Vector field statistical analysis of kinematic and force trajectories
Introduction
Measurements of motion and the forces underlying that motion are fundamental to biomechanical experimentation. These measurements are often manifested as one-dimensional (1D) scalar trajectories yi(q), where i represents a particular physical body, joint, axis or direction, and where q represents 1D time or space. Experiments typically involve repeated measurements of yi(q) followed by registration (i.e. homologously optimal temporal or spatial normalization) to a domain of 0%–100% (Sadeghi et al., 2003). This paper pertains to analysis of registered data yi(q).
Given that many potential sources of bias exist in yi(q) analysis (Rayner, 1985, James and Bates, 1997, Mullineaux et al., 2001, Knudson, 2009), a non-trivial challenge is to employ statistical methods that are consistent with one's null hypothesis. Consider first ‘directed’ null hypotheses: those which claim response equivalence in particular vector components i, and in particular points q or windows [q0, q1]:
Example ‘directed’ null hypothesis: Controls and Patients exhibit identical maximum knee flexion during walking between 20% and 30% stance.
To test this hypothesis only maximum knee flexion should be assessed, and only in the specified time window. Testing other time windows, joints, or joint axes in a post hoc sense would constitute bias. This is because increasing the number of statistical tests increases our risk of incorrectly rejecting the null hypothesis (see Supplementary Material – Appendix A). In other words, it is biased to expand the scope of one's null hypothesis after seeing the data. We refer to this type of bias as ‘post hoc regional focus bias’.
Next consider ‘non-directed’ null hypotheses: those which broadly claim kinematic or dynamic response equivalence:
Example ‘non-directed’ null hypothesis: Controls and Patients exhibit identical hip and knee kinematics during stance phase.
To address this hypothesis both hip and knee joint rotations should be assessed, about all three orthogonal spatial axes, and from 0% to 100% stance (i.e. the entire dataset yi(q)). It would be biased to assess only maximum hip flexion, for example, in a post hoc sense but for the opposite reason: it is biased to reduce the scope of one's null hypothesis after seeing the data.
Non-directed hypotheses expose a second potential source of bias: covariance among the I vector components. Scalar analyses ignore covariance and are therefore coordinate-system dependent (see Supplementary Material – Appendix B). This is important because a particular coordinate system — even one defined anatomically and local to a moving segment — may not reflect underlying mechanical function (Kutch and Valero-Cuevas, 2011). Joint rotations, for example, may not be independent because muscle lines of action are generally not parallel to externally defined axes (Jensen and Davy, 1975). Joint moments may also not be independent because endpoint force control, for example, requires coordinated joint moment covariance (Wang et al., 2000). Under a non-directed hypothesis this covariance must be analyzed because separate analysis of the I components is equivalent to an assumption of independence, an assumption which may not be justified (see Supplementary Material – Appendix B). We refer to this source of bias as ‘inter-component covariance bias’.
Both post hoc regional focus bias and inter-component covariance bias have been acknowledged previously (Rayner, 1985, James and Bates, 1997, Mullineaux et al., 2001, Knudson, 2009). However, to our knowledge no study has proposed a comprehensive solution.
The purpose of this paper is to show that a method called Statistical Parametric Mapping (SPM) (Friston et al., 2007) greatly mitigates both bias sources. The method begins by regarding the data yi(q) as a vector field , a multi-component vector whose values change in time or space q (Fig. 1). When regarding the data in this manner, it is possible to use random field theory (RFT) (Adler and Taylor, 2007) to calculate the probability that observed vector field changes resulted from chance vector field fluctuations.
We use SPM and RFT to conduct formalized hypothesis testing on three separate, publicly available biomechanical vector field datasets. We then contrast these results with the traditional scalar extraction approach. Based on statistical disagreement between the two methods we infer that, by definition, at least one of the methods is biased. We finally use mathematical arguments (Supplementary Material) and logical interpretations of the original data to conclude that scalar extraction constitutes a biased approach to non-directed hypothesis testing, and that SPM overcomes these biases.
Section snippets
Datasets
We reanalyzed three publicly available datasets (Table 1):
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Dataset A (Neptune et al., 1999) (http://isbweb.org/data/rrn/): stance-phase lower extremity dynamics in ten subjects performing ballistic side-shuffle and v-cut tasks (Fig. 2). Present focus was on within-subject mean three dimensional knee rotations for the eight subjects whose data were labeled unambiguously in the public dataset.
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Dataset B (Besier et al., 2009) (https://simtk.org/home/muscleforces): stance-phase knee-muscle forces
Dataset A: knee kinematics
The knee appeared to be comparatively more flexed (Fig. 2a) and somewhat more externally rotated (Fig. 2c) in the side-shuffle vs. v-cut tasks, with slightly more abduction at 0% stance (Fig. 2b). Statistical tests on the extracted scalars found significant differences between the two tasks for both maximal knee flexion (t=3.093, p=0.018) and abduction at 0% stance (t=3.948, p=0.006).
SPM vector field analysis (Fig. 5) found significant kinematic differences between the two tasks at
Discussion
The current vector field SPM and scalar extraction results all agreed qualitatively with the data, yet the two approaches yielded different results and even incompatible statistical conclusions. This, by definition, indicates that at least one of the methods is biased. For non-directed hypotheses testing we contend that scalar extraction is susceptible to two non-trivial bias sources:
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Post hoc regional focus bias — Type I or Type II error (i.e. false positives or false negatives) resulting from
Conflict of interest statement
The authors report no conflict of interest, financial or otherwise.
Glossary
Category Symbol Other Description Index: Counts I i Vector components J j Responses (i.e. experimental recordings) K k Predictor variables N Extracted scalars (e.g. maximum force) Q q Field measurement nodes (e.g. 100 points in time) Mean, Variance: Responses yi y, s2 Scalar response (withst.dev.) yi(q) y(q), s2(q) Scalar field response (with st.dev. field) y(q) y(q), W (q) Vector field response (with covariance field) Field: Test statistics t SPM{t}≡t(q) Student’s t statistic F SPM{F}≡F(q) Variance ratio (e.g. from ANOVA) T2 SPM{T
Acknowledgments
Financial support for this work was provided in part by JSPS Wakate B Grant no. 22700465.
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