Motivated by the analysis of the dependence of knee movement patterns during functional tasks on subject-specific covariates, we introduce a distribution-free procedure for testing a ...functional-on-scalar linear model with fixed effects. The procedure does not only test the global hypothesis on the entire domain but also selects the intervals where statistically significant effects are detected. We prove that the proposed tests are provided with an asymptotic control of the intervalwise error rate, that is, the probability of falsely rejecting any interval of true null hypotheses. The procedure is applied to one-leg hop data from a study on anterior cruciate ligament injury. We compare knee kinematics of three groups of individuals (two injured groups with different treatments and one group of healthy controls), taking individual-specific covariates into account.
Abstract Background Despite interventions, anterior cruciate ligament ruptures can cause long-term deficits. To assist in identifying and treating deficiencies, 3D-motion analysis is used for ...objectivizing data. Conventional statistics are commonly employed to analyze kinematics, reducing continuous data series to discrete variables. Conversely, functional data analysis considers the entire data series. Methods Here, we employ functional data analysis to examine and compare the entire time-domain of knee-kinematic curves from one-leg hops between and within three groups. All subjects (n = 95) were part of a long-term follow-up study involving anterior cruciate ligament ruptures treated ~ 20 years ago conservatively with physiotherapy only or with reconstructive surgery and physiotherapy, and matched knee-healthy controls. Findings Between-group differences (injured leg, treated groups; non-dominant leg, controls) were identified during the take-off and landing phases, and in the sagittal (flexion/extension) rather than coronal (abduction/adduction) and transverse (internal/external) planes. Overall, surgical and control groups demonstrated comparable knee-kinematic curves. However, compared to controls, the physiotherapy-only group exhibited less flexion during the take-off (0–55% of the normalized phase) and landing (44–73%) phase. Between-leg differences were absent in controls and the surgically treated group, but observed during the flight (4–22%, injured leg > flexion) and the landing (57–85%, injured leg < internal rotation) phases in the physiotherapy-only group. Interpretation Functional data analysis identified specific functional knee-joint deviations from controls persisting 20 years post anterior cruciate ligament rupture, especially when treated conservatively. This approach is suggested as a means for comprehensively analyzing complex movements, adding to previous analyses.
Simultaneous inference for functional data in sports biomechanics Pataky, Todd Colin; Abramowicz, Konrad; Liebl, Dominik ...
Advances in statistical analysis : AStA : a journal of the German Statistical Society,
03/2023, Letnik:
107, Številka:
1-2
Journal Article
Recenzirano
The recent sports science literature conveys a growing interest in robust statistical methods to analyze smooth, regularly-sampled functional data. This paper focuses on the inferential problem of ...identifying the parts of a functional domain where two population means differ. We considered four approaches recently used in sports science: interval-wise testing (IWT), statistical parametric mapping (SPM), statistical nonparametric mapping (SnPM) and the Benjamini-Hochberg (BH) procedure for false discovery control. We applied these procedures to both six representative sports science datasets, and also to systematically varied simulated datasets which replicated ten signal- and/or noise-relevant parameters that were identified in the experimental datasets. We observed generally higher IWT and BH sensitivity for five of the six experimental datasets. BH was the most sensitive procedure in simulation, but also had relatively high false positive rates (generally > 0.1) which increased sharply (> 0.3) in certain extreme simulation scenarios including highly rough data. SPM and SnPM were more sensitive than IWT in simulation except for (1) high roughness, (2) high nonstationarity, and (3) highly nonuniform smoothness. These results suggest that the optimum procedure is both signal and noise-dependent. We conclude that: (1) BH is most sensitive but also susceptible to high false positive rates, (2) IWT, SPM and SnPM appear to have relatively inconsequential differences in terms of domain identification sensitivity, except in cases of extreme signal/noise characteristics, where IWT appears to be superior at identifying a greater portion of the true signal.
Pini and Vantini (2017) introduced the interval-wise testing procedure which performs local inference for functional data defined on an interval domain, where the output is an adjusted p-value ...function that controls for type I errors. We extend this idea to a general setting where domain is a Riemannian manifolds. This requires new methodology such as how to define adjustment sets on product manifolds and how to approximate the test statistic when the domain has non-zero curvature. We propose to use permutation tests for inference and apply the procedure in three settings: a simulation on a "chameleon-shaped" manifold and two applications related to climate change where the manifolds are a complex subset of \(S^2\) and \(S^2 \times S^1\), respectively. We note the tradeoff between type I and type II errors: increasing the adjustment set reduces the type I error but also results in smaller areas of significance. However, some areas still remain significant even at maximal adjustment.
Within the framework of Gaussian graphical models, a prior distribution for the underlying graph is introduced to induce a block structure in the adjacency matrix of the graph and learning ...relationships between fixed groups of variables. A novel sampling strategy named Double Reversible Jumps Markov chain Monte Carlo is developed for block structural learning, under the conjugate G-Wishart prior. The algorithm proposes moves that add or remove not just a single link but an entire group of edges. The method is then applied to smooth functional data. The classical smoothing procedure is improved by placing a graphical model on the basis expansion coefficients, providing an estimate of their conditional independence structure. Since the elements of a B-Spline basis have compact support, the independence structure is reflected on well-defined portions of the domain. A known partition of the functional domain is exploited to investigate relationships among the substances within the compound.
Since Benjamini and Hochberg introduced false discovery rate (FDR) in their seminal paper, this has become a very popular approach to the multiple comparisons problem. An increasingly popular topic ...within functional data analysis is local inference, i.e., the continuous statistical testing of a null hypothesis along the domain. The principal issue in this topic is the infinite amount of tested hypotheses, which can be seen as an extreme case of the multiple comparisons problem. In this paper we define and discuss the notion of false discovery rate in a very general functional data setting. Moreover, a continuous version of the Benjamini-Hochberg procedure is introduced along with a definition of adjusted p-value function. Some general conditions are stated, under which the functional Benjamini-Hochberg procedure provides control of the functional FDR. Two different simulation studies are presented; the first study has a one-dimensional domain and a comparison with another state of the art method, and the second study has a planar two-dimensional domain. Finally, the proposed method is applied to satellite measurements of Earth temperature. In detail, we aim at identifying the regions of the planet where temperature has significantly increased in the last decades. After adjustment, large areas are still significant.
Motivated by the analysis of spectrometric data, we introduce a Gaussian graphical model for learning the dependence structure among frequency bands of the infrared absorbance spectrum. The spectra ...are modeled as continuous functional data through a B-spline basis expansion and a Gaussian graphical model is assumed as a prior specification for the smoothing coefficients to induce sparsity in their precision matrix. Bayesian inference is carried out to simultaneously smooth the curves and to estimate the conditional independence structure between portions of the functional domain. The proposed model is applied to the analysis of infrared absorbance spectra of strawberry purees.
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