## 26 Nov Denote what the study was about. Discuss how random-field theory was used in the case study. What were the results of the

- Denote what the study was about.
- Discuss how random-field theory was used in the case study.
- What were the results of the false recovery rate in the study?

Ensure there are at least two-peer reviewed sources to support your work. The paper should be at least two pages of content (this does not include the cover page or reference page).

Submitted 11 July 2019 Accepted 11 November 2019 Published 10 December 2019

Corresponding author Todd C. Pataky, [email protected]

Academic editor Andrew Gray

Additional Information and Declarations can be found on page 15

DOI 10.7717/peerj.8189

Copyright 2019 Naouma and Pataky

Distributed under Creative Commons CC-BY 4.0

OPEN ACCESS

A comparison of random-field-theory and false-discovery-rate inference results in the analysis of registered one- dimensional biomechanical datasets Hanaa Naouma1,2 and Todd C. Pataky2

1 Bioengineering Course/Graduate School of Science and Technology, Shinshu University, Ueda, Nagano, Japan 2 Department of Human Health Sciences/Graduate School of Medicine, Kyoto University, Kyoto, Japan

ABSTRACT Background. The inflation of falsely rejected hypotheses associated with multiple hypothesis testing is seen as a threat to the knowledge base in the scientific literature. One of the most recently developed statistical constructs to deal with this problem is the false discovery rate (FDR), which aims to control the proportion of the falsely rejected null hypotheses among those that are rejected. FDR has been applied to a variety of problems, especially for the analysis of 3-D brain images in the field of Neuroimaging, where the predominant form of statistical inference involves the more conventional control of false positives, through Gaussian random field theory (RFT). In this study we considered FDR and RFT as alternative methods for handling multiple testing in the analysis of 1-D continuum data. The field of biomechanics has recently adopted RFT, but to our knowledge FDR has not previously been used to analyze 1-D biomechanical data, nor has there been a consideration of how FDR vs. RFT can affect biomechanical interpretations. Methods. We reanalyzed a variety of publicly available experimental datasets to understand the characteristics which contribute to the convergence and divergence of RFT and FDR results. We also ran a variety of numerical simulations involving smooth, random Gaussian 1-D data, with and without true signal, to provide complementary explanations for the experimental results. Results. Our results suggest that RFT and FDR thresholds (the critical test statistic value used to judge statistical significance) were qualitatively identical for many experimental datasets, but were highly dissimilar for others, involving non-trivial changes in data interpretation. Simulation results clarified that RFT and FDR thresholds converge as the true signal weakens and diverge when the signal is broad in terms of the proportion of the continuum size it occupies. Results also showed that, while sample size affected the relation between RFT and FDR results for small sample sizes (<15), this relation was stable for larger sample sizes, wherein only the nature of the true signal was important. Discussion. RFT and FDR thresholds are both computationally efficient because both are parametric, but only FDR has the ability to adapt to the signal features of particular datasets, wherein the threshold lowers with signal strength for a gain in sensitivity. Additional advantages and limitations of these two techniques as discussed further. This article is accompanied by freely available software for implementing FDR analyses involving 1-D data and scripts to replicate our results.

How to cite this article Naouma H, Pataky TC. 2019. A comparison of random-field-theory and false-discovery-rate inference results in the analysis of registered one-dimensional biomechanical datasets. PeerJ 7:e8189 http://doi.org/10.7717/peerj.8189

Subjects Bioengineering, Kinesiology, Statistics Keywords Time series analysis, Random field theory, False discovery rate, Type I error rate, Dynamics, Kinematics, Forces, Biological systems, Biomechanics

INTRODUCTION Multiple testing refers to performing many tests on the same dataset. This scenario is common in experimental research fields such as bioinformatics (Fernald et al., 2011), Molecular biology (Pollard, Pollard & Pollard, 2019), and medicine (Banerjee, Jadhav & Bhawalkar, 2009) which consider multiple dependent variables when drawing statistical conclusions. Usually an acceptable cutoff probability α of 0.05 or 0.01 (Type I error rates) is used for decision making. However, with the growing number of hypotheses being simultaneously tested, the probability of falsely rejecting hypotheses has become high (James Hung & Wang, 2010). In biomechanics, multiple testing problems are one of the major causes of a ‘‘confidence crisis of results’’ emerging in the field (Knudson, 2017), with 73% to 81% of applied biomechanics original research reports employing uncorrected multiple statistical analyses (Knudson, 2009). There is therefore an urgent need to both adopt multiple testing procedures and consider the differences amongst them.

The simplest method for handling multiple testing is the Bonferroni adjustment. However, this adjustment assumes independence (i.e., zero correlation) amongst the multiple tests, so is an extreme way to control false positives which can increase the likelihood of false negatives, especially amongst non-independent tests (Nichols & Hayasaka, 2003; Abdi, 2007; Pataky, Vanrenterghem & Robinson, 2015). In neuroimaging, for example, Bonferroni adjustments fail to consider correlation due to spatiotemporal data smoothness. Thus, there is a need for an alternative multiple testing procedure to restore the balance between false positives and false negatives.

Biomechanics is a scientific field which uses mechanical principles to understand the dynamics of biological systems. Measurements of motion and the forces underlying that motion are often analyzed as temporal one-dimensional (1-D) continua. Prior to analysis, these data are often registered to a common temporal domain, resulting in homologous data representation over a 1-D domain of 0%–100% (Sadeghi et al., 2003). 1-D biomechanical datasets like these are used in a large variety of studies. For example: to assess wearable technology effects on spine movement (Papi, Koh & McGregor, 2017), to understand arm swing contributions to vertical jump dynamics (Lees, Vanrenterghem & Clercq, 2004) and to study tendon-to-bone healing in dogs (Rodeo et al., 1993).

In biomechanics literature, the most common analysis method is to extract zero- dimensional (0-D) metrics such as local extrema (Pataky, Vanrenterghem & Robinson, 2015), integrals or means from 1-D measurements. Reducing 1-D data, which often represents complex temporal dynamics, to a single discrete number is non-ideal, not only because it ignores many aspects of the 1-D data, but also because this approach is often inconsistent with the experiment’s null hypothesis, which usually pertains to kinematics or dynamics in general, and not specifically to the extracted 0-D metrics. For instance, gait researchers who collect knee flexion/extension data often record this variable over time

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(e.g., 0–100% gait cycle), but rarely make hypotheses regarding the specific times at which scientifically relevant signals are expected to occur, or specific time series features like range of motion (Pataky, Robinson & Vanrenterghem, 2013). They instead extract 0-D scalars like maximum flexion angle, often because, when the data are visualized, these features appear to embody the instants of maximum effect size (Morgan & O’Connor, 2019; Le Sant et al., 2019). This scalar extraction approach not only fails to consider the whole movement, but also increases the probability of creating/ eliminating statistical significance. This approach has been termed ‘‘regional focus bias’’, or ad hoc feature selection, and it can greatly increase the risk of incorrectly rejecting the null hypothesis (Pataky, Robinson & Vanrenterghem, 2013).

An alternative to 0-D metrics extraction, whole-trajectory 1-D analyses, emerged in the Biomechanics literature over the last two decades. The main 1-D techniques include: functional data analysis (FDA) (Ramsay & Silverman, 2005), principle component analysis (PCA) (Daffertshofer et al., 2004) and statistical parametric mapping (SPM) (Pataky, Robinson & Vanrenterghem, 2013). PCA is a dimensionality reduction technique and does not provide a method for hypothesis testing, so cannot easily be compared to the other two methods. FDA encompasses a variety of inferential procedures used to analyze 1-D data, including nonparametric permutation methods (Ramsay & Silverman, 2005; Warmenhoven et al., 2018). Since there are many existing FDA procedures of varying complexity, in this study we consider only SPM, which is simpler than FDA because it utilizes a relatively simple random field theory (RFT) inferential procedure, which requires just two parameters: sample size and smoothness. The smoothness parameter is the full-width-at-half-maximum (FWHM) of a Gaussian kernel which, when convolved with uncorrelated 1-D Gaussian data, would yield the same temporal smoothness as the average smoothness of the given dataset’s residuals. A robust procedure for estimating FWHM was introduced for n-dimensional data in (Kiebel et al., 1999) and has been validated for 1-D data in (Pataky, 2016).

Exactly as 0-D parametric inference assumes 0-D Gaussian randomness, RFT assumes 1-D Gaussian randomness. 0-D Gaussian randomness is parameterized by sample size, or more precisely: degrees of freedom, and 1-D Gaussian randomness is additionally parameterized by a smoothness parameter, the FWHM (Kiebel et al., 1999). However, since this assumption might be violated researchers are encouraged to check the normality of their data before conducting RFT analyses. One way is to use the D’Agostino-Person normality test (D’Agostino, Belanger & D’Agostino, 1990), which can be RFT-corrected (Pataky, 2012).

SPM’s applied use of RFT was developed in neuroimaging (Worsley et al., 1992; Friston et al., 2007) to control the false positive rate. SPM and RFT have recently spread to various fields such as Electrophysiology (Kiebel & Friston, 2004) and Biomechanics (Pataky, 2012) and have been validated for hypothesis testing for 1-D data (Pataky, 2016). Example uses of SPM in biomechanics include: dynamic comparisons of elite and recreational athletes (Mei et al., 2017), effects of chronic ankle instability on landing kinematics (De Ridder et al., 2015), and effects of shoe ageing on running dynamics in children (Herbaut et al., 2017).

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A viable alternative to SPM’s false positive control during multiple hypothesis testing is to instead control the false discovery rate (FDR). The FDR represents the proportion of falsely rejected null hypotheses amongst all rejected null hypotheses when simultaneously testing multiple hypotheses (Benjamini & Hochberg, 1995). FDR inference uses the highest p-value satisfying the inequality p (i) ≤ i α/Q as a critical threshold, where α is the Type I error rate, usually 0.05, i is the index of the ordered p-values, and Q is the total number of tests. Thus, the FDR control procedure of (Benjamini & Hochberg, 1995) computes node-wise p-values and orders them to calculate the p threshold that ensures that the FDR is less than α over a large number of experiments. Usually inter-test independence is assumed (Benjamini & Hochberg, 1995) even if the assumption has little practical impact on the results (Benjamini, 2010; Chumbley et al., 2010).

Moreover, FDR procedures are generally less conservative than Type I error control across the Q tests and the adaptability of FDR thresholds to the data allow a balance of Type I and Type II errors (Benjamini & Hochberg, 1995; Storey & Tibshirani, 2003). FDR has been used as a thresholding technique for functional neuroimaging (Genovese, Lazar & Nichols, 2002; Chumbley & Friston, 2009; Schwartzman & Telschow, 2019) and has been described as a method that has the potential to eclipse competing multiple testing methods (Nichols & Hayasaka, 2003; Pike, 2011).

In biomechanics literature, FDR procedures have been used to correct multiple testing problems involving 0-D metrics (Matrangola et al., 2008; Horsak & Baca, 2013). However, to the best of our knowledge, no previous study has used FDR control to analyze 1-D data.

Although RFT inference is considered the most popular method to control family wise error rates in the neuroimaging literature (Lindquist & Mejia, 2015), the breakthrough FDR control paper (Benjamini & Hochberg, 1995) has led FDR control to become widely adopted in diverse fields such as: Neuroimaging (Genovese, Lazar & Nichols, 2002), bioinformatics (Reiner, Yekutieli & Benjamini, 2003), genomics (Storey & Tibshirani, 2003), metabolomics (Denery et al., 2010) and ecology (Pike, 2011). It has been argued that FDR control is more appealing than Type I error control because the former is more scientifically relevant than the latter (Genovese & Wasserman, 2002). That is, scientists are generally more interested in the proportion of nodes that are reported as false positives (FDR) than if there are any false positives (Type I error control). Thus, FDR has higher probability that the results declared significant correspond to an actual effect and not to chance.

The primary purpose of this study was to compare FDR and RFT thresholds in the analyses of 1-D data, and in particular to check whether these procedures could lead to qualitatively different interpretations of experimental datasets. To this end we reanalyzed a variety of publicly available datasets representing diverse experimental tasks (running, walking, cutting) and data modalities which span the breadth of biomechanical data including forces, kinematics and electrical muscle signals. These types of data have very different physical natures, are measured using very different equipment, and are generally processed in very different manners. For reporting purposes, we selected two datasets that most clearly illustrate the most relevant scientific implications of choosing between RFT and FDR. We also performed complementary numerical simulations, involving random

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Table 1 Experimental datasets.

Dataset Source J Q Model Task Variable

A Caravaggi et al., 2010 10 101 Paired t-test Walking Plantar arch deformation B Dorn, Schache & Pandy, 2012 7 100 Linear regression Running/Sprinting Ground reaction force

Notes. J, sample size; Q, number of time nodes.

(Gaussian) 1-D data, to explain the RFT and FDR results’ convergence and divergence that we observed in the experimental datasets.

MATERIALS & METHODS Experimental datasets Across a range of six public datasets in the spm1d software package (Pataky, 2012) from the Biomechanics literature (Neptune, Wright & van den Bogert, 1999; Pataky et al. 2008; Pataky et al. 2014; Besier et al. 2009; Caravaggi et al., 2010; Dorn, Schache & Pandy, 2012) we selected two datasets to report in the main manuscript (Table 1). The criteria for inclusion were: (1) one dataset exhibiting RFT-FDR convergence, (2) one dataset exhibiting RFT-FDR divergence, and (3) adherence of these two datasets to RFT’s normality assumption, so that the RFT results could reasonably be considered valid. Information regarding the remaining datasets, including the statistical analysis results, are available in Appendix E.

Dataset A (Caravaggi et al., 2010) consisted of plantar arch deformation data with the purpose of studying the relationship between the longitudinal arch and the passive stabilization of the plantar aponeurosis. Ground reaction force (GRF) data were collected from ten participants during walking at different speeds: slow, normal and fast walking. For each speed, participants performed ten trials over a wooden walkway with an integrated force plate to record stance-phase GRF. Here we consider only two of the study’s categorical speeds: ‘‘normal’’ and ‘‘fast’’. Since each participant performed both speeds, the underlying experimental design was paired.

Dataset B (Dorn, Schache & Pandy, 2012) consisted of three-dimensional GRF data from seven participants during running and sprinting at four different speeds, slow running at 3.56 m/s, medium-paced running at 5.20 m/s, fast running 7.00 m/s and maximal sprinting at 9.49 m/s. Over a 110 m track, the participant accelerated to a steady state up to 60 m, held the steady state for 20 m and decelerated over the remaining 30 m. The data were collected during the steady state phase. Only two trials per speed were available, and only the mediolateral GRF component was analyzed. Speed effects were examined using linear regression analysis.

Data analysis The analyses in this paper were conducted in Python 3.6 (Van Rossum, 2018), using Anaconda 4.4.10 (Anaconda, Inc.) and the open source software packages: spm1d (Pataky, 2012) and power1d (Pataky, 2017). Software implementing FDR inferences for 1-D data (see text below) are available in this project’s repository: https://github.com/0todd0000/fdr1d.

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For both datasets, first the test statistic (t value) was computed at each time node, yielding an ‘‘SPM{t}’’ as detailed elsewhere (Kiebel et al., 1999). Statistical inferences regarding this SPM{t} were then conducted by computing critical domain-wide thresholds. These thresholds were calculated using two different procedures: Type I error rate control using RFT and FDR control, the ratio of Type I errors to the number of significant tests. The two methods yielded two thresholds per dataset. The RFT thresholds were calculated based on estimated temporal smoothness (Kiebel et al., 1999) as detailed elsewhere (Friston et al., 2007 ; Pataky, 2016). The FDR thresholds were calculated according to (Benjamini & Hochberg, 1995) as detailed elsewhere (Genovese, Lazar & Nichols, 2002) and as described in this article’s Supplemental Information.

Both statistical methods (RFT and FDR) have corrected for Q comparisons, where Q is the number of time nodes in each dataset (Table 1). We also briefly considered 0-D (‘‘Uncorrected’’) and Bonferroni procedures in the context of 1-D smooth data to demonstrate the limitations of both in the analysis of 1-D measurements.

Simulations Numerical simulations involving smooth, random 1-D data were conducted with the goal of explaining the similarities and differences between the aforementioned RFT and FDR thresholds. Two sets of simulations were conducted: (i) qualitative experimental results replication, and (ii) RFT/FDR threshold divergence exploration.

Replicating the experimental results Two simulations were conducted, one per dataset (Fig. 1), involving both 1-D signal (Fig. 2A) and smooth 1-D noise (Fig. 2B). The signal was modeled as a Gaussian pulse, and parameterized by pulse center (q), pulse breadth (σ , standard deviation units) and amplitude (amp). These three parameters were manually adjusted so that, when added to random 1-D noise, the resulting simulated dataset (Fig. 2C) yielded statistical results that qualitatively replicated the experimental datasets’ results. (Table 2) lists the selected parameters which were estimated from experimental datasets as FWHM = 20.37 and 7.94 for Datasets A and B, respectively.

The 1-D noise was created using a previously validated 1-D random number generator (Pataky, 2016). This generator accepted three parameters: sample size (J), number of continuum nodes (Q), and smoothness estimate (FWHM). All noise parameters were selected to follow the experimental datasets (Tables 1–2). For these ‘‘replication’’ simulations, only two realizations of noise (and thus only two simulated datasets) were produced. These datasets were analyzed identically to the experimental datasets.

Exploring threshold divergence Both datasets without signal (Fig. 2B) and datasets with signal (Fig. 2C) were simulated. The aforementioned simulations were conducted using Monte Carlo simulations, involving manipulation of J and FWHM for datasets without signal and all aforementioned parameters except Q (i.e., J, q, σ , amp and FWHM) for the simulated datasets with signal. Sample size J was varied between 5 and 50, representing the small to moderate sample sizes typical in Biomechanics research (Knudson, 2017). Signal position q was

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0 20 40 60 80 100 Time (%)

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Figure 1 Experimental datasets. (A) Dataset A (Caravaggi et al., 2010): plantar arch angle in 10 par- ticipants during normal and fast walking (means and standard deviation clouds). (B) Dataset B (Dorn, Schache & Pandy, 2012): mediolateral ground reaction force during running/sprinting at four different speeds for one participant, means of two trials were shown.

Full-size DOI: 10.7717/peerj.8189/fig-1

Table 2 Qualitatively estimated simulation parameters which yielded similar results to the experi- mental datasets. Sample sizes were the same as in the original datasets. Noise smoothness (FWHM) was estimated from the experimental datasets following (Kiebel et al., 1999).

Characteristic Symbol (Simulated) Dataset A

(Simulated) Dataset B

Sample size J 10 7 Signal center q 101 17 Signal breadth σ 3 19 Signal amplitude amp 2.3 1.2 Noise smoothness FWHM 20.37 7.94

varied between 0 and Q. Signal breadth σ was varied between 0 and 20. Signal amplitude was varied between 0 and 4; since the standard deviation of the noise is one, the latter corresponds to approximately four times the noise amplitude. The smoothness (FWHM) was varied between 10% and 30%, representing the set of previously reported smoothness values for biomechanical data (Pataky, Vanrenterghem & Robinson, 2015) that were found in this study to be sufficient to illustrate RFT/FDR divergence.

For each parameter combination, 10,000 simulation iterations were conducted, each involving a new noise realization. FDR thresholds were computed for each dataset, then averaged across the 10,000 iterations. For simulations without signal, RFT thresholds were also computed for every iteration. Convergence/divergence of the RFT and FDR thresholds were judged qualitatively, by plotting them as functions of the other simulation parameters. In interest of space we report only key simulation findings. Moreover, additional details and results, including code necessary to produce our results, are provided as Supplemental Information in this project’s public repository (https://github.com/0todd0000/fdr1d/).

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Figure 2 Simulated dataset example (DV = dependent variable). (A) Gaussian pulse, representing the true signal, and characterized by amplitude amp and standard deviation σ . (B) One-dimensional smooth Gaussian fields, representing the dataset residuals, and characterized by the smoothness parameter FWHM =20% (full-width-at-half-maximum) (Kiebel et al., 1999). (C) Simulated dataset (signal+noise).

Full-size DOI: 10.7717/peerj.8189/fig-2

RESULTS Experimental datasets results Dataset A: plantar arch deformation during walking Plantar arch deformation in early-to mid-stance increased with walking speed (Fig. 1A). It reached its maximum deformation during late stance with fast walking exhibiting less deformation compared to normal walking (Fig. 1A). Statistical results suggested a rejection of the null hypothesis of no speed effects, with significant differences over 95% to 100% stance (Fig. 3A). The four critical thresholds were related as follows: Uncorrected <RFT<FDR<Bonferroni

Dataset B: GRF during sprinting As running speed increased, the mediolateral GRF magnitude increased systematically, with relatively high magnitude during the first 50% of stance time (Fig. 1B). The four different analysis approaches (Uncorrected, FDR, RFT and Bonferroni) yielded contradictory

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Figure 3 Experimental results, two tailed-tests. (A) Dataset A, (B) Dataset B. Four different thresholds are depicted: Bonferroni, false discovery rate (FDR), random field theory (RFT) and uncorrected. The null hypothesis is rejected if the t value traverses a threshold.

Full-size DOI: 10.7717/peerj.8189/fig-3

results (Fig. 3B). While RFT and Bonferroni inferences failed to find significant correlation between speed and mediolateral GRF, uncorrected and FDR inferences yielded significance.

Simulation results Replicating the experimental results Simulating two experimental-like datasets (Fig. 4) yielded qualitatively similar results to the real experimental results (Fig. 3). In particular, a convergence of RFT and FDR thresholds was observed in both experimental and simulated analyses of the paired datasets (Figs. 3A–4A), and a divergence of thresholds was observed for the other dataset (Figs. 3B–4B). The signal parameters used to create these results (Table 2) suggest that RFT and FDR thresholds converge for small-amplitude, small-breadth signal, and that they diverge for large-amplitude, large-breadth signal. However, this result pertains to just two specific cases, so more systematic simulation results were used to verify these observations, as described below.

Exploring threshold divergence Systematic simulations of datasets without signal (Fig. 5) found that FDR thresholds were slightly lower than RFT thresholds, irrespective of smoothness. Moreover, both RFT and FDR thresholds decreased with both smoothness (FWHM) and sample size. Since any given dataset embodies a single sample size and a single smoothness, these results suggest that FDR thresholds will be marginally smaller than RFT thresholds when there is no true signal.

For simulations with true signal, signal breadth (σ) (Fig. 6A) and signal amplitude (Fig. 6B) did not affect RFT thresholds, which are, by definition, dependent only on noise characteristics. FDR thresholds, in contrast, reduced with both signal breadth and signal amplitude. For low signal breadth values (σ < 5), the decrease of FDR thresholds was pronounced, from 9.15 to 3.42, and the FDR thresholds were greater than the RFT

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