- Email: [email protected]
Response to letter to the editor regarding “Application of principal component analysis in clinical gait research”
Letters to the Editor / Journal of Biomechanics 47 (2014) 1554–1556
Conflict of interest statement The authors have no financial or personal relationships with other people or organisations that could inappropriately influence (bias) our work.
1555
Huber, P., 1981. Robust Statistics, 1st ed. Wiley, New York. MacCallum, R., Widaman, K., Zhang, S., Hong, S., 1999. Sample size in factor analysis. Psychol. Methods 4, 84–89. Nunnally, J.C., 1987. Psychometric Theory, 2nd ed. McGraw Hill, New York. Richter, C., O'Connor, N.E., Marshall, B., Moran, K., 2014. Analysis of characterizing phases on waveforms – an application to vertical jumps. J. Appl. Biomech. (in press), PMID: 24042053.
References Bernstein, N.A. 1967. The Co-ordination and Regulation of Movements, Pergamon Press. Boyer, K.A., Angst, M.S., Asay, J., Giori, N.J., Andriacchi, T.P., 2012. Sensitivity of gait parameters to the effects of anti‐inflammatory and opioid treatments in knee osteoarthritis patients. J. Orthop. Res. 30 (7), 1118–1124. Comrey, A., Lee, H., 1992. A First Course in Factor Analysis. Erlbaum, Hillsdale, NJ. Federolf, P.A., Boyer, K.A., Andriacchi, T.P., 2013. Application of principal component analysis in clinical gait research: identification of systematic differences between healthy and medial knee-osteoarthritic gait. J. Biomech. 46 (13), 2173–2178. Harrison, A.J., Ryan, W., Hayes, K., 2007. Functional data analysis of joint coordination in the development of vertical jump performance. Sports Biomech./Int. Soc. Biomech. Sports 6 (2), 199–214.
Kieran Moran n, Chris Richter Applied Sports Performance Research, School of Health and Human Performance, Dublin City University, Dublin, Ireland E-mail address: [email protected] (K. Moran)
Noel E. O'Connor Insight Centre for Data Analytics, Dublin City University, Dublin, Ireland 3 January 2014 n
http://dx.doi.org/10.1016/j.jbiomech.2014.01.057
Response to letter to the editor regarding “Application of principal component analysis in clinical gait research” We would like to thank Moran and colleagues for their questions and for the opportunity to further discuss and explain our approach. We agree on the shortcomings of the traditional discrete point analysis and on the need to develop new analysis techniques that evaluate in an unbiased way the whole information obtained from marker-based motion tracking. One step towards using more information from motion capture data is the waveform analysis as Moran and colleagues (Richter et al., in press) and several other author groups (Deluzio and Astephen, 2007; Landry et al., 2007) applied. This analysis approach considers the information provided by the temporal structure in the waveform of each individual variable, but does not take advantage of the information provided by the specific interdependences between the variables. Troje (2002) and Daffertshofer et al. (2004) applied principal component analysis (PCA) to determine the correlation structure between movement variables, but did not evaluate in the same analysis the information provided by the temporal structure in the data. Approaches that consider the information by both, the betweenvariable correlation structure and the temporal evolution structure, have been reported in several publications, for example, Boyer et al. (2012), Eskofier et al. (2013), Nigg et al. (2012), Pataky et al. (2013), von Tscharner et al. (2013), or the study in question (Federolf et al., 2013) – each with its own advantages and limitations: Support vector machines (Eskofier et al., 2013; Nigg et al., 2012; von Tscharner et al., 2013) are particularly sensitive to group differences (von Tscharner et al., 2013) and statistical parametric mapping (Pataky et al., 2013) offers a stringent, step-wise statistical analysis comparable with performing an ANOVA followed by a post hoc test on discrete variables. The advantage of the approach we have suggested is that we explicitly use and interpret the PC-vectors as manifestations of the segment-coordination structure in a movement, i.e. as inherent biomechanical features that provide essential information about the internal constraints governing how a subject carries out a specific movement task (Federolf, 2013). Furthermore, we conceptually explain why the natural movement variability affects the coordination patterns less than it affects the standard movement variables (Fig. 1 in the paper). Each PC-vector quantifies a specific pattern of how the movement variability covariates among the available subjects. If the projection of two subject groups' data onto the lower order PC-vectors
Corresponding author at: Applied Sports Performance Research, School of Health and Human Performance, Dublin City University, Dublin, Ireland.
differs, then this indicates that the pattern of how the motion variables are coordinated differ between the two groups. Moran and colleagues are correct to point out that only one single statistical test was conducted. This test does not prove that individual variables differ “significantly” between the two groups, but it suggests (with p¼ 0.017) that the coordination structure between the movement variables differed between the two groups. Figs. 2 and 3 in the paper visualize the discriminant vector, i.e. how the coordination structure differed between the two groups. We did not claim that the individual features that we discussed were “significant”, but they are features of a general coordination pattern that, as a whole, differed between the two specific groups tested in our study. In our opinion, this is an objective, scientific result. Furthermore, we believe that studying the coordination structure is particularly promising for studies with small sample sizes since conventional statistical analyses typically yield inconclusive results due to the natural movement variability, while the coordination patterns investigated in our study are less affected. Moran et al. further criticize the selection of specific features from the multifaceted components of the discriminant vector. In our opinion, it is an essential step in research to evaluate results and to compare them to previously reported findings. To some degree this is a subjective process. We tried to provide a comprehensive and accurate summary of the important features in the discriminant vector, but Moran and colleagues are correct to point out that some features were not included. We agree that the selection of which features are discussed might be subjective, but the features themselves are objective results. Another issue raised in this letter is that the study was conducted using marker coordinates in the external coordinate system in which they were measured instead of joint angles determined from these marker coordinates. Various sources of artifacts and issues associated with the calculation of joint angles or with choosing appropriate reference frames are well documented in the literature (Della Croce et al., 2005). The directly measured marker coordinates and GRF force components encompass the most accurate and most comprehensive dataset quantifying the movement of a subject. Moran and colleagues further state that the interpretation of differences in more distal marker positions was more challenging because these differences are “geometrically dependent upon differences at more proximal joints.” As an example, they point out that “differences at the toe/
1556
Letters to the Editor / Journal of Biomechanics 47 (2014) 1554–1556
ankle joint can be geometrically exaggerated because of small accumulative differences at the hip/pelvis and knee.” – We fully agree, but we disagree that these differences should be isolated. On the contrary: joint angles are not independent and it is therefore not correct to treat them that way. Furthermore, if these small differences occur in the same way and at the same time in various gait cycles then they are likely part of the actual differences between the two groups. If each joint angle is analyzed individually, then these differences might be small compared to the general movement variability and might therefore not be detected. However, the PCA that we conducted determined patterns of correlated deviations from the mean motion pattern in all variables and could therefore pick up such differences that co-vary between several variables. In this context, Moran and colleagues also suggest that how the stick figures were centrally aligned was not entirely clear. This was described in Section 2.4 of the paper: “the characteristic differences between the healthy and pathologic gait patterns” (were visualized) “by arbitrarily selecting the normalized data of one healthy subject, adding the discriminant vector, and then retracing the normalization steps.” In other words, the stick figures do not compare OA patients with their matched controls, instead, they visualize the gait of one healthy subject and show how this gait pattern would change according to those differences in the gait patterns that were common between all subjects in the OA group and all subjects in the healthy reference group. The last issue raised by Moran and colleagues addresses repeatability and questions in general if calculation of PCA is appropriate if only small samples are available in high-dimensional vector spaces. The repeatability of the measurements themselves were investigated in a separate study using conventional analysis methods (Asay et al., 2013). We thank Moran and colleagues for their suggestions regarding further validation of the analysis procedure. How the proposed procedures perform in the specific case of small datasets needs to be further investigated. The concerns regarding calculation of PCA using few samples in high-dimensional datasets are well noted, however, the 1:50 or 1:10 rules of thumb (Nunnally, 1987; Comrey and Lee, 1992) were not set up for the specific case of gait data, which shows particularly high covariation between variables and large number of redundant dimensions (Troje, 2002). It is indeed questionable if subspaces within a highly dimensional vector space can be determined consistently from small datasets (Jung and Marron, 2009), however, the projection of the trial vectors onto a specific vector (i.e. the discriminant vector) and the result of the statistical test on the resultant scores is unambiguous. Hence, the results of our study are specific for the given dataset and should not be generalized. Furthermore, the results show one aspect of systematic differences in the movement coordination patterns between the two groups which is not necessarily the only aspect that differs systematically. If large datasets are available, then other analysis methods such as support vector machines can be applied and may yield more divers and more reliable results than the method described here (von Tscharner et al., 2013). However, in our opinion there are many applications – for example, when the available sample sized is limited – where the specific correlation structure in a given dataset is an important source of information. Determining this correlation structure by applying a PCA might then be a method that identifies systematic differences where other methods fail or are less reliable.
References Asay, J.L., Boyer, K.A., Andriacchi, T.P., 2013. Repeatability of gait analysis for measuring knee osteoarthritis pain in patients with severe chronic pain. J. Orthop. Res. 31, 1007–1012. Boyer, K.A., Federolf, P., Lin, C., Nigg, B.M., Andriacchi, T.P., 2012. Kinematic adaptations to a variable stiffness shoe: mechanisms for reducing joint loading. J. Biomech. 45, 1619–1624. Comrey, A., Lee, H., 1992. A First Course in Factor Analysis. Erlbaum, Hillsdale, NJ Della Croce, U., Leardini, A., Chiari, L., Cappozzo, A., 2005. Human movement analysis using stereophotogrammetry: Part 4: assessment of anatomical landmark misplacement and its effects on joint kinematics. Gait Posture 21, 226–237. Deluzio, K.J., Astephen, J.L., 2007. Biomechanical features of gait waveform data associated with knee osteoarthritis: an application of principal component analysis. Gait Posture 25, 86–93. Daffertshofer, A., Lamoth, C.J.C., Meijer, O.G., Beek, P.J., 2004. PCA in studying coordination and variability: a tutorial. Clin. Biomech. 19, 415–428. Eskofier, B.M., Federolf, P., Kugler, P.F., Nigg, B.M., 2013. Marker-based classification of young–elderly gait pattern differences via direct PCA feature extraction and SVMs. Comput. Methods Biomech. Biomed. Eng. 16, 435–442. Federolf, P.A., 2013. A novel approach to solve the “missing marker problem” in marker-based motion analysis that exploits the segment coordination patterns in multi-limb motion data. PLoS One 8 (10), e78689, http://dx.doi.org/10.1371/ journal.pone.0078689. Federolf, P.A., Boyer, K.A., Andriacchi, T.P., 2013. Application of principal component analysis in clinical gait research: identification of systematic differences between healthy and medial knee-osteoarthritic gait. J. Biomech. 46, 2173–2178. Jung, S., Marron, J.S., 2009. PCA consistency in high dimension, low sample size context. Ann. Stat. 37, 4104–4130. Landry, S.C., McKean, K.A., Hubley-Kozey, C.L., Stanish, W.D., Deluzio, K.J., 2007. Knee biomechanics of moderate OA patients measured during gait at a selfselected and fast walking speed. J. Biomech. 40, 1754–1761. Nigg, B.M., Baltich, J., Maurer, C., Federolf, P., 2012. Shoe midsole hardness, sex and age effects on lower extremity kinematics during running. J. Biomech. 45, 1692–1697. Nunnally, J.C., 1987. Psychometric Theory, 2nd ed. McGraw-Hill, New York Pataky, T.C., Robinson, M.A., Vanrenterghem, J., 2013. Vector field statistical analysis of kinematic and force trajectories. J. Biomech. 46, 2394–2401. Richter C., O’Connor N.E., Marshall B. and Moran K., Analysis of characterizing phases on waveforms – an application to vertical jumps, Journal of Applied biomechanicsJ. Appl. Biomech. in press. Troje, N.F., 2002. Decomposing biological motion: a framework for analysis and synthesis of human gait patterns. J. Vis. 2, 371–387. von Tscharner, V., Enders, H., Maurer, C., 2013. Subspace identification and classification of healthy human gait. PLoS One 8 (7), e65063, http://dx.doi. org/10.1371/journal.pone.0065063.`
Peter Federolf n Department of Neuroscience, Faculty of Medicine, Norwegian University of Science and Technology NTNU, 7489 Trondheim, Norway Department of Physical Performance, Norwegian School of Sport Sciences, Oslo, Norway E-mail address: [email protected] Katharine Boyer Department of Kinesiology, University of Massachusetts-Amherst, Amherst, MA, USA Thomas Andriacchi Department of Mechanical Engineering, Stanford University, USA Department of Orthopedic Surgery, Stanford University, USA Veterans Administration, Palo Alto, CA, USA 13 February 2014
Conflict of interest statement No external funding was received for this study. We have no financial interests or other forms of conflicts of interest. http://dx.doi.org/10.1016/j.jbiomech.2014.02.013
n Corresponding author at: Department of Neuroscience, Faculty of Medicine, Norwegian University of Science and Technology NTNU, 7489 Trondheim, Norway. Tel.: þ47 23262322.
Conflict of interest statement The authors have no financial or personal relationships with other people or organisations that could inappropriately influence (bias) our work.
1555
Huber, P., 1981. Robust Statistics, 1st ed. Wiley, New York. MacCallum, R., Widaman, K., Zhang, S., Hong, S., 1999. Sample size in factor analysis. Psychol. Methods 4, 84–89. Nunnally, J.C., 1987. Psychometric Theory, 2nd ed. McGraw Hill, New York. Richter, C., O'Connor, N.E., Marshall, B., Moran, K., 2014. Analysis of characterizing phases on waveforms – an application to vertical jumps. J. Appl. Biomech. (in press), PMID: 24042053.
References Bernstein, N.A. 1967. The Co-ordination and Regulation of Movements, Pergamon Press. Boyer, K.A., Angst, M.S., Asay, J., Giori, N.J., Andriacchi, T.P., 2012. Sensitivity of gait parameters to the effects of anti‐inflammatory and opioid treatments in knee osteoarthritis patients. J. Orthop. Res. 30 (7), 1118–1124. Comrey, A., Lee, H., 1992. A First Course in Factor Analysis. Erlbaum, Hillsdale, NJ. Federolf, P.A., Boyer, K.A., Andriacchi, T.P., 2013. Application of principal component analysis in clinical gait research: identification of systematic differences between healthy and medial knee-osteoarthritic gait. J. Biomech. 46 (13), 2173–2178. Harrison, A.J., Ryan, W., Hayes, K., 2007. Functional data analysis of joint coordination in the development of vertical jump performance. Sports Biomech./Int. Soc. Biomech. Sports 6 (2), 199–214.
Kieran Moran n, Chris Richter Applied Sports Performance Research, School of Health and Human Performance, Dublin City University, Dublin, Ireland E-mail address: [email protected] (K. Moran)
Noel E. O'Connor Insight Centre for Data Analytics, Dublin City University, Dublin, Ireland 3 January 2014 n
http://dx.doi.org/10.1016/j.jbiomech.2014.01.057
Response to letter to the editor regarding “Application of principal component analysis in clinical gait research” We would like to thank Moran and colleagues for their questions and for the opportunity to further discuss and explain our approach. We agree on the shortcomings of the traditional discrete point analysis and on the need to develop new analysis techniques that evaluate in an unbiased way the whole information obtained from marker-based motion tracking. One step towards using more information from motion capture data is the waveform analysis as Moran and colleagues (Richter et al., in press) and several other author groups (Deluzio and Astephen, 2007; Landry et al., 2007) applied. This analysis approach considers the information provided by the temporal structure in the waveform of each individual variable, but does not take advantage of the information provided by the specific interdependences between the variables. Troje (2002) and Daffertshofer et al. (2004) applied principal component analysis (PCA) to determine the correlation structure between movement variables, but did not evaluate in the same analysis the information provided by the temporal structure in the data. Approaches that consider the information by both, the betweenvariable correlation structure and the temporal evolution structure, have been reported in several publications, for example, Boyer et al. (2012), Eskofier et al. (2013), Nigg et al. (2012), Pataky et al. (2013), von Tscharner et al. (2013), or the study in question (Federolf et al., 2013) – each with its own advantages and limitations: Support vector machines (Eskofier et al., 2013; Nigg et al., 2012; von Tscharner et al., 2013) are particularly sensitive to group differences (von Tscharner et al., 2013) and statistical parametric mapping (Pataky et al., 2013) offers a stringent, step-wise statistical analysis comparable with performing an ANOVA followed by a post hoc test on discrete variables. The advantage of the approach we have suggested is that we explicitly use and interpret the PC-vectors as manifestations of the segment-coordination structure in a movement, i.e. as inherent biomechanical features that provide essential information about the internal constraints governing how a subject carries out a specific movement task (Federolf, 2013). Furthermore, we conceptually explain why the natural movement variability affects the coordination patterns less than it affects the standard movement variables (Fig. 1 in the paper). Each PC-vector quantifies a specific pattern of how the movement variability covariates among the available subjects. If the projection of two subject groups' data onto the lower order PC-vectors
Corresponding author at: Applied Sports Performance Research, School of Health and Human Performance, Dublin City University, Dublin, Ireland.
differs, then this indicates that the pattern of how the motion variables are coordinated differ between the two groups. Moran and colleagues are correct to point out that only one single statistical test was conducted. This test does not prove that individual variables differ “significantly” between the two groups, but it suggests (with p¼ 0.017) that the coordination structure between the movement variables differed between the two groups. Figs. 2 and 3 in the paper visualize the discriminant vector, i.e. how the coordination structure differed between the two groups. We did not claim that the individual features that we discussed were “significant”, but they are features of a general coordination pattern that, as a whole, differed between the two specific groups tested in our study. In our opinion, this is an objective, scientific result. Furthermore, we believe that studying the coordination structure is particularly promising for studies with small sample sizes since conventional statistical analyses typically yield inconclusive results due to the natural movement variability, while the coordination patterns investigated in our study are less affected. Moran et al. further criticize the selection of specific features from the multifaceted components of the discriminant vector. In our opinion, it is an essential step in research to evaluate results and to compare them to previously reported findings. To some degree this is a subjective process. We tried to provide a comprehensive and accurate summary of the important features in the discriminant vector, but Moran and colleagues are correct to point out that some features were not included. We agree that the selection of which features are discussed might be subjective, but the features themselves are objective results. Another issue raised in this letter is that the study was conducted using marker coordinates in the external coordinate system in which they were measured instead of joint angles determined from these marker coordinates. Various sources of artifacts and issues associated with the calculation of joint angles or with choosing appropriate reference frames are well documented in the literature (Della Croce et al., 2005). The directly measured marker coordinates and GRF force components encompass the most accurate and most comprehensive dataset quantifying the movement of a subject. Moran and colleagues further state that the interpretation of differences in more distal marker positions was more challenging because these differences are “geometrically dependent upon differences at more proximal joints.” As an example, they point out that “differences at the toe/
1556
Letters to the Editor / Journal of Biomechanics 47 (2014) 1554–1556
ankle joint can be geometrically exaggerated because of small accumulative differences at the hip/pelvis and knee.” – We fully agree, but we disagree that these differences should be isolated. On the contrary: joint angles are not independent and it is therefore not correct to treat them that way. Furthermore, if these small differences occur in the same way and at the same time in various gait cycles then they are likely part of the actual differences between the two groups. If each joint angle is analyzed individually, then these differences might be small compared to the general movement variability and might therefore not be detected. However, the PCA that we conducted determined patterns of correlated deviations from the mean motion pattern in all variables and could therefore pick up such differences that co-vary between several variables. In this context, Moran and colleagues also suggest that how the stick figures were centrally aligned was not entirely clear. This was described in Section 2.4 of the paper: “the characteristic differences between the healthy and pathologic gait patterns” (were visualized) “by arbitrarily selecting the normalized data of one healthy subject, adding the discriminant vector, and then retracing the normalization steps.” In other words, the stick figures do not compare OA patients with their matched controls, instead, they visualize the gait of one healthy subject and show how this gait pattern would change according to those differences in the gait patterns that were common between all subjects in the OA group and all subjects in the healthy reference group. The last issue raised by Moran and colleagues addresses repeatability and questions in general if calculation of PCA is appropriate if only small samples are available in high-dimensional vector spaces. The repeatability of the measurements themselves were investigated in a separate study using conventional analysis methods (Asay et al., 2013). We thank Moran and colleagues for their suggestions regarding further validation of the analysis procedure. How the proposed procedures perform in the specific case of small datasets needs to be further investigated. The concerns regarding calculation of PCA using few samples in high-dimensional datasets are well noted, however, the 1:50 or 1:10 rules of thumb (Nunnally, 1987; Comrey and Lee, 1992) were not set up for the specific case of gait data, which shows particularly high covariation between variables and large number of redundant dimensions (Troje, 2002). It is indeed questionable if subspaces within a highly dimensional vector space can be determined consistently from small datasets (Jung and Marron, 2009), however, the projection of the trial vectors onto a specific vector (i.e. the discriminant vector) and the result of the statistical test on the resultant scores is unambiguous. Hence, the results of our study are specific for the given dataset and should not be generalized. Furthermore, the results show one aspect of systematic differences in the movement coordination patterns between the two groups which is not necessarily the only aspect that differs systematically. If large datasets are available, then other analysis methods such as support vector machines can be applied and may yield more divers and more reliable results than the method described here (von Tscharner et al., 2013). However, in our opinion there are many applications – for example, when the available sample sized is limited – where the specific correlation structure in a given dataset is an important source of information. Determining this correlation structure by applying a PCA might then be a method that identifies systematic differences where other methods fail or are less reliable.
References Asay, J.L., Boyer, K.A., Andriacchi, T.P., 2013. Repeatability of gait analysis for measuring knee osteoarthritis pain in patients with severe chronic pain. J. Orthop. Res. 31, 1007–1012. Boyer, K.A., Federolf, P., Lin, C., Nigg, B.M., Andriacchi, T.P., 2012. Kinematic adaptations to a variable stiffness shoe: mechanisms for reducing joint loading. J. Biomech. 45, 1619–1624. Comrey, A., Lee, H., 1992. A First Course in Factor Analysis. Erlbaum, Hillsdale, NJ Della Croce, U., Leardini, A., Chiari, L., Cappozzo, A., 2005. Human movement analysis using stereophotogrammetry: Part 4: assessment of anatomical landmark misplacement and its effects on joint kinematics. Gait Posture 21, 226–237. Deluzio, K.J., Astephen, J.L., 2007. Biomechanical features of gait waveform data associated with knee osteoarthritis: an application of principal component analysis. Gait Posture 25, 86–93. Daffertshofer, A., Lamoth, C.J.C., Meijer, O.G., Beek, P.J., 2004. PCA in studying coordination and variability: a tutorial. Clin. Biomech. 19, 415–428. Eskofier, B.M., Federolf, P., Kugler, P.F., Nigg, B.M., 2013. Marker-based classification of young–elderly gait pattern differences via direct PCA feature extraction and SVMs. Comput. Methods Biomech. Biomed. Eng. 16, 435–442. Federolf, P.A., 2013. A novel approach to solve the “missing marker problem” in marker-based motion analysis that exploits the segment coordination patterns in multi-limb motion data. PLoS One 8 (10), e78689, http://dx.doi.org/10.1371/ journal.pone.0078689. Federolf, P.A., Boyer, K.A., Andriacchi, T.P., 2013. Application of principal component analysis in clinical gait research: identification of systematic differences between healthy and medial knee-osteoarthritic gait. J. Biomech. 46, 2173–2178. Jung, S., Marron, J.S., 2009. PCA consistency in high dimension, low sample size context. Ann. Stat. 37, 4104–4130. Landry, S.C., McKean, K.A., Hubley-Kozey, C.L., Stanish, W.D., Deluzio, K.J., 2007. Knee biomechanics of moderate OA patients measured during gait at a selfselected and fast walking speed. J. Biomech. 40, 1754–1761. Nigg, B.M., Baltich, J., Maurer, C., Federolf, P., 2012. Shoe midsole hardness, sex and age effects on lower extremity kinematics during running. J. Biomech. 45, 1692–1697. Nunnally, J.C., 1987. Psychometric Theory, 2nd ed. McGraw-Hill, New York Pataky, T.C., Robinson, M.A., Vanrenterghem, J., 2013. Vector field statistical analysis of kinematic and force trajectories. J. Biomech. 46, 2394–2401. Richter C., O’Connor N.E., Marshall B. and Moran K., Analysis of characterizing phases on waveforms – an application to vertical jumps, Journal of Applied biomechanicsJ. Appl. Biomech. in press. Troje, N.F., 2002. Decomposing biological motion: a framework for analysis and synthesis of human gait patterns. J. Vis. 2, 371–387. von Tscharner, V., Enders, H., Maurer, C., 2013. Subspace identification and classification of healthy human gait. PLoS One 8 (7), e65063, http://dx.doi. org/10.1371/journal.pone.0065063.`
Peter Federolf n Department of Neuroscience, Faculty of Medicine, Norwegian University of Science and Technology NTNU, 7489 Trondheim, Norway Department of Physical Performance, Norwegian School of Sport Sciences, Oslo, Norway E-mail address: [email protected] Katharine Boyer Department of Kinesiology, University of Massachusetts-Amherst, Amherst, MA, USA Thomas Andriacchi Department of Mechanical Engineering, Stanford University, USA Department of Orthopedic Surgery, Stanford University, USA Veterans Administration, Palo Alto, CA, USA 13 February 2014
Conflict of interest statement No external funding was received for this study. We have no financial interests or other forms of conflicts of interest. http://dx.doi.org/10.1016/j.jbiomech.2014.02.013
n Corresponding author at: Department of Neuroscience, Faculty of Medicine, Norwegian University of Science and Technology NTNU, 7489 Trondheim, Norway. Tel.: þ47 23262322.