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  • doi: 10.1152/jn.00447.2012109:591-602, 2013. First published 24 October 2012;J Neurophysiol

    J. Lucas McKay, Torrence D. J. Welch, Brani Vidakovic and Lena H. TingANOVAwaveforms revealed using wavelet-based functional Statistically significant contrasts between EMG

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  • Innovative Methodology

    Statistically significant contrasts between EMG waveforms revealed usingwavelet-based functional ANOVA

    J. Lucas McKay, Torrence D. J. Welch, Brani Vidakovic, and Lena H. TingThe Wallace H. Coulter Department of Biomedical Engineering, Emory University and the Georgia Institute of Technology,Atlanta, Georgia

    Submitted 28 May 2012; accepted in final form 20 October 2012

    McKay JL, Welch TD, Vidakovic B, Ting LH. Statisticallysignificant contrasts between EMG waveforms revealed using wave-let-based functional ANOVA. J Neurophysiol 109: 591602, 2013.First published October 24, 2012; doi:10.1152/jn.00447.2012.Wedeveloped wavelet-based functional ANOVA (wfANOVA) as a novelapproach for comparing neurophysiological signals that are functionsof time. Temporal resolution is often sacrificed by analyzing such datain large time bins, increasing statistical power by reducing the numberof comparisons. We performed ANOVA in the wavelet domainbecause differences between curves tend to be represented by a fewtemporally localized wavelets, which we transformed back to the timedomain for visualization. We compared wfANOVA and ANOVAperformed in the time domain (tANOVA) on both experimentalelectromyographic (EMG) signals from responses to perturbationduring standing balance across changes in peak perturbation acceler-ation (3 levels) and velocity (4 levels) and on simulated data withknown contrasts. In experimental EMG data, wfANOVA revealed thecontinuous shape and magnitude of significant differences over timewithout a priori selection of time bins. However, tANOVA revealedonly the largest differences at discontinuous time points, resulting infeatures with later onsets and shorter durations than those identifiedusing wfANOVA (P 0.02). Furthermore, wfANOVA requiredsignificantly fewer (; P 0.015) significant F tests thantANOVA, resulting in post hoc tests with increased power. In simu-lated EMG data, wfANOVA identified known contrast curves with ahigh level of precision (r2 0.94 0.08) and performed better thantANOVA across noise levels (P 0.01). Therefore, wfANOVAmay be useful for revealing differences in the shape and magnitude ofneurophysiological signals (e.g., EMG, firing rates) across multipleconditions with both high temporal resolution and high statisticalpower.

    electromyogram; time series analysis; repeated measurements; bal-ance

    WE OFTEN WANT TO COMPARE the shapes of waveforms that arefunctions of time, but traditional statistical methods cannotreveal differences between curves without sacrificing temporalresolution or power. Certain features of the waveforms that areclearly identifiable based on visual inspection may not berevealed by traditional statistical tests such as t-tests orANOVA applied across time points due to the large number ofcomparisons (Fig. 1A; Abramovich et al. 2004; Fan and Lin1998). For example, many studies present clear differences inmean waveforms across conditions in electromyograms(EMGs; Hiebert et al. 1994), H-reflex responses (Frigon 2004),kinematics (Ivanenko et al. 2005), and neural firing rates

    (Mushiake et al. 1991), but statistical tests are often notperformed or reported due to low power. A common approachto overcome this problem is to apply statistical analyses tomean values over a time bin of interest (Frigon 2004; Welchand Ting 2009). However, this approach sacrifices the temporalresolution of the interesting feature by approximating it as asingle data point. To address this undesirable trade-off betweentemporal resolution and statistical power, we propose perform-ing statistical inference outside of the time domain.

    The wavelet transform is a versatile tool for the analysis ofwaveforms with time-varying frequency content because itreveals not only the different frequency components of thewaveform, but also the temporal structure of those compo-nents. Because of its power to describe signals containingevents throughout the range of time-frequency localization, thewavelet transform has been used in many biomedical applica-tions, including analysis of electroencephalogram and electro-cardiogram signals and processing for positron emission to-mography and MRI (see Unser and Aldroubi 1996 for areview). Like the Fourier transform, the wavelet transformdecomposes the signal of interest into orthogonal basis func-tions with different frequency characteristics that additivelyrepresent the original signal. In contrast to the Fourier trans-form, the basis functions used in the wavelet transform aretemporally localized. This property allows the representationof both frequency and temporal information within the trans-formed signal and provides a particularly rich description ofbiomedical signals, which often have nonstationary frequencycomposition and burstlike temporal structure (Cohen and Ko-vacevic 1996).

    Previously, the wavelet transform has been used to identifyfeatures of EMG signals, but it has not been used for statisticalcomparison of EMG waveform across conditions (Cohen andKovacevic 1996; Unser and Aldroubi 1996). For example, thewavelet transform has been used to quantify automatically thetiming of bursts within EMG signals in a manner similar totemplate matching (Flanders 2002). Wavelet packets have alsobeen used to quantify the temporal location and width of unitbursts in EMG signals (Hart and Giszter 2004) as well as forextraction and classification of motor unit action potentialsfrom EMG records (Fang et al. 1999; Ostlund et al. 2006; Renet al. 2006). Wavelet analyses have also been used to estimatelatent information within EMG signals. For example, time-dependent power spectra (Huber et al. 2011; von Tscharner2000; Wakeling and Horn 2009) as well as changes in thesespectra, for example, with fatigue (Kumar et al. 2003), havebeen estimated to offer information about underlying musclefiber conduction velocity (Stulen and De Luca 1981). How-

    Address for reprint requests and other correspondence: L. H. Ting, TheWallace H. Coulter Dept. of Biomedical Engineering, Emory Univ. and theGeorgia Institute of Technology, 313 Ferst Dr., Atlanta, GA 30332-0535(e-mail:

    J Neurophysiol 109: 591602, 2013.First published October 24, 2012; doi:10.1152/jn.00447.2012.

    5910022-3077/13 Copyright 2013 the American Physiological

    by Lena Ting on January 15, 2013

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  • ever, to our knowledge, the wavelet transform has not beenpreviously applied in the context of hypothesis testing forcomparison of waveform shapes (Unser and Aldroubi 1996).

    Here, we leveraged the beneficial properties of the wavelettransform to develop a generalized technique called wavelet-based functional ANOVA (wfANOVA) to compare statisti-cally the shapes of waveforms that are functions of timewithout sacrificing temporal resolution or statistical power.Our method is related to functional data analysis (Ramsay andSilverman 2005) because each wavelet is a function of timerather than a single time point. When expressed in the waveletdomain, temporally localized waveform features tend to bewell-represented by a few wavelets rather than by many cor-related time samples (Fig. 1B) or by many independent sinu-soids as in Fourier analysis. Therefore, applying statistical teststo wavelet coefficients reduces the number of statistical testsrequired while retaining statistical power (Angelini and Vida-kovic 2003). In contrast to many wavelet-domain techniquesthat have been applied to EMG data, wfANOVA does not useinformation about the wavelet coefficients themselves to sup-pose possible physiological mechanisms that may have gener-ated the EMG signals (e.g., subpopulations of motor units ormuscle fiber types). Rather, wavelets are used only as basisfunctions that correspond to these localized time differencesthat are easily identifiable by eye. The unique and generaliz-able attribute of wfANOVA is the reconstruction of the wave-lets into the time domain after performing functional analysesto allow the visualization of the effects of experimental ma-nipulation as contrast curves in the time domain (Fig. 1B)rather than reporting these effects in the wavelet domain.

    As proof of principle, we applied wfANOVA to previouslypublished EMG data to demonstrate its ability as a generalmethod for determining differences in the time courses ofwaveforms. We first compared the ability of wfANOVA andtime-point ANOVA (tANOVA) to identify contrast curves inpreviously published EMG waveforms in response to pertur-bations during standing balance (Welch