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Submitted 13 July 2016

Accepted 4 October 2016

Published 2 November 2016

Corresponding author

Todd C. Pataky,

tpataky@shinshu-u.ac.jp

Academic editor

John Hutchinson

Additional Information and

Declarations can be found on

page 10

DOI 10.7717/peerj.2652

Copyright

2016 Pataky et al.

Distributed under

Creative Commons CC-BY 4.0

OPEN ACCESS

Region-of-interest analyses of one-

dimensional biomechanical trajectories:

bridging 0D and 1D theory, augmenting

statistical power

Todd C. Pataky1, Mark A. Robinson2and Jos Vanrenterghem3

1Institute for Fiber Engineering, Department of Bioengineering, Shinshu University, Ueda, Nagano, Japan

2Research Institute for Sport and Exercise Sciences, Liverpool John Moores University,

Liverpool, United Kingdom

3Department of Rehabilitation Sciences, Katholieke Universiteit Leuven, Belgium

ABSTRACT

One-dimensional (1D) kinematic, force, and EMG trajectories are often analyzed

using zero-dimensional (0D) metrics like local extrema. Recently whole-trajectory 1D

methods have emerged in the literature as alternatives. Since 0D and 1D methods can

yield qualitatively different results, the two approaches may appear to be theoretically

distinct. The purposes of this paper were (a) to clarify that 0D and 1D approaches are

actually just special cases of a more general region-of-interest (ROI) analysis framework,

and (b) to demonstrate how ROIs can augment statistical power. We first simulated

millions of smooth, random 1D datasets to validate theoretical predictions of the 0D,

1D and ROI approaches and to emphasize how ROIs provide a continuous bridge

between 0D and 1D results. We then analyzed a variety of public datasets to demonstrate

potential effects of ROIs on biomechanical conclusions. Results showed, first, that

a priori ROI particulars can qualitatively affect the biomechanical conclusions that

emerge from analyses and, second, that ROIs derived from exploratory/pilot analyses

can detect smaller biomechanical effects than are detectable using full 1D methods.

We recommend regarding ROIs, like data filtering particulars and Type I error rate, as

parameters which can affect hypothesis testing results, and thus as sensitivity analysis

tools to ensure arbitrary decisions do not influence scientific interpretations. Last, we

describe open-source Python and MATLAB implementations of 1D ROI analysis for

arbitrary experimental designs ranging from one-sample ttests to MANOVA.

Subjects Animal Behavior, Bioengineering, Kinesiology, Statistics

Keywords Time series analysis, Kinematics, Constrained hypotheses, Statistical parametric

mapping, Dynamics, Random field theory, Hypothesis testing, Biomechanics, Human movement

INTRODUCTION

Many biomechanical measurements may be regarded as ‘n-dimensional m-dimensional’

(nDmD) continua, where nand mare the dimensionalities of the measurement domain

and dependent variable, respectively. Common examples include: joint flexion (1D1D),

ground reaction force (1D3D), plantar pressure distribution (2D1D) and bone strain tensor

distributions (3D6D). These data are often analyzed using 0D1D metrics from regions of

How to cite this article Pataky et al. (2016), Region-of-interest analyses of one-dimensional biomechanical trajectories: bridging 0D and

1D theory, augmenting statistical power. PeerJ 4:e2652; DOI 10.7717/peerj.2652

interest (ROIs) which summarize particular continuum features. In this paper ‘ROI’ refers

to a geometrical subset of a continuum dataset, and ‘ROI analysis’ refers to the analysis

of data extracted from an ROI. More explicit definitions for these terms with literature

context are provided in Appendix A.

In n>1 datasets ROIs are often explicitly constructed based on anatomical

rationale, especially for plantar pressure (Cavanagh & Ulbrecht, 1994) and finite element

analyses (Radcliffe & Taylor, 2007). In n=1 datasets ROIs tend to be used both explicitly

(e.g., with phase labels including: ‘‘early stance,’’ ‘‘push off,’’ ‘‘swing,’’ etc.) (Blanc et al.,

1999) and implicitly (e.g., local extrema are used without explicitly labeled continuum

regions) (Cavanagh & Lafortune, 1980). Regardless, the ultimately analyzed metrics are

often n=0 scalars, so we refer to this class of methods as ‘0D.’ For simplicity the remainder

of this paper focusses on n=1 datasets and corresponding ‘1D methods’ (Appendix A).

Recently a variety of 1D methodologies have emerged in the Biomechanics literature

including functional data analysis (FDA) (Ramsay & Silverman, 2005), principal

component analysis(PCA) (Daffertshofer et al., 2004) and statistical parametric mapping

(SPM) (Pataky, Robinson & Vanrenterghem, 2013), each of which afford whole-field 1DmD

analysis. SPM in particular is ideal for ROI-related hypothesis testing because it is valid for

arbitrary 1D geometries including broken or segmented regions of arbitrary size (Pataky,

2016). FDA is less ideal because it employs continuous basis functions and not, to our

knowledge, piecewise continuous ones. PCA can easily handle arbitrary ROI data, but it is

predominantly a data reduction technique and not a hypothesis testing technique.

This paper therefore focusses on SPM, a methodology that was initially developed in the

Neuroimaging literature in the 1990s (Friston et al., 1995), that spread to Electrophysiology

through the 2000s (Kiebel & Friston, 2004;Kilner, Kiebel & Friston, 2005), and which has

more recently appeared in the Biomechanics literature (Pataky, 2012;Pataky, Robinson

& Vanrenterghem, 2013). In Neuroimaging and Electrophysiology SPM has grown into

a comprehensive suite of techniques capable of handling all aspects of n-dimensional

continuum analysis including univariate and multivariate continuum analysis, parametric

and non-parametric probability density utilization, classical and Bayesian inference, and

multi-modal analysis among other functionality (Friston et al., 2007). In the context of this

paper, SPM’s classical hypothesis testing ability is key. Briefly, and considering only 1D

data, SPM first computes a 1D test statistic continuum (often the tstatistic continuum)

from a set of experimentally measured 1D continua. This step is effectively equivalent

to 1D mean and standard deviation continuum computation, which is frequently done

in the Biomechanics literature. SPM then conducts statistical inference at a Type I error

rate of ↵by calculating the critical test statistic value above which random test statistic

continua (generated by smooth, 1D Gaussian continua) would traverse in only (1↵)%

of an infinite number of identical experiments; if the experimentally observed continuum

exceeds that critical value the null hypothesis is rejected. This general approach to classical

hypothesis testing has been validated extensively in the Neuroimaging literature for 3D

(and 4D) continua (Friston et al., 1995;Friston et al., 2007) and has also been validated for

1D univariate and multivariate data (Pataky, 2016).

Pataky et al. (2016), PeerJ , DOI 10.7717/peerj.2652 2/12

Despite the validity of 1D approaches, a variety of conceptual difficulties may arise when

attempting to corroborate 0D and 1D approaches. In particular, 0D statistical results are

typically tabulated using single numbers for the test statistic and pvalues, so test statistic

continua may appear odd. Another apparent discrepancy between 0D and 1D techniques

is that the former requires continuum summary metric extraction but the latter does not,

so 0D techniques may appear to be somewhat more subjective than 1D techniques. A final

discrepancy is that 1D techniques involve multiple comparison corrections, so they may

appear to be less powerful than 0D techniques. All of these real or perceived discrepancies

could lead one to infer that 0D and 1D approaches are fundamentally different.

The primary purpose of this paper was to clarify the theoretical consistency between

0D and 1D techniques as special cases of ROI analysis. To that end we describe 1D ROI

theory then validate its predictions using numerical simulations of random datasets with

temporal scopes ranging from single points to large 1D continua. The second purpose was

to demonstrate how ROIs can be used to augment statistical power in both exploratory

and hypothesis-driven experiments. The final purpose was to introduce an open-source

software implementation of ROI analysis (in Appendix C) which emphasizes how 0D, 1D

and ROI analyses can all be executed using a common software interface.

METHODS

All analyses were implemented in Python 2.7 (Van Rossum, 2014) using Canopy 1.6

(Enthought Inc., Austin, TX, USA) and spm1d (Pataky, 2012;Pataky, 2016)(http:

//www.spm1d.org). All datasets described below are included in the spm1d package

and are accessible using the spm1d.data interface as described in Appendix D. High-level

Python and MATLAB (The MathWorks, Natick, MA, USA) interfaces for ROI analyses are

now available in spm1d as described in Appendix C.

ROI theory and validation

In classical hypothesis testing the null hypothesis is rejected if the experimentally observed

test statistic texceeds a critical threshold t⇤, which can be computed according to:

P(t>t⇤)=↵(1)

where ↵is the Type I error rate (usually 0.05) and P(t>t⇤) is the probability that the test

statistic exceeds t⇤if the null hypothesis is true.

For an ROI of size S(S0), Eq. (1) can be written as (Friston et al., 2007;Pataky, 2016):

P(tmax >t⇤)=1exp"P0D (t>t⇤)S

Wp4log2

2⇡✓1+(t⇤)2

⌫◆(⌫1)/2#=↵(2)

where tmax is the maximum value of the tstatistic inside the ROI, P0D(t>t⇤) is the proba-

bility under the null hypothesis that 0D random Gaussian data will produce a tvalue greater

than t⇤,Wis the FWHM representing trajectory smoothness (Appendix B), and ⌫is the

degrees of freedom. Note that P(tmax >t⇤) converges to P0D(t>t⇤) as Sapproaches zero,

and that t⇤must increase as Sincreases to maintain a given ↵. In other words, the larger the

Pataky et al. (2016), PeerJ , DOI 10.7717/peerj.2652 3/12

Figure 1 Simulated datasets. Datasets A and B are identical except in Dataset B the signal at time =75%

is amplified. Three regions of interest (ROIs) are depicted, centered at time =75% and spanning time

windows of 10%, 20% and 30%, respectively.

ROI, the more likely smooth, purely random 1D Gaussian data are to produce high tvalues.

Also note that Eq. (2) can accommodate multiple ROIs using a relatively simple correction

(Friston et al., 2007).

We computed t⇤for a range of ROI sizes (S), smoothness values (FWHM) and sample

sizes (⌫). ROI size was systematically varied over the full range of possibilities (i.e., from

0 to 100% trajectory length). FWHM and ⌫values were selected to span a range of

values observed in representative open-access biomechanical datasets (FWHM =[5, 50],

⌫=[5,49])(Pataky, Vanrenterghem & Robinson, 2016).

To validate Eq. (2) for arbitrary ROI sizes we simulated 100,000 smooth, purely random

Gaussian 1D datasets using ‘rft1d’ (Pataky, 2016), and repeated for each combination of S,

FWHM and ⌫values. For each random dataset we computed tmax , thereby producing one

distribution of 100,000 tmax values for each combination of parameters. We then estimated

t⇤for each distribution as the 95th percentile of the distribution, then qualitatively

compared to the theoretical result (Eq. (2)).

Example ROI analyses

Simulated datasets

Datasets A and B (Fig. 1)(Pataky, Robinson & Vanrenterghem, 2013) consisted of ten

simulated, smooth, random 1D Gaussian fields to which a Gaussian pulse was added at

time =75%. The degrees of freedom and number of time nodes were ⌫=9 and Q=101,

respectively, for both datasets. The pulse was slightly larger in Dataset B than in Dataset A.

Both were analyzed using six procedures, in order of increasing conservativeness: (1) 0D

analyses on the local maxima at time =75%, (2–4) 1D ROI analysis with ROIs centered

at time =75% and with temporal sizes of ±5%, ±10% and ±15%, respectively, (5)

1D full-field analysis (i.e., with ROI size =100%), and (6) 1D full-field analysis with a

Bonferroni correction. The latter assumes independence amongst adjacent trajectory nodes

so is overly conservative for smooth data (Pataky, Robinson & Vanrenterghem, 2013).

Pataky et al. (2016), PeerJ , DOI 10.7717/peerj.2652 4/12

Figure 2 Experimental datasets; error clouds depict standard deviations. (A) Knee kinematics during

side-shuffle and v-cut maneuvers (Neptune, Wright & Van den Bogert, 1999). (B) Pilot data from N=1

subject comparing vertical ground reaction force (VGRF) in ‘‘Normal’’ vs. ‘‘Fast’’ walking (Pataky et al.,

2008). (C) Subsequent identical VGRF experiment conducted on N=6 subjects (Pataky et al., 2008).

Experimental datasets

Dataset C (Fig. 2A)(Neptune, Wright & Van den Bogert, 1999)(⌫=7,Q=101) contained

stance-phase sagittal plane knee angles from eight participants who each performed both

side-shuffle and v-cut maneuvers. We started with 0D analysis of maximal knee flexion

(i.e., S=0), then conducted three ROI analyses with ROIs centered approximately on

maximal flexion (time =50%) and with temporal extents of 10%, 40% and 80%,

respectively. Finally, the 1D full-field ROI and Bonferroni procedures were applied as

in the simulated datasets.

Dataset D (Figs. 2B and 2C)(Pataky, Robinson & Vanrenterghem, 2013)(Q=101)

contained stance-phase body-weight-normalized vertical ground reaction forces (VGRF)

from seven subjects who each performed normal, self-paced walking and fast walking in

a randomized order. Analysis proceeded in two-stages to demonstrate how ROIs can be

used to increase analysis sensitivity in exploratory experiments (Pataky, Vanrenterghem

& Robinson, 2016, Fig.7). First, the chronologically first subject was separated as a pilot

subject (Fig. 2B). This subject’s results were examined qualitatively and were used to define

ROIs. Finally, those ROIs were used as an a priori constraint in analysis of the six remaining

subjects (Fig. 2C)(⌫=5). Results of this ROI-driven two-stage procedure were compared

to a full-field 1D analysis.

RESULTS

Critical thresholds t⇤necessary to maintain ↵=0.05 (Eq. (2)) increased nonlinearly as ROI

size increased and t⇤values for 1D data converged to 0D t⇤values as ROI size approached

zero (Fig. 3). Numerical simulations validated theoretical t⇤values for arbitrary 1D field

smoothness values and arbitrary sample sizes. These results emphasize that 0D analyses are

a special case of 1D analysis for which ROI size is zero.

In Dataset A, the test statistic exceeded the critical threshold for 0D analysis and also for

a narrow ROI of 10%, but failed to reach the thresholds for wider ROIs of 20% and 30%,

and also failed to reach the full-field threshold (Fig. 4A). In contrast, Dataset B’s slightly

amplified signal at time =75% exceeded all thresholds except for the highly conservative

Pataky et al. (2016), PeerJ , DOI 10.7717/peerj.2652 5/12

Figure 3 Validation of theoretical critical thresholds for region of interest (ROI) analysis. (A) Three

different degrees of freedom (⌫) and one smoothness (FWHM) value. (B) Three different FWHM values

and one ⌫value. The analytical 0D results assume 0D Gaussian randomness and the 1D results assume

smooth 1D Gaussian randomness.

Figure 4 Simulated dataset hypothesis testing results. ‘‘SPM{t}’’ denotes the tvalue extended in time to

form a ‘statistical parametric map.’ Six critical thresholds are depicted (see text). The width of each thresh-

old depicts the temporal extent of the null hypothesis.

full-field Bonferroni correction (Fig. 4B). These results imply that small ROIs can identify

relatively small effects like in Dataset A, and that large or even full-field ROIs can identify

large effects. ROIs thus embody a trade-off between statistical power and the temporal

scope of the null hypothesis; statistical power decreases as ROI size increases and vice versa.

For Dataset C, 0D analysis conducted on maximum knee flexion passed the critical

threshold, as did a moderately broad ROI of 40% (Fig. 5). However, an ROI of 80% and

full-field analysis failed to cross the critical threshold in the vicinity of maximum flexion.

The null hypothesis was nevertheless rejected for both an ROI size of 80% and full-field

analysis, but in this case the statistical conclusion pertains to regions other than maximum

flexion.

Pataky et al. (2016), PeerJ , DOI 10.7717/peerj.2652 6/12

Figure 5 Dataset C (knee flexion/extension) hypothesis testing results. ‘‘SPM{t}’’ denotes the tvalue.

Critical thresholds are presented as in Fig. 4, with both the height and temporal extent of each threshold

depicted.

For Dataset D the pilot subject’s data were used to identify two relatively narrow ROIs

in the vicinity of the first local maximum and the local minimum at mid-stance (Fig. 6A).

These ROIs led to null hypothesis rejection for both ROIs in the independent six-subject

dataset (Fig. 6C). Had the ROIs not been defined the null hypothesis would not have been

rejected (Fig. 6B).

DISCUSSION

The main purpose of this paper was to clarify the theoretical consistency between ‘‘0D’’

and ‘‘1D’’ analyses, and to emphasize that both are actually just special cases of ROI

analysis. In particular, the ROI size (S) and 1D smoothness (FWHM) parameters together

construct a continuous theoretical bridge between 0D and 1D methodologies (Eq. (2),

Fig. 3). ROI-based statistical analysis of nD continua was formalized in the Neuroimaging

literature (Brett et al., 2002;Poldrack, 2006;Friston et al., 2007) where established ROI

software tools exist, including especially MarsBar (Hammers et al., 2002;Brett et al., 2002)

for SPM99 (Wellcome Trust Centre for Neuroimaging, University College London, UK).

Although ROI analyses are common in biomechanical applications like plantar pressure

analysis (Cavanagh & Ulbrecht, 1994) and finite element analysis (Radcliffe & Taylor, 2007),

statistical ROI theory has not, to our knowledge, been addressed in the Biomechanics

literature.

The key theoretical point to consider when implementing ROI analyses is that small

ROIs can more readily detect true within-region effects than large ROIs (Figs. 3 and 4).

However, ROI analysis are not necessarily more sensitive than full-field 1D analysis because

effects may exist outside the ROI (Fig. 5, narrowest ROI). An investigator must therefore

balance local signal detectability (via narrow a priori ROI definition) with full-field signal

Pataky et al. (2016), PeerJ , DOI 10.7717/peerj.2652 7/12

Figure 6 Dataset D (VGRF) sensitivity augmentation procedure and results. (A) Pilot study’s mean

inter-condition VGRF difference (see Fig. 2B) for N=1 subject along with user-defined ROIs. The ROIs

specify both the direction and temporal extent of an omnibus a priori hypothesis to be tested in an inde-

pendent experiment. (B) Experimental analysis on N=6 independent subjects using full-field inference

(i.e., had no ROIs been selected); ‘‘SPM{t}’’ denotes the tvalue. The experimental differences between

Normal and Fast are insufficiently large to reject the full-field null hypothesis (because the tvalue fails to

traverse the critical threshold—depicted in red—at ↵=0.05). (C) Experimental analysis using ROI-based

inference: the omnibus ROI-based null hypothesis is rejected.

Pataky et al. (2016), PeerJ , DOI 10.7717/peerj.2652 8/12

detectability (via larger ROIs). In the absence of an a priori hypothesis regarding a specific

portion of the continuum it has been suggested that full-field analyses should be conducted

(Pataky, Vanrenterghem & Robinson, 2016), and this paper extends that suggestion to

include a caveat for exploratory work in a two-stage procedure involving an initial full-field

analysis on an exploratory dataset followed by more precise ROI-based testing (Fig. 6).

An apparent limitation of ROIs is that, since they can create / eradicate statistical

significance (Figs. 4 and 5), they have the potential to be abused. We don’t regard this

limitation as unique to ROI analysis because ROI size, like ↵, can be manipulated to

artificially raise / lower critical thresholds. We would therefore recommend that, instead

of choosing a single ROI size or a single ↵, investigators should actively manipulate

all parameters that might affect ultimate conclusions including: ROI size, ↵, data

filtering, coordinate system definitions, etc., and then actively report the results of those

manipulations in a sensitivity analysis as has been done elsewhere (Pataky et al., 2014). If

the reported results are robust to those manipulations then one can be more confident that

the reported results are neither false positives (Pataky, Vanrenterghem & Robinson, 2016)

nor false negatives. Such manipulations may be especially important for biomechanical

datasets considering that these data can be sensitive to ROI definitions (Figs. 5–6), and

also considering that some controversies regarding the relative merits of 0D and ROI

vs. full-field analysis exist in other literatures (Kubicki et al., 2002;Giuliani et al., 2005;

Furutani et al., 2005;Friston et al., 2006;Saxe, Brett & Kanwisher, 2006;Snook, Plewes &

Beaulieu, 2007;Kilner, 2013).

More formally, ROI definitions and procedures should be considered from the

perspective of ‘circular analysis’ (Kriegeskorte et al., 2009), an umbrella term encompassing

bias-generating factors in scientific intepretations of processed data. In particular, the

approach we have recommended with Dataset D — ROI generation based on independent-

subject pilot studies—is analogous to ‘functional localizers’ in the Neuroimaging literature

for which substantial benefits but also substantial cicular risks exist (Friston et al., 2006;

Saxe, Brett & Kanwisher, 2006;Kilner, 2013). As a simple example of circular analysis and

its dangers consider the local maximum in Dataset A near time =15% (Fig. 4A). Defining

an ROI about this maximum after seeing this result would lead to null hypothesis rejection

even for relatively broad ROIs of 20%. This would be a circular conclusion because the

observed result has directly affected the conclusion’s assumption that the identified ROI

was not of a priori interest but is now. In contrast, the 1D (full-field) threshold correctly

fails to reject the null hypothesis; the 1D approach’s conclusion is not circular because the

a priori assumption of full-field effects was not affected by the observed result.

The literature contains a variety of recommendations regarding avoiding bias associated

with circularity in ROI analyses, and recent developments in particular show that it is

possible to use algorithmic data-driven ROI selection in an unbiased manner to increase

statistical power and also maintain control over Type I error rates (Brooks, Zoumpoulaki

& Bowman, in press). This result is limited to specific cases of experimental variance, but

may nevertheless be promising for researchers concerned regarding inadequate power in

analyses of large 1D datasets with potentially many 1D variables. Regardless, if one wishes

to conduct ROI analysis or otherwise reduce the recorded 1D dataset, they should be aware

Pataky et al. (2016), PeerJ , DOI 10.7717/peerj.2652 9/12

of circularity and how circular (il)logic can produce a vast array of biased conclusions

following dataset reduction.

A technical point not addressed in the analyses above is how to handle within-ROI

signals that extend beyond the ROI. This situation is observable in Fig. 6C, when the

test statistic value is greater than the threshold on the ROI boundaries. From a classical

hypothesis perspective the point is moot because the null hypothesis is rejected regardless

of the temporal scope of a supra-threshold signal. It has been recommended elsewhere that,

if a within-ROI signal is detected, the entire signal be reported even if it extends outside of

the ROI (Friston et al., 2007). Our recommended approach of manipulating ROI location

and size in a sensitivity analysis is consistent with that approach in that ROI boundaries

should generally be regarded as soft.

In summary, this paper has introduced and validated an ROI approach for analyzing 1D

biomechanical trajectories which clarifies the consistency between common 0D approaches

and recent 1D approaches. Since biomechanical interpretations can be sensitive not only to

ROI size but also to other data processing particulars like filtering and coordinate system

definitions, it is recommended that ROIs be used only when there is adequate a priori

justification for ignoring other regions of the 1D continuum, and that when they are used

ROI sensitivity results are also reported.

ADDITIONAL INFORMATION AND DECLARATIONS

Funding

This work was supported by Wakate A Grant 15H05360 from the Japan Society for the

Promotion of Science. The funders had no role in study design, data collection and analysis,

decision to publish, or preparation of the manuscript.

Grant Disclosures

The following grant information was disclosed by the authors:

Japan Society for the Promotion of Science: 15H05360.

Competing Interests

The authors declare there are no competing interests.

Author Contributions

•Todd C. Pataky conceived and designed the experiments, analyzed the data, wrote the

paper, prepared figures and/or tables, reviewed drafts of the paper.

•Mark A. Robinson and Jos Vanrenterghem conceived and designed the experiments,

analyzed the data, wrote the paper, reviewed drafts of the paper.

Data Availability

The following information was supplied regarding data availability:

All raw data analyzed in this paper are available in the ‘‘spm1d’’ software package

available at: http://www.spm1d.org.

Pataky et al. (2016), PeerJ, DOI 10.7717/peerj.2652 10/12

Supplemental Information

Supplemental information for this article can be found online at http://dx.doi.org/10.7717/

peerj.2652#supplemental-information.

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Appendix A Nomenclature

This Appendix proposes a set of terms and deﬁnitions for use in analyses of 1D biomechanical

continua (Table A.1) and relates these terms to analogous, established terms from the neuroimaging

literature (Table A.2). Note that the neuroimaging terminology cannot easily be used for 1D applications

because the meaning of ‘volume’ is unclear for 1D data.

Table A.1: Proposed nomenclature for statistical parametric mapping (SPM) analyses of 1D continua.

Category Term AbbreviationDescription

Geometry

Point of in-

terest

POI A single position in a 1D continuum. (e.g. time = 10%)

Region of in-

terest

ROI A continuous portion of a 1D continuum. (e.g. time =

20–50%).

Statistical

inference

0D inference ⇤Inference procedures applied to 0D univariate or multi-

variate data which do not account for correlation amongst

adjacent continuum points.

1D inference Inference procedures applied to univariate or multivariate

1D data which use multiple comparisons corrections to ac-

count for correlation amongst adjacent continuum points.

RFT procedures are assumed unless otherwise stated.

Random

Field Theory

RFT Provides analytical parametric solutions for probabilities

associated with smooth 1D Gaussian continua, and in par-

ticular the probability that small samples of 1D Gaussian

continua will produce test statistic continua that reach cer-

tain heights in particular experiments. Reference: Adler

and Taylor (2007)

Small region

correction

SRC A multiple comparisons correction across within-ROI val-

ues using (parametric) RFT or a nonparametric alternative.

RFT is assumed unless otherwise stated.

General

Methodology

0D analysis Analysis of univariate or multivariate 0D data using 0D

inference procedures.

1D analysis Analysis of univariate or multivariate 1D data using 1D

inference procedures

0D method A method which conducts 0D analysis of 0D data.

1D method A method which conducts 1D analysis of 1D data.

Statistical

Parametric

Mapping

SPM A methodology for analyzing nD continua, often using clas-

sical hypothesis testing. SPM’s classical hypothesis testing

involves: (i) test statistic continuum computation from a

set of registered nD continua, (ii) continuum smoothness

estimation based on the nD gradient of the model residuals,

(iii) critical threshold computation using RFT and the esti-

mated smoothness, and (iv) probability value computation

using RFT and the estimated smoothness for threshold-

surviving clusters. Reference: Friston et al. (2007)

Region of in-

terest analy-

sis

ROIA A set of procedures involving (i) ROI deﬁnition and (ii)

statistical analyses of ROI data.

Speciﬁc

Methodology

0D metric-

based ROI

analysis

ROIA-0D ROIA which conducts 0D analysis on a 0D summary metric

(e.g. mean, median, maximum, etc.) that is meant to sum-

marize or otherwise represent all values in an ROI. ROIA-

0D with a local extremum summary metric (minimum or

maximum) is common in the Biomechanics literature.

1D ROI

analysis

ROIA-1D ROIA which conducts 1D analysis of an ROI’s data using a

small region correction (SRC). This is the focus of the main

manuscript. Equivalent to ‘SRC-based ROI analysis’.

⇤Note: Bonferroni and other corrections applied across multiple continuum nodes but which do not

consider inter-node correlation are included in ‘0D inference’.

Table A.2: Relevant nomenclature from the neuroimaging literature. Terms without abbreviations do

not explicitly appear in the literature and are instead introduced here to clarify connections to the

proposed nomenclature in Table A.1.

Term Abbreviation Description

Region of interest ROI A continuous portion of an nD continuum. Also called a ‘vol-

ume of interest’ (VOI) in 3D.

Region of interest

analysis

⇤0D analyses conducted on a 0D metric extracted from an

ROI. Reference: Brett et al. (2002). Equivalent to ‘volume of

interest (VOI) analysis’.

Small volume cor-

rection

SVC A multiple comparisons correction across within-ROI val-

ues using (parametric) RFT or a nonparametric alterna-

tive. RFT is assumed unless otherwise stated. Ref-

erence: SPM12 Manual (FIL Methods Group, Wellcome

Trust Centre for Neuroimaging, University College London)

http://www.ﬁl.ion.ucl.ac.uk/spm/software/spm12/

Appendix B Field smoothness: FWHM

The full width at half maximum (FWHM) parameter describes the smoothness of 1D random ﬁelds.

Most precisely, the FWHM speciﬁes the breadth of a Gaussian kernel (Fig.B.1) which, when convolved

with uncorrelated (perfectly rough) Gaussian 1D data yields smooth Gaussian 1D ﬁelds (Fig.B.2).

Random ﬁeld theory (RFT) (Adler and Taylor, 2007) uses the FWHM value to describe the prob-

abilistic behavior of smooth ﬁelds. The most important probability for classical hypothesis testing is

the probability that the random ﬁelds will reach a certain height (Friston et al., 2007); setting that

probability to ↵= 0.05 yields the critical RFT thresholds depicted in Fig.3 (main manuscript).

Figure B.1: Breadth parameters for Gaussian kernels: and FWHM.

Figure B.2: One-dimensional Gaussian random ﬁelds. The FWHM parameterizes ﬁeld smoothness.

The smaller the FWHM the rougher the ﬁeld, and the more likely random ﬁelds are to reach a speciﬁed

height. Inﬁnitely smooth ﬁelds (FWHM=1) are probabilistically equivalent to 0D scalars. It has been

shown that Biomechanics datasets generally tend to lie in the range FWHM = [5%, 50%], including

(processed) kinematics, force and processed EMG (Pataky et al., 2016).

Appendix C ROI analysis in Python and MATLAB

This Appendix describes the Python and MATLAB interfaces for region of interest (ROI) analysis

in spm1d (www.spm1d.org). Below Python and MATLAB code snippets are presented in green and

orange, respectively.

C.1 Example ROI analysis

In spm1d, ROI analysis can be conducted using the keyword “roi” in all statistical routines from

spm1d.stats including t tests, regression, ANOVA, etc. As an example, a two-sample t test with an

ROI spanning from time = 10% to 40% can be conducted as follows:

import numpy as np

import spm1d

#(0) Load dataset:

YA,YB = spm1d.data.uv1d.t2.SimulatedTwoLocalMax().get_data()

#(1) Define SR:

roi = np.array( [False]*101 )

roi[10:40] = True

#(2) Conduct t test:

t = spm1d.stats.ttest2(YB, YA, roi=roi)

ti = t.inference(0.05)

ti.plot()

%(0) Load dataset:

dataset = spm1d.data.uv1d.t2.SimulatedTwoLocalMax();

[YA,YB] = deal(dataset.YA, dataset.YB);

%(1) Define ROI:

roi = false( 1, 101 );

roi(11:40) = true;

%(2) Conduct t test:

t = spm1d.stats.ttest2(YB, YA, 'roi',roi);

ti = t.inference(0.05);

ti.plot()

C.2 Example analysis in detail

After importing the necessary packages, the next commands retrieve one of spm1d’s built-in datasets:

YA,YB = spm1d.data.uv1d.t2.SimulatedTwoLocalMax().get_data()

dataset = spm1d.data.uv1d.t2.SimulatedTwoLocalMax();

[YA,YB] = deal(dataset.YA, dataset.YB);

Here the variables YA and YB are both (J⇥Q) arrays, where Jand Qare the number of observations

and the number of nodes in the 1D continuum, respectively. They can be visualized as follows:

from matplotlib import pyplot

pyplot.plot(YA.T, color="k")

pyplot.plot(YB.T, color="r")

plot(YA', 'color','r')

hold on

plot(YB', 'color','k')

This produces the following ﬁgure:

Figure C.1: Example dataset “SimulatedTwoLocalMax”.

In this example the ROI is speciﬁed as a binary vector of length Q, where True indicates the ROI:

roi = np.array( [False]*101 )

roi[10:40] = True

roi = false( 1, 101 );

roi(11:40) = true;

Since the boolean values False and True numerically evaluate to 0 and 1, respectively, the ROI can

be visualized using a standard plotting command (Fig.C.2a):

pyplot.plot(roi, color="b")

pyplot.ylim(-0.1, 1.1)

plot(roi, 'color', 'b')

ylim( [-0.1 1.1] )

Alternatively the ROI can be visualized using spm1d’s “plot roi” function — currently only available

in Python (Fig.C.2b):

pyplot.plot(YA.T, color="k")

pyplot.plot(YB.T, color="r")

spm1d.plot.plot_roi(roi, facecolor="b", alpha=0.3)

Figure C.2: Example ROI visualization (a) using pyplot.plot and (b) using spm1d.plot.plot roi

ROI-based statistical analysis proceeds as follows (Fig.C.3a)

t = spm1d.stats.ttest2(YB, YA, roi=roi)

ti = t.inference(0.05)

ti.plot()

t = spm1d.stats.ttest2(YB, YA, 'roi', roi)

ti = t.inference(0.05)

ti.plot()

To conduct the same analysis without an ROI simply drop the “roi” keyword as follows (Fig.C.3b)

t = spm1d.stats.ttest2(YB, YA)

ti = t.inference(0.05)

ti.plot()

t = spm1d.stats.ttest2(YB, YA)

ti = t.inference(0.05)

ti.plot()

Figure C.3: Example analysis (a) with one ROI and (b) without any ROIs. “SPM{t}” denotes the t

statistic extended in time to form a ‘statistical parametric map’.

Note the following:

•When conducting ROI analysis spm1d masks portions of the t statistic curve which lie outside

the speciﬁed ROIs. This masking is replicated in the statistical continuum itself (the “z” attribute

of the spm object), as indicated below. Note: in MATLAB, masked elements “- -” are replaced

by “NaN”.

print( t.z )

[-- -- -- -- -- -- -- -- -- -- -0.22776187136870144 -0.35193394183457477

-0.4454435693960439 -0.4866562443204332 -0.45343849771681227

-0.32945352380074494 -0.10481565221034789 0.22601109469000205

0.6659607107024268 1.2186672527052544 1.8786714310449357 2.624090708123374

3.3977458417276942 4.09182563625598 4.557446151350232 4.676118446926481

4.4442233781988865 3.976243703618212 3.414764145392637 2.864043743188491

2.379655562970072 1.9818420780205368 1.6712848172222807 1.4407743903014176

1.2783724255448865 1.17159899332 1.1081081854538306 1.0770125187802424

1.0676148836585948 1.0710020774519424 -- -- -- -- -- -- -- -- -- -- -- --

-- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- --

-- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- --]

•Critical thresholds (hashed horizontal lines in Fig.C.3) are always lower in ROI vs. non-ROI

analysis because the search volume is smaller.

•When conducting ROI analysis there may be regions outside of the speciﬁed ROI which would

have reached signiﬁcance had whole-ﬁeld analysis been conducted (Fig.C.3b) and/or had the ROIs

been chosen di↵erently. This issue is discussed in the main manuscript.

C.3 Multiple ROIs in a single analysis

To conduct statistical analysis using multiple ROIs simply add multiple portions to the boolean

vector:

roi = np.array( [False]*101 )

roi[10:40] = True

roi[60:75] = True

roi = false( 1, 101 );

roi(11:40) = true;

roi(61:75) = true;

When there are multiple ROIs spm1d automatically raises the critical threshold to account for the

expanded search volume (Fig.C.4). Note that the critical threshold for two ROIs of breadths b1and b2

is not equivalent to the the critical threshold for a single ROI of breadth (b1+b2). For example, using

the two-ROIs deﬁned above yields a critical threshold of t⇤=3.115. If, instead a single ROI of the same

total breadth had been deﬁned as follows:

roi = np.array( [False]*101 )

roi[10:55] = True

roi = false( 1, 101 );

roi(11:55) = true;

the critical threshold would be slightly lower: t⇤=3.050. The reason is that within-ROI variance is

smooth (across the 1D continuum), but between-ROI variance is generally not smooth. The spm1d

software accounts for this using the Euler characteristic of the speciﬁed ROI (see Pataky 2016).

Figure C.4: Example analysis (a) with one ROI and (b) with two ROIs.

C.4 Directional ROIs

‘Directional ROIs’ can be used to simultaneously test a set of one-tailed hypotheses. As depicted

in Fig.5 (main manuscript), di↵erent ROIs can embody one-tailed hypotheses. Directional ROIs are

speciﬁed in spm1d as an integer vector containing the values: -1, 0 and +1 as follows:

roi = np.array( [0]*101 )

roi[10:40] = +1

roi[60:75] = -1

roi = zeros( 1, 101 );

roi(11:40) = +1;

roi(61:75) = -1;

Results for this particular set of directional ROIs as applied to the dataset above are depicted in

Fig.C.5b. Interpreting this result is somewhat complex and warrants discussion. First let us compare

this result with the case where both ROIs are positive (Fig.C.5a):

roi = np.array( [0]*101 )

roi[10:40] = +1

roi[60:75] = +1

roi = zeros( 1, 101 );

roi(11:40) = +1;

roi(61:75) = +1;

There are two results to consider:

1. Omnibus result: The omnibus result involves simultaneous testing of all ROIs. In this example

there is suﬃcient evidence to reject the omnibus null hypothesis for both Fig.C.5a and C.5b because

the SPM{t}crosses the critical threshold in the predicted direction in both cases.

2. Individual ROI results: These results may be regarded as a post hoc qualiﬁcation of the omnibus

test, much in the same way post hoc t tests work in ANOVA. In Fig.C.5a the null hypotheses are

rejected for both ROIs, but in Fig.C.5b the null hypothesis is rejected only for the ﬁrst ROI.

In other words, there is suﬃcient evidence to reject the omnibus null hypothesis for both Fig.C.5a

and Fig.C.5b. In Fig.C.5a there is suﬃcient evidence to reject the null hypotheses pertaining to both

individual ROIs in post hoc analysis. However, in Fig.C.5a there is only suﬃcient evidence to reject the

null hypothesis pertaining to the ﬁrst ROI.

Figure C.5: Example directional ROI analysis. (a) Both ROIs are positive-directed. (b) The ﬁrst and

second ROIs are positive- and negative-directed, respectively.

Appendix D Accessing datasets

All datasets analyzed in this paper are available in the spm1d software package (www.spm1d.org)

and can be accessed via the spm1d.data interface as described below. Python and MATLAB code

snippets appear in green and orange font, respectively.

All datasets in spm1d.data are organized as follows:

spm1d.data.XXXX.ZZZZ.DatasetName

where XXXX refers to the data modaility (e.g. univariate, one-dimensional), ZZZZ refers to the original

or appropriate statistical test (e.g. one-sample t test), and DatasetName is a unique name assigned to

each dataset, containing the main author’s family name and publication year where appropriate.

Datasets A and B (Pataky et al., 2015)

Datasets A and B contain univariate, one-dimensional data (“uv1d”), the appropriate test for both

is a one-sample t-test (“’t1”), and the dataset names are “SimulatedPataky2015a” and “Simulated-

Pataky2015b”. They can be accessed as indicated below. The YA and YB variables are both (10 x 101)

arrays, where 10 is the number of responses and 101 is the number of time nodes used to approximate

the continuum.

import spm1d

datasetA = spm1d.data.uv1d.t1.SimulatedPataky2015a()

datasetB = spm1d.data.uv1d.t1.SimulatedPataky2015b()

YA = datasetA.Y

YB = datasetB.Y

datasetA = spm1d.data.uv1d.t1.SimulatedPataky2015a();

datasetB = spm1d.data.uv1d.t1.SimulatedPataky2015b();

YA = datasetA.Y;

YB = datasetB.Y;

Dataset C (Neptune et al., 1999)

Dataset C contains multivariate, one-dimensional data (“mv1d”), the appropriate test is a paired

Hotellings T2test (“hotellings paired”), and the dataset name is “Neptune1999kneekin”. They can be

accessed as indicated below. The YA and YB variables are both (8 x 101 x 3), where 8 is the number

of responses, 101 is the number of time nodes, and 3 is the number of vector components. Although

these data are multivariate, in this study univariate analyses were conducted on only the ﬁrst vector

component (variables yA and yB).

import spm1d

dataset = spm1d.data.mv1d.hotellings_paired.Neptune1999kneekin()

YA,YB = dataset.YA, dataset.YB

yA,yB = YA[:,:,0], YB[:,:,0]

dataset = spm1d.data.mv1d.hotellings_paired.Neptune1999kneekin();

[YA,YB] = deal( dataset.YA, dataset.YB );

[YA,YB] = deal( YA(:,:,1), YB(:,:,1) );

Dataset D (Pataky et al., 2008)

Dataset D contains univariate, one-dimensional data (“uv1d”), the appropriate test is one-way

ANOVA (“anova1”), and the dataset name is “SpeedGRFcategorical”. Individual subjects’ data can

be loaded using the “subj” keyword as indicated below. The Y and A variables in each dataset are (60

x 101), and (60 x 1), respectively, where 60 is the number of responses and 101 is the number of time

nodes. Y contains the GRF data and A contains a vector of condition labels, where the labels “1”, “2”,

and “3” refer to slow, normal and fast walking, respectively. Note that 20 repetitions of each condition

were conducted and that the conditions were presented in a randomized order. In this paper only the

normal and fast conditions were considered.

The Y00 and Y01 variables are both (20 x 101) and were used in the Stage 1 analyses (Fig.6a). The

Y0 and Y1 variables are both (6 x 101) and contain within-subject mean trajectories for each of the six

subjects, and were used in the State 2 analyses (Fig.6b–c).

# load pilot subject's data:

subj0 = 0

dataset = spm1d.data.uv1d.anova1.SpeedGRFcategorical(subj=subj0)

Y,A = dataset.Y, dataset.A

Y00 = Y[A==2]

Y01 = Y[A==3]

# load experiment subjects' data:

SUBJ = [2, 3, 4, 5, 6, 7]

Y0 = []

Y1 = []

for subj in SUBJ:

dataset = spm1d.data.uv1d.anova1.SpeedGRFcategorical(subj=subj)

Y,A = dataset.Y, dataset.A

Y0.append( Y[A==2].mean(axis=0) )

Y1.append( Y[A==3].mean(axis=0) )

Y0,Y1 = np.array(Y0), np.array(Y1)

% load pilot subject's data:

subj0 = 0;

dataset = spm1d.data.uv1d.anova1.SpeedGRFcategorical(subj0);

[Y,A] = deal( dataset.Y, dataset.A );

Y00 = Y(A==2,:);

Y01 = Y(A==3,:);

% load experiment subjects' data:

SUBJ = [2 3 4 5 6 7];

Y0 = zeros(6, 101);

Y1 = zeros(6, 101);

for i = 1:6

dataset = spm1d.data.uv1d.anova1.SpeedGRFcategorical( SUBJ(i) );

[Y,A] = deal( dataset.Y, dataset.A );

Y0(i,:) = mean( Y(A==2,:), 1);

Y1(i,:) = mean( Y(A==3,:), 1);

end

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