch and
Valero-Cuevas, 2011). Joint rotations, for example, may not be independent because muscle lines of action
are generally not parallel to externally-defined axes (Jensen and Davy, 1975). Joint moments may also not
be independent because endpoint force control, for example, requires coordinated joint moment covariance
(Wang et al., 2000). Under a non-directed hypothesis this covariance must be analyzed because separate
analysis of the Icomponents is equivalent to an assumption of independence, an assumption which may not
be justified (see Supplementary Material – Appendix B). We refer to this source of bias as ‘inter-component
covariance bias’.
Both post hoc regional focus bias and inter-component covariance bias have been acknowledged previously
(Rayner, 1985; James and Bates, 1997; Mullineaux et al., 2001; Knudson, 2009). However, to our knowledge
no study has proposed a comprehensive solution.
The purpose of this paper is to show that a method called Statistical Parametric Mapping (SPM) (Friston
et al., 2007) greatly mitigates both bias sources. The method begins by regarding the data yi(q) as a vector
field y(q), a multi-component vector ywhose values change in time or space q(Fig.1). When regarding the
data in this manner, it is possible to use random field theory (RFT) (Adler and Taylor, 2007) to calculate
the probability that observed vector field changes resulted from chance vector field fluctuations.
We use SPM and RFT to conduct formalized hypothesis testing on three separate, publicly available
biomechanical vector field datasets. We then contrast these results with the traditional scalar extraction
approach. Based on statistical disagreement between the two methods we infer that, by definition, at least
one of the methods is biased. We finally use mathematical arguments (Supplementary Material) and logical
interpretations of the original data to conclude that scalar extraction constitutes a biased approach to non-
directed hypothesis testing, and that SPM overcomes these biases.
2 Methods
2.1 Datasets
We reanalyzed three publicly available datasets (Table 1):
•Dataset A (Neptune et al., 1999) (http://isbweb.org/data/rrn/): stance-phase lower extremity dy-
namics in ten subjects performing ballistic side-shu✏e and v-cut tasks (Fig.2). Present focus was on
4
within-subject mean three dimensional knee rotations for the eight subjects whose data were labeled
unambiguously in the public dataset.
•Dataset B (Besier et al., 2009) (https://simtk.org/home/muscleforces): stance-phase knee-muscle
forces during walking and running in 16 Controls and 27 Patello-Femoral Pain (PFP) patients, as
estimated from EMG-driven forward-dynamics simulations. Present focus was on walking and abso-
lute forces (newtons) (Fig.3).
•Dataset C (Dorn et al., 2012) (https://simtk.org/home/runningspeeds): one subject’s full-body kine-
matics and ground reaction forces (GRF) during running at four di↵erent speeds: 3.56, 5.20, 7.00, and
9.49 ms1. Present focus was on three-dimensional left-foot GRF (Fig.4), for which a total of eight
responses were available. We linearly interpolated the GRF data across stance phase to Q=100 time
points.
These three datasets were chosen, first, to represent a range of biomechanical data modalities: kinematics,
modeled (internal) muscle forces, and external forces. Second, they were chosen to demonstrate how vector
field analysis applies to a range of statistical tests: (A) paired ttests, (B) two-sample ttests, and (C) linear
regression.
2.2 Traditional scalar extraction analysis
Two, ten, and three scalars were respectively extracted from the three datasets (Table 1). These particular
scalars were chosen either because they appeared to be most a↵ected by the experiment (Datasets A and
C), or because they were physiologically relevant (Dataset B: maximum force). As indicated above, Dataset
A’s task e↵ects were assessed using paired ttests, Dataset B’s group e↵ects were assessed using two-sample
(independent) ttests, and Dataset C’s speed e↵ects were assessed using linear regression.
Since we conducted one test for each scalar, we performed N=2, N=10 and N=3 tests on Datasets A, B
and C, respectively, where Nis the number of extracted scalars. To retain a family-wise Type I error rate
of ↵=0.05 we adopted ˇ
Sid´ak thresholds of p=0.0253, p=0.0051, and p=0.0170 respectively, where the ˇ
Sid´ak
threshold is:
pcritical =1(1 ↵)1/N (1)
These scalar analyses superficially appear to be legitimate analysis options. However, through comparison
5
with the equivalent vector field analyses (§2.3), we will show how and why scalar extraction is biased for
non-directed null hypothesis testing.
2.3 Statistical Parametric Mapping (SPM)
SPM analyses (Friston et al., 2007) were conducted using vector field analogs to the aforementioned
univariate tests (§2.2). Before detailing SPM procedures, we note that they are conceptually identical to
univariate procedures: conducting a one-sample ttest on ten scalar values, for example, is nearly identical
to conducting a one-sample ttest on ten vector fields. The only di↵erences are that SPM: (i) considers
vector covariance when computing the test statistic, (ii) considers field smoothness and size when computing
the critical test statistic threshold, and (iii) considers random field behavior when computing pvalues (see
Appendix A and B – Supplementary Material).
Ultimately each SPM test results in a test statistic field (e.g. the tstatistic as a function of time), and
RFT is used to assess the significance of this statistical field. §2.3.1–§2.3.3 below detail test statistic field
computations for the current datasets, §2.3.4 describes RFT computations of critical test statistic values and
pvalues, and §2.3.5 suggests a post hoc procedure for scrutinizing vector field test results.
2.3.1 Paired Hotelling’s T2test (Dataset A)
SPM’s vector field analog to the paired ttest is the paired Hotelling’s T2test, which is given by the
one-sample T2statistic (Cao and Worsley, 1999):
SPM{T2}⌘T2(q)=Jy(q)>W(q)1y(q) (2)
where Jis the number of vector fields (Table 1) and y(q) is the mean vector field or — in the case of a
paired test — the vector field di↵erence y(q) (see Supplementary Material – Appendix C). Wis the (I⇥I)
sample covariance matrix:
W(q)= 1
J1
0
@
J
X
j=1
⇣yj(q)y(q)⌘⇣yj(q)y(q)⌘>1
A(3)
representing the variances-within and correlations-between vector components across the Jresponses (Sup-
plementary Material - Appendix D).
The notation “SPM{T2}” (Friston et al., 2007) indicates that the test statistic T2varies in continu-
ous time (or space), forming a temporal (or spatial) statistical ‘map’. To clarify: “SPM” refers to the
6
methodology, and “SPM{T2}” to a specific variable.
2.3.2 Two-sample Hotelling’s T2test (Dataset B)
SPM’s vector field analog to the two-sample ttest is the Hotelling’s T2test (Cao and Worsley, 1999):
SPM{T2}⌘T2(q)= J1J2
J1+J2⇣y1(q)y2(q)⌘>
W(q)1⇣y1(q)y2(q)⌘(4)
where subscripts “1” and “2” index the two groups. Here Wis the pooled covariance matrix:
W=1
J1+J22
0
@
J1
X
j=1
(y1jy1)(y1jy1)>+
J2
X
j=1
(y2jy2)(y2jy2)>1
A(5)
where the domain “(q)” is dropped for compactness.
2.3.3 Canonical correlation analysis (Dataset C)
SPM’s vector field analog to linear regression is canonical correlation analysis (CCA) (Hotelling, 1936;
Worsley et al., 2004). The goal of CCA is to determine the strength of linear correlation between a set of
predictor variables xj(K-component vectors) and a set of response variables yj(I-component vectors). We
provide a brief technical summary of CCA. An extended discussion is provided as Supplementary Material
(Appendix E).
Following Worsley et al. (2004), the test statistic of interest was the maximum canonical correlation (R),
a single correlation coefficient which varies over q, and which transforms to the Fstatistic via the identity:
SPM{F}⌘F(q)=R(q)J1
1R(q)(6)
To compute R, one must first assemble three covariance matrices:
•CXX —the(K⇥K) predictors covariance matrix
•CYY —the(I⇥I) responses covariance matrix
•CXY —the(K⇥I) predictor-response covariance matrix
The maximum canonical correlation (R) is the maximum eigenvalue of the (K⇥K) canonical correlation
matrix (C) (Worsley et al., 2004):
7
C=C1
XXCXY C1
YYC>
XY (7)
An equivalent interpretation is that Ris the maximum correlation coefficient obtainable when the pre-
dictor and response coordinate systems are permitted to mutually rotate. K=2 predictors (running speed
and an intercept) were employed to model the I=3 force vector components of Dataset C.
2.3.4 Statistical inference
To determine the significance of the aforementioned test statistic fields, first field smoothness was es-
timated from the temporal gradients of the residuals (Friston et al., 2007). Next, given this smoothness,
RFT (Adler and Taylor, 2007) was used to determine the critical test statistic threshold that retained a
family-wise error rate of ↵=0.05 (Cao and Worsley, 1999; Worsley et al., 2004). Last, the probability with
which suprathreshold clusters could have been produced by chance (i.e. by random fields with the same
temporal smoothness) was calculated using analytic expectation (Cao and Worsley, 1999). In other words,
rather than controlling the false-positive rate at each point in time, we presently controlled the false-positive
rate of the test statistic field’s sample-rate invariant topological features (Friston et al., 2007). For additional
details refer to Appendix A (Supplementary Material).
2.3.5 Post hoc scalar field SPM
When testing non-directed hypotheses regarding biomechanical vector fields, we propose that SPM should
be implemented in a hierarchical manner, analogous to ANOVA with post hoc t testing. One should first
use SPM to analyze the entire vector field y(q), and particular vector components (scalar field yi(q)) should
only be tested, in a post hoc manner, if statistical significance is reached at the vector-field level.
Following vector field analyses, pos t hoc tests were conducted on each vector component separately (i.e.
on scalar fields). For scalar fields the aforementioned tests (§2.3.1–§2.3.3) reduce to: the paired tstatistic
(Dataset A), the two-sample tstatistic (Dataset B), and the linear regression tstatistic (Dataset C). Each
scalar field test produced one SPM{t}, whose significance was determined as described above §.2.3.1. To
maintain a family-wise error of ↵=0.05, Sid´ak thresholds (Eqn.1) of p=0.0170, p=0.0051, and p=0.0170 were
used to correct for the I=3, I=10, and I=3 vector components of Datasets A, B, and C, respectively (Table
1). All aforementioned analyses were implemented in Python 2.7 using Enthought Canopy 1.0 (Enthought
Inc., Austin, USA).
8
3 Results
3.1 Dataset A: knee kinematics
The knee appeared to be comparatively more flexed (Fig.2a) and somewhat more externally rotated
(Fig.2c) in the side-shu✏e vs. v-cut tasks, with slightly more abduction at 0% stance (Fig.2b). Statistical
tests on the extracted scalars found significant di↵erences between the two tasks for both maximal knee
flexion (t=3.093, p=0.018) and abduction at 0% stance (t=3.948, p=0.006).
SPM vector field analysis (Fig.5) found significant kinematic di↵erences between the two tasks at ap-
proximately 1%, 10%, 20%, 30-35% and 95-100% stance. Post hoc t tests revealed that the e↵ects over
30-35% and 95-100% stance resulted primarily from increased flexion (p=0.015) and increased external ro-
tation (p=0.004), respectively, in the side-shu✏e vs. v-cut tasks (Fig.6). Apparent discrepancies amongst
vector field SPM, scalar field SPM, and scalar extraction (both here and in the remainder of the Results),
are addressed in the Discussion.
3.2 Dataset B: muscle forces
Most muscles appeared to exhibit higher forces in PFP vs. Controls over most of stance (Fig.3). Nonethe-
less, none of the statistical tests on the extracted scalars reached significance; the medial gastrocnemius force
exhibited the strongest e↵ect (t=2.617, p=0.013), but like the nine other muscles (t<1.91, p>0.063) this failed
to reach the ˇ
Sid´ak significance threshold of p=0.0051.
In contrast, SPM vector field analyses found significance for the entire stance phase (Fig.7; p<0.001).
Post hoc ttests on individual muscle trajectories found significantly greater forces in PFP only for the medial
gastrocnemius, and only over scattered time regions (maximum p=0.002) (Fig.8).
3.3 Dataset C: ground reaction forces
Forces appeared to increase systematically with running speed, particularly in the vicinity of 30% and
75% stance (Fig.4). Linear regression found that all three extracted scalars surpassed the ˇ
Sid´ak threshold
for significance (p=0.0170); analysis of maximum propulsion, vertical and lateral forces yielded r2=0.951,
0.691 and 0.737, and p=0.00004, 0.001, and 0.006, respectively.
SPM vector field analysis (Fig.9) found that GRF was significantly correlated with running speed in three
intervals with approximate windows of: 10–18%, 20–43% and 60–88% stance. Post hoc scalar field analysis
9
revealed that GRFxwas primarily responsible for the 10–18% and 60–88% e↵ects (Fig.10a), and that GRFy
was primarily responsible for the 20–43% e↵ect (Fig.10b).
4 Discussion
The current vector field SPM and scalar extraction results all agreed qualitatively with the data, yet the
two approaches yielded di↵erent results and even incompatible statistical conclusions. This, by definition,
indicates that at least one of the methods is biased. For non-directed hypotheses testing we contend that
scalar extraction is susceptible to two non-trivial bias sources:
1. Post hoc regional focus bias — Type I or Type II error (i.e. false positives or false negatives) resulting
from the failure to consider the entire measurement domain.
2. Inter-component covariation bias — Type I or Type II error resulting from the failure to consider the
covariance amongst vector components.
We further contend that vector field testing overcomes both bias sources because it uses the entire
measurement domain and all vector components to maintain a constant error rate of ↵. The remainder of
the Discussion is devoted to justifying these claims.
4.1 Bias in scalar extraction analyses
Dataset A exhibited Type I error due to post hoc regional focus bias. Scalar extraction analysis of
maximum flexion (at ⇠50% stance) reached significance (§3.1) but neither vector field analysis (Fig.5) nor
post hoc scalar field analysis (Fig.6a) reached significance in this field region. Similarly, scalar extraction
found a significant ab-/adduction e↵ect at 0% stance, but SPM did not. These discrepancies are resolved
through multiple comparisons theory (Knudson, 2009); it is highly likely that at least one of Dataset A’s
303 vector field points will exceed an (uncorrected) threshold of p=0.05 simply by chance. By extracting
only scalars which appeared to exhibit maximum e↵ects (Fig.2) we e↵ectively conducted 303 tests and then
chose to report the results of only two.
The opposite e↵ect (Type II error) was also present in Dataset A. Scalar extraction focussed on only two
scalars, and thus failed to identify the other e↵ects present in the dataset (Fig.5), and in particular the large
late-stance internal/external rotation e↵ect (Fig.6c). A simple example (Supplementary Material – Appendix
10
A) clarifies how it is possible for scalar extraction and SPM to yield opposite statistical conclusions, and
that the scalar extraction results can’t be trusted because they fail to honor the ↵error rate.
Dataset B exhibited Type II error due to covariation bias: scalar extraction failed to reach significance
(§3.2) even though SPM found substantial evidence for muscular di↵erences between Controls and PFP
(Fig.7). This is resolved by correlation amongst muscles like the vasti (Fig.3). A simple example (Supple-
mentary Material – Appendix B) clarifies how it is possible for vector resultant changes to reach significance
when vector component changes do not. Scalar analysis of vector data cannot be trusted because it fails to
account for vector component covariance.
Scalar extraction analysis of Dataset C exhibited both Type I and Type II error due to regional focus
bias. Scalar extraction analyses of lateral forces exhibited Type I error because there is insufficient field-wide
evidence to support its conclusion of significance (Fig.10c). Scalar extraction also exhibited Type II error by
failing to analyze braking forces and therefore failing to identify positive correlation between running speed
and braking forces at 15% stance (Fig.10a).
4.2 Bias in scalar field SPM analyses
Scalar field SPM solves regional focus bias (because it tests the entire domain q), but it remains susceptible
to covariance bias because it separately tests the Ivector components. Scalar field analysis of Dataset
A exhibited Type II error by failing to identify all field e↵ects, and particularly the large early-stance
e↵ect (Fig.5,6). Appendix B clarifies that this was caused by scalar field analysis’ failure to consider inter-
component covariance; it regards trajectory variance as a 1D time-varying ‘cloud’ (Fig.2) when in fact it is
an I-D time-varying hyper-ellipsoid (Fig.1) representing both within- and between-component (co)variance.
In Dataset B, vector field analysis reached significance (Fig.7), so scalar field analysis, had it not been
conducted in a post hoc manner, would have exhibited Type II error by underestimating the temporal scope
of e↵ects (Fig.8). This is also explained by covariance (Appendix B); the e↵ect was manifested more strongly
in the resultant 10-component muscle force vector than it was in each muscle independently.
In Dataset C vector field e↵ect timing (Fig.9) agreed with scalar field e↵ect timing (Fig.10), so the latter
would not have been biased had they not been conducted in a post hoc sense. Nevertheless, by failing to
consider covariance the scalar field results fail to capture the full temporal extent of the vector e↵ects.
A separate but notable trend was that Dataset C’s covariance ellipses all tended to be narrow and to point
toward the origin (Fig.1). This suggests that vector magnitude was far more variable than vector direction.
A plausible mechanical explanation is friction: to avoid slipping normal forces must increase when tangential
11
forces increase. Regardless of the mechanism, this observation reinforces our contention that non-directed
hypotheses must consider vector changes.
4.3 SPM’s solution to regional focus and covariance bias
SPM solves both regional focus bias and covariance bias by considering the covariance of all vector com-
ponents (i) across the entire measurement domain (q), while simultaneously handling the inherent problem
of multiple comparisons (Knudson, 2009) in a theoretically robust manner. Specifically, SPM uses a RFT
correction (Adler and Taylor, 2007; Worsley et al., 2004) to ensure that no more than ↵% of the points in the
(I⇥Q) vector field reach significance simply by chance; this RFT correction is embodied in the thresholds
depicted in Figs.5-10.
Non-RFT corrections like the ˇ
Sid´ak correction (Eqn.1) can partially solve the problem of multiple compar-
isons, but only partially because they fail to consider the (spatiotemporal) smoothness of the measurement
domain q, and therefore overestimate the number of independent tests. This ultimately leads to an overly
conservative threshold (i.e. inflated Type II error rate) except for very rough fields (Friston et al., 2007).
Non-RFT corrections also fail to solve covariance bias because they assume that vector components vary
independently (Supplementary Material – Appendix B). While covariance bias could partially be solved with
a principal axis rotation prior to statistical testing (Cole et al., 1994; Knudson, 2009); we’d argue that: (i)
Hotelling’s T2and CCA are simpler solutions because their results are identical for all coordinate system
definitions, and (ii) principal axis rotations of only the response vectors do not necessarily maximize the
mutual correlation between predictors and responses (§2.3.3).
We acknowledge that many additional important sources of bias exist (James and Bates, 1997), (Mullineaux
et al., 2001), (Knudson, 2009). However, none of these is unique to SPM. Trajectory mis-registration (Sadeghi
et al., 2003) and unit normalization (e.g. absolute vs. relative muscle forces), for example, pose common
problems to scalar extraction and vector field analyses. We contend only that SPM addresses two bias
sources.
4.4 SPM generalizability
Although SPM was originally developed to analyze 3D brain function (Friston et al., 2007), it has been
shown that SPM is generalizable to a variety of biomechanical scalar datasets including 1D trajectories
(Pataky, 2012), 2D pressure fields and 3D strain fields (Pataky, 2010). The current study is the first, in
any scientific field, to have shown that SPM is also applicable to a large class of practical 1D vector field
12
problems. SPM theory suggests that generalizations to biomechanical vector/tensor fields in nD spaces are
also possible (Xie et al., 2010).
SPM encompasses the entire family of parametric hypothesis testing (Worsley et al., 2004; Friston et al.,
2007). It also accommodates all non-parametric variants (Nichols and Holmes, 2002; Lenho↵et al., 1999),
which may be useful if one’s data do not adhere to the parametric assumption that the residuals are normally
distributed (Friston et al., 2007). This hypothesis testing generalization is apparent when one considers the
following hierarchy: vector fieldCCA simplifies to the vector field Hotelling’s T2test when the predictors are
binary (Worsley et al., 2004), which in turn simplifies to scalar field ttests when there is only one vector
component i, which in turn simplifies to the univariate Student’s ttest when the scalar field reduces to
a single point q. Thus SPM, through CCA, generalizes to all statistical tests of Idimensional vectors on
arbitrarily sized fields Qof arbitrary dimensionality (Worsley et al., 2004).
For readers interested in implementing SPM analyses, we note that constructing test statistic trajectories
is straightforward; it is trivial to combine mean and standard deviation trajectories to form an SPM{t}, for
example. The non-trivial step is statistical inference. As a first approximation it is easy to implement a
ˇ
Sid´ak correction (Eqn.1), which will (very) conservatively reduce the Type I error rate, but which will also
unfortunately inflate the Type II error rate. For more precise control of both error rates (via RFT) the
reader is directed to the literature (Friston et al., 2007) and open source software packages (Pataky, 2012).
4.5 Conclusions
Ad hoc reduction of vector trajectories through scalar extraction can non-trivially bias non-directed
biomechanical hypothesis testing, most notably via regional focus and coordinate system bias sources. This
paper shows that SPM overcomes both sources of bias by treating the vector field as the fundamental, initially
indivisible unit of observation. Grounded in random field theory, SPM appears to be a useful, generalized
tool for the analysis of often-complex biomechanical datasets.
Acknowledgments
Financial support for this work was provided in part by JSPS Wakate B Grant#22700465.
13
Conflict of Interest
The authors report no conflict of interest, financial or otherwise.
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15
Table 1: Dataset and scalar extraction overview. I,J,Qand Nare the numbers of vector com-
ponents, responses, time points, and extracted scalars, respectively. For vector field analyses,
post hoc scalar field analyses, and extracted scalar analyses we conducted one, Iand Ntests,
respectively. ˇ
Sid´ak thresholds of p=0.0253, p=0.0170 and p=0.0051 maintained a family-wise
error rate of ↵=0.05 across 2, 3, and 10 tests, respectively (see Eqn.1).
I J Q N Extracted scalars
Dataset A 3 8 101 2 (1) Max. flexion (at ⇠50% stance)
(2) Ad-abduction at 0% stance
Dataset B 10 43 100 10 Max. force for each muscle (J1=16, J2=27)
Dataset C 3 8 100 3
(1) Max. propulsion force (GRFx,⇠75% stance)
(3) Max. vertical force (GRFy,⇠30–50% stance)
(3) Max. lateral force (GRFz,⇠15% stance)
FIGURES
Figure 1. Vector field schematic: a two-component vector varying in time. Depicted are mean
ground reaction force (GRF) vectors F = [Fx Fy]T from one subject during running (Dorn et al.,
2012), where +x and +y represent the anterior and vertical directions, respectively. These vectors,
when projected on the (Time, Fx) and (Time, Fy) planes, produce common GRF plots (see Fig.
4a,b); here vertical dotted lines depict standard deviation ‘clouds’. When F is projected on the
(Fx, Fy) plane these standard deviations are revealed to arise from covariance ellipses, where
ellipse orientation indicates the direction of maximum covariance between Fx and Fy (see
Supplementary Material - Appendix B).
Figure 2. Dataset A (Neptune et al., 1999) depicting knee kinematics in side-shuffle vs. v-cut
tasks. Cross-subject mean trajectories with standard deviation clouds (dark: side-shuffle, light: v-
cut) are depicted. Each of the eight subjects has three (scalar) trajectories yi(q) for each task, and
these were combined into a single (I=3, Q=101) vector field y(q) for each subject and each task.
Figure 3. Dataset B (Besier et al., 2009) depicting muscle forces during walking in Control vs.
Patello-Femoral Pain (PFP) subjects; 16 and 27 subjects, respectively. Cross-subject mean
trajectories with standard deviation clouds (dark: Control, light: PFP). These ten scalar
trajectories were combined into a single (I=10, Q=100) vector field y(q) for each subject.
Figure 4. Dataset C (Dorn et al., 2012) depicting ground reaction forces (GRF) during running/
sprinting at various speeds. Single-subject cross-trial means; standard deviation clouds are not
depicted in interest of visual clarity. These data form one (I=3, Q=100) vector field y(q) for each
trial.
Figure 5. Dataset A, Hotelling’s T2 trajectory (SPM{T2}). The horizontal dotted line indicates the
critical random field theory threshold of T2 = 29.39.
Figure 6. Dataset A, post hoc scalar field t tests (SPM{t}), depicting where side-shuffle angles
were greater (+) and less (-) than v-cut angles. At a Sidak threshold of p=0.017 (Eqn.1), the thin
dotted lines indicate the critical RFT thresholds for significance: |t| > 4.52, 5.24, 5.26 for (a), (b),
and (c) respectively. The thresholds are different because each vector component has different
temporal smoothness (Fig.2); less smooth trajectories have higher thresholds because there are
more ‘processes’ present between 0 and 100% time. Probability (p) values indicate the likelihood
with which each suprathreshold cluster is expected to have been produced by a random field
process with the same temporal smoothness.
Figure 7. Dataset B, Hotelling’s T2 trajectory (SPM{T2}), depicting where muscle forces differed
between Controls and PFP. The horizontal dotted line indicates the critical RFT threshold of T2 =
9.35. The entire trajectory has exceeded the threshold, so the single suprathreshold cluster has a
very low p value.
Figure 8. Dataset B, post hoc scalar trajectory t tests (SPM{t}), depicting where Control forces
were greater than (+) and less than (-) PFP forces. Thin dotted lines indicate the critical RFT
thresholds for significance.
Figure 9. Dataset C, canonical correlation analysis results, with SPM{F} depicting where ground
reaction forces were correlated with running speed. Critical RFT threshold: F = 38.1.
Figure 10. Dataset C, post hoc scalar trajectory linear regression tests (SPM{t}), depicting the
strength of positive (+) and negative (-) correlation between ground reaction forces (GRF) and
running speed.
Appendix A. Scalar extraction vs. scalar field statistics
The purpose of this Appendix is to demonstrate how scalar extraction can bias non-directed
hypothesis testing. To this end we developed and analyzed an arbitrary dataset (Fig.S1). We
caution readers that we have constructed these data specifically to demonstrate particular con-
cepts. The reader is therefore left to judge the relevance of this discussion to real (experimental)
datasets.
The specific goal of this Appendix is to scrutinize the similarities and di↵erences between:
(a) a typical univariate two-sample ttest, and (b) a scalar field two-sample ttest.
Consider the simulated scalar field dataset in Fig.S1. In Fig.S1a, arbitrary true mean fields
are defined for two experimental conditions: “Cond A” and “Cond B”. The Cond B mean was
produced using a half sine cycle. The Cond A mean was produced by adding a small Gaussian
pulse (at time= 85%) to the Cond B mean. This Gaussian pulse is evident in the true mean
field di↵erence (Fig.S1b).
Figure S1: Simulated scalar field dataset depicting two experimental conditions: “Cond A”
and “Cond B” (arbitrary units).
We next simulate smooth random fields: five for each condition (Fig.S1c). These random
fields were constructed by generating ten fields, each containing 100 random, uncorrelated
and normally distributed numbers, then smoothing them using a Gaussian kernel. Adding
the random fields to the true field means (Fig.S1a) produced the final simulated responses
(Fig.S1d). For interpretive convenience, let us assume that these data represent joint flexion.
Imagine next that we wish to test the following (non-directed) null hypothesis: “Cond A
and Cond B yield identical kinematics”. Consider first scalar extraction: after observing the
data (Fig.S1d) one might decide to extract and analyze the maximum flexion, which occurs
near time = 50%:
yA=⇥100.0 91.2 92.2 95.5 97.1⇤
yB=⇥97.2 101.9 104.8 106.3 111.7⇤
A two-sample ttest on these data yields: t=3.16, p=0.013. We would reject the null
hypothesis at ↵=0.05, and we would conclude that Cond B produces significantly greater
maximal flexion than Cond A.
An alternative is to use Statistical Parametric Mapping (SPM) (Fig.S2). The SPM pro-
cedures are conceptually identical to univariate procedures (Table S1). The only apparent
di↵erence is that SPM uses a di↵erent probability distribution (Steps 4 and 5). This probabil-
ity distribution is actually not di↵erent because it reduces to the univariate distribution when
Q=1 (i.e. if there is only one time point).
SPM results find significant di↵erences between the two conditions near time = 85% (Fig.S2d).
We would therefore reject our null hypothesis, with the caveat that significant di↵erences were
only found near time = 85%.
Although univariate ttesting and SPM ttesting are conceptually identical, they have yielded
(e↵ectively) opposite results. The univariate test found significantly greater maximal flexion
in Cond B, but SPM found significantly greater flexion in Cond A (near time=85%).
Table S1: Comparison of computational steps for univariate and SPM two-sample ttests
(“st.dev.” = standard deviation).
Step (a) Univariate two-sample ttest (b) SPM two-sample ttest Figure
1 Compute mean values yAand yB. Compute mean fields yA(q) and yB(q) S2(b)
2 Compute st.dev. values sAand sB. Compute st.dev. fields sA(q) and sB(q) S2(b)
3 Compute the ttest statistic:
t=yByA
q1
J(s2
A+s2
B)
Compute the ttest statistic field:
SPM{t}⌘t(q)= yB(q)yA(q)
q1
J(s2
A(q)+s2
B(q))
S2(c)
4 Conduct statistical inference. First use ↵
and the univariate tdistribution to com-
pute tcritical.Ift>t
critical, then reject null
hypothesis.
Conduct statistical inference. First use ↵
and the random field theory tdistribution
to compute tcritical.IfSPM{t}exceeds
tcritical, then reject null hypothesis for the
suprathreshold region(s).
S2(d)
5 Compute exact pvalue using tand the uni-
variate tdistribution.
Compute exact pvalue(s) for each
suprathreshold cluster using cluster size
and random field theory distribution(s) for
SPM{t}topology.
S2(d)
Figure S2: Scalar field analysis using Statistical Parametric Mapping (SPM). In panel (d)
the thin dotted lines depict the critical random field theory threshold of |tcritical |=3.533. The
(incorrect) ˇ
Sid´ak threshold is |tcritical|=5.595.
This discrepancy can be resolved through standard probability theory regarding multiple
comparisons, through a consideration of ‘corrected’ and ‘uncorrected’ thresholds. First consider
conducting one statistical test at ↵=0.05. The choice: “↵=0.05” means that we are accepting
a 5% chance of incorrectly rejecting the null hypothesis, or, equivalently, a 5% chance of a
‘false positive’. If we conduct more than one test, there is a greater-than 5% chance of a false
positive. Specifically, if we conduct Nstatistical tests, the probability of at least one false
positive is given by the family-wise error rate ↵:
↵=1(1 ↵)N
For N=2 tests, there is a ↵=9.75% chance that at least one test will produce a false positive.
For N=100 tests, ↵=99.4%.
To protect against false positives, and to maintain a constant family-wise error rate of
↵=0.05, we must adopt a corrected threshold. One option is the ˇ
Sid´ak threshold:
pcritical =1(1 ↵)1/N
For N=2 and N=100 tests, the ˇ
Sid´ak thresholds are pcritical=0.0253 and pcritical=0.000513,
respectively.
Herein lies one problem: our scalar extraction analysis has used an uncorrected threshold
of pcritical=0.05. Even though we have formally conducted only one statistical test, the data
were extracted from a dataset that is 100 times as large. Since we observed the data before
choosing which scalar to extract, we e↵ectively conducted N=100 tests, albeit visually, then
chose to focus on only one test. By failing to adopt a corrected threshold, we have biased our
analyses.
Although the ˇ
Sid´ak correction helps to avoid false positives, it is not generally a good choice
because it assumes that there are 100 independent tests (i.e. one for each time point in our
dataset). The points in this dataset are clearly not independent because the curves are smooth,
changing only gradually over time. Thus the ˇ
Sid´ak correction is too severe, lowering ↵well
below 0.05. An overly severe threshold produces the opposite bias: an increased chance of false
negatives.
SPM employs a random field theory (RFT) correction to more accurately maintain ↵=0.05.
The precise threshold is based not only on field size (Q=100), but also on field smoothness —
which is estimated from temporal derivatives. Computational details for RFT corrections are
provided in the SPM literature.
Unfortunately, even if our scalar analysis had employed a corrected threshold, it still would
have been biased, but for a separate reason. By focussing only on maximal flexion (which did
not appear in our null hypothesis), we have neglected to consider the signal at time = 85%,
and have therefore not detected the true field di↵erence (Fig.S1a). In contrast, SPM was able
to uncover the true signal because it both adopted a corrected threshold and considered the
entire field simultaneously (Fig.S1d).
The aforementioned sources of bias — (1) failing to adopt a corrected threshold, and (2)
failing to consider the entire field — are referred to collectively in the main manuscript as
‘regional focus bias’.
Last, we reiterate that this Appendix is relevant only to non-directed hypotheses. If we
had formulated a (directed) hypothesis regarding only maximal flexion — prior to observing
the data — then our scalar extraction analyses would not have been biased because our null
hypothesis would not have pertained to the entire time domain 0–100%.
In summary, regional focus bias can be avoided by:
1. Specifying a directed null hypothesis — before observing the data — and then extracting
only those scalars which are specified in the null hypothesis.
2. Analyzing the data using SPM or another field technique which both considers the entire
temporal domain and which adopts a corrected threshold.
Appendix B. Univariate vs. vector analysis
The purpose of this Appendix is to demonstrate how univariate testing of vector data can
bias non-directed hypothesis testing. To this end we developed and analyzed an arbitrary
dataset (Table S2). As in Appendix A, we caution readers that we have constructed these
data specifically to demonstrate particular concepts. The reader is therefore left to judge the
relevance of this discussion to real (experimental) datasets.
The specific goal of this Appendix is to compare and contrast the (univariate) ttest and
its (multivariate) vector equivalent: the Hotelling’s T2test.
Table S2: A simulated dataset exhibiting biased univariate testing. (a) Two-component force
vector responses F=[Fx,Fy]>. (b)-(d) Scalar (univariate) testing. (e)-(g) Vector (multivari-
ate) testing. Sources of bias and further details are discussed in the text. Technical overviews
of covariance matrices (W) and the Hotelling’s T2statistic are provided in Appendix D and
§2.3 (main manuscript), respectively.
Group A Group B Inter-Group
(a) Responses
FA1 = [159, 719]>FB1 = [143, 759]>
FA2 = [115, 762]>FB2 = [172, 734]>
FA3 = [177, 681]>FB3 = [161, 735]>
FA4 = [138, 694]>FB4 = [195, 733]>
FA5 = [98, 697]>FB5 = [168, 706]>
Univariate
(b) Means (Fx)A= 137.4 (Fx)B= 167.8 Fx= 30.4
(Fy)A= 710.6 (Fy)B= 733.4 Fy= 22.8
(c) St.dev. (sx)A= 28.6 (sx)B= 16.8 sx= 23.5
(sy)A= 28.5 (sy)B= 16.8 sy= 23.4
(d) ttests tx=1.832; px=0.104
ty=1.380; py=0.205
Vector
(e) Means FA= [137.4, 710.6]>FB= [167.8, 733.4]>F= [30.4, 22.8]>
(f) Covariance WA="817.8323.2
323.2809.8#WB="283.8131.9
131.9281.8#W="550.8227.6
227.6545.8#
(g) T2test T2=7.113; p=0.028
In Table S2(a) above there are five force vector responses (F=[Fx,Fy]>) for each of two
groups: “A” and “B”. Their means and standard deviations are shown in Table S2(b)-(c). In
Table S2(d) we see that ttests pertaining to both Fxand Fyfail to reach significance; pvalues
are greater than (even an uncorrected) threshold of p=0.05. An adequate interpretation is
that the mean force component changes (Fxand Fy) are not unexpectedly large given their
respective variances (i.e. standard deviations: sxand sy).
We next jump ahead to the final results of the vector procedure in Table S2(g): here we
see that the Hotelling’s T2test reached significance (p=0.032). An adequate interpretation is
that the mean force vector change (F) was unexpectedly large given its (co)variance (W).
Let us now backtrack and consider why the univariate and vector procedures yield di↵erent
results.
The first step of the vector procedure is to compute mean vectors; in Table S2(e) we can
see that the vector means have the same component values as the univariate means from Table
S2(b). However, there is already one critical discrepancy to note: the vector procedure assesses
F,whichistheresultant vector connecting the Group A and Group B means (Fig.S3).
From Pythagoras’ theorem:
|F|2=F2
x+F2
y(B.1)
it is clear that the magnitude of the resultant will always be greater than the magnitude of its
components — except in the experimentally unlikely cases of Fx=0 and/or Fy=0. This is
non-trivial for two reasons. First, since the vector procedure assesses the maximum di↵erence
between the two groups, it is more robust to Type II error than univariate procedures (note:
the univariate tests in Table S2 exhibit Type II error by failing to reach significance). Second,
the vector technique’s assessment of di↵erences is independent of the xy coordinate system
definition; whereas the component e↵ects (Fxand Fy) can change when the xy coordinate
system definition changes, both the resultant and the variance along the resultant direction will
always have the same magnitude. This may have non-trivial implications for biomechanical
datasets that employ difficult-to-define coordinate systems (e.g. joint rotation axes).
Figure S3: Graphical depiction of the data from Table S2. Small circles depict individual
responses. Thick colored arrows depict the mean force vectors for the two groups. The thick
black arrow depicts the (vector) di↵erence between the two groups, and thin black lines indicate
its xand ycomponents. The ellipses depict within-group (co)variance; their principal axes (thin
dotted lines) are the eigenvectors of the covariance matrices in Table S2(f ). Here covariance
ellipse radii are scaled to two principal axis standard deviations (to encompass all responses).
The next step of the vector procedure is to compute covariance matrices W(Appendix D).
The diagonal elements of WAand WBin Table S2(f) are simply the variances (i.e. squared
standard deviations) s2
xand s2
yfrom Table S2(c). The o↵-diagonal terms are equal and represent
the covariance (i.e. correlation) between Fxand Fy.IfFxtends to increase when Fyincreases
then the o↵-diagonal terms would be positive, but in this case they are negative, indicating
that Fxtends to decrease when Fyincreases. This tendency can be seen in the raw data (small
circles) in Fig.S3.
The presence of non-zero o↵-diagonal terms thus has a critical implication: changes in Fx
and Fyare not independent. This is critical because univariate tests implicitly assume that Fx
and Fyare independent.
To appreciate this point it is useful to recognize that covariance matrices may be interpreted
geometrically as ellipses: the eigenvectors of Wrepresent the ellipse’s principal axes, and its
eigenvalues represent the variance along each principal direction. This is perfectly analogous to
inertia matrices: the eigenvectors of an inertia matrix define a body’s principal axes of inertia,
and eigenvalues specify the principal moments of inertia.
The importance of this geometric interpretation becomes clear when visualizing covariance
ellipses. In Fig.S3 we can see that the principal axes of the covariance matrices are not aligned
with the xy coordinate system, implying that changes in Fxand Fyare not independent.
Critically, we can also see that the direction of minimum variance is very similar to the direction
of F. Thus the standard deviations sxand sy(used in the univariate analyses) are larger
than the standard deviation in the direction of F.
In summary, vector statistical testing more objectively detects vector changes because : (a)
it is coordinate system-independent, (b) it considers both the maximum di↵erence between
groups (i.e. the resultant di↵erence) and the variation along this direction. This Appendix has
demonstrated how univariate testing of vector data can lead to Type II error. With a di↵erent
dataset it would also be possible to demonstrate Type I error, but in interest of space we end
here. The most important point, the main paper contends, is that non-directed hypothesis
testing mustn’t assume vector component independence.
Appendix C. Mean vector field calculation
An I-component vector ywhich varies over Qpoints in space or time may be regarded as
an (I⇥Q) vector field response y(q). Given Jresponses, the mean vector field is:
y(q)= 1
J
J
X
j=1
yj(q) (C.1)
For the paired Hotelling’s T2test (Dataset A: §2.3.1, main manuscript), one must first
compute pairwise di↵erences:
yj(q)=yBj(q)yAj (q) (C.2)
where “A” and “B” represent the two tasks (v-cut and side-shu✏e) and jindexes the subjects.
A paired Hotelling’s T2test is thus equivalent to a one-sample Hotelling’s T2test conducted
on the pairwise di↵erences y(q). The same is true in the univariate case: a paired ttest is
equivalent to a one-sample ttest on pairwise di↵erences.
Appendix D. Covariance matrices
Although the concepts presented below apply identically to vector fields, for brevity present
discussion is limited to simple vectors.
Consider a two-component force vector response F:
Fj=⇥Fxj Fyj⇤>(D.1)
where jindexes the responses, and there are a total of Jresponses. After computing the mean
force vector Fas:
F=
2
6
6
4
Fx
Fy
3
7
7
5
=1
J
J
X
j=1
Fj(D.2)
the covariance matrix Wcan be assembled as follows:
W=
2
6
6
4
Wxx Wxy
Wyx Wyy
3
7
7
5
(D.3)
where the elements of Ware:
Wxx =1
J1
J
X
j=1
(Fxj Fx)2(D.4)
Wyy =1
J1
J
X
j=1
(Fyj Fy)2(D.5)
Wxy =Wyx =1
J1
J
X
j=1
(Fxj Fx)(Fyj Fy) (D.6)
Thus the diagonal elements Wxx and Wyy are the intra-component variances (i.e. squared
standard deviations), and the o↵-diagonal elements Wxy and Wyx are the inter-component
covariances between Fxand Fyover multiple responses. Importantly, changes in Fxand Fyare
completely uncorrelated if and only if Wxy=0.
One contention of this paper is that separate (univariate) analysis of Fxand Fyis biased
when testing non-directed hypotheses. The main reason is that Fxanalysis considers only Wxx
and Fyanalysis considers only Wyy. This is equivalent to assuming Wxy=0, an assumption
which may not be valid (Appendix B).
A geometric interpretation of Wis useful both for visualizing vector variance (Fig.S3) and
for appreciating canonical correlation analysis (Appendix E). Consider that Wrepresents an
ellipse whose geometry is defined by the solutions to the eigenvalue problem:
Wv =v(D.7)
Here vand are the eigenvectors and eigenvalues, respectively, and there are two unique
eigensolutions unless both (Wxx =Wyy) and (Wxy = 0), in which case there is only one
eigensolution and Wrepresents a circle. When there are two solutions the eigenvectors repre-
sent the ellipse axes (or equivalently: principal axes), and the eigenvalues represent the axes’
lengths (or variance in the direction of the principal axes). An equivalent interpretation is that
one eigenvector of Wrepresents the direction of maximum variance within the dataset. This
means that we can rotate our original coordinate system xy to a new coordinate system x0y0so
that variance along the new x0axis is the maximum possible variance obtainable for all possible
x0.
Appendix E. Canonical correlation analysis (CCA)
CCA aims to quantify the amount of variance that a multivariate predictor (i.e. vector)
Xcan explain in a multivariate response Y. One type of CCA useful for hypothesis testing is
to find the maximum possible correlation coefficient that can be obtained when the coordinate
systems defining Xand Yare permitted to mutually rotate.
Consider a response variable Ythat describes three orthogonal force components F:
Yj=⇥F1jF2jF3j⇤>(E.1)
where “1”, “2” and “3” represent orthogonal axes and where jindexes a total of Jresponses.
Next consider a predictor variable Xthat describes the rotations ✓about two orthogonal axes
at a given joint:
Xj=⇥✓1j✓2j⇤>(E.2)
where “1” and “2” indicate the two joint axes. The relevant null hypothesis is: Xand Yare
not linearly related.
To test this hypothesis one needs to assemble three covariance matrices. The first is a (3
⇥3) response covariance matrix WYY which describes variance within and the co-variation
between the three force components (see Appendix D). The second is a (2 ⇥2) predictor
covariance matrix WXX which describes the variance and covariance of the two joint angles.
The third is a (2 ⇥3) predictor-response covariance matrix WXY which describes how each of
the predictor variables co-varies with each of the response variables.
The predictor-response covariance matrix WXY is relevant to the null hypothesis because it
embodies the strength of linear correlation between Xand Y. For completion, in the example
above WXY has six elements, corresponding to:
1. The linear correlation between ✓1and F1
2. The linear correlation between ✓1and F2
3. The linear correlation between ✓1and F3
4. The linear correlation between ✓2and F1
5. The linear correlation between ✓2and F2
6. The linear correlation between ✓2and F3
Initially these correlations refer only to X’s and Y’s original coordinate systems. Since
arbitrary coordinate systems can bias non-directed hypothesis testing (Appendix B), we must
allow the coordinate systems to rotate in order to most objectively test our null hypothesis.
One CCA solution is to choose the Xand Ycoordinate systems that mutually maximize
a single correlation coefficient. The logic is that all other coordinate systems underestimate
correlation strength. In other words, as the coordinate systems rotate the elements of WXY
change, and one (not necessarily unique) coordinate system combination maximizes an element
of WXY . CCA solves this problem efficiently using the maximum eigenvalue of the canonical
correlation matrix (Eqn.7, main manuscript).
As an aside, we note that the K=2 model in the main manuscript is equivalent to a K=1
model (i.e. only a running speed regressor) because only one (diagonal) element of WXX is
non-zero. For generalizability the main manuscript treats CCA in its K>1 form.
