High-dimensional multivariate analysis of variance via geometric median and bootstrapping.

The geometric median, which is applicable to high-dimensional data, can be viewed as a generalization of the univariate median used in 1-dimensional data. It can be used as a robust estimator for identifying the location of multi-dimensional data and has a wide range of applications in real-world scenarios. This paper explores the problem of high-dimensional multivariate analysis of variance (MANOVA) using the geometric median. A maximum-type statistic that relies on the differences between the geometric medians among various groups is introduced. The distribution of the new test statistic is derived under the null hypothesis using Gaussian approximations, and its consistency under the alternative hypothesis is established. To approximate the distribution of the new statistic in high dimensions, a wild bootstrap algorithm is proposed and theoretically justified. Through simulation studies conducted across a variety of dimensions, sample sizes, and data-generating models, we demonstrate the finite-sample performance of our geometric median-based MANOVA method. Additionally, we implement the proposed approach to analyze a breast cancer gene expression dataset.

Postural behaviour in people with multiple sclerosis: A complexity paradox.

BACKGROUND: Balance deficits are a major concern for people with multiple sclerosis (pwMS). Measuring complexity of motor behaviour can offer an insight into MS-related changes in adaptability of the balance control system when dealing with increasingly complex tasks. QUESTION: Does postural behaviour complexity differ between pwMS at early stages of the disease and healthy controls (HC)? Does postural behaviour complexity change across increasingly complex tasks? METHODS: Forty-eight pwMS and 24 HC performed four increasingly complex postural tasks with eyes open (EO), eyes closed (EC), on firm (FS) and compliant surface (CS). Lumbar and sternum sensors recorded 3D acceleration, from which complexity index (CI) was calculated using multiscale sample entropy (MSE) in the frontal and sagittal planes. RESULTS: We found that only the complexity index in both planes during the eyes closed on compliant surface (EC-CS) task was significantly lower in pwMS compared to HC. We also found that complexity in pwMS was significantly lower during EC-CS compared to the other three tasks when using both lumbar and sternum sensors. SIGNIFICANCE: Increasing the complexity of postural tasks reduces the complexity of postural behaviour in pwMS. This paradox may reflect reduced adaptability of the sensorimotor integration processes at early stages of MS. CI can provide a different perspective on balance deficits and could potentially be a more sensitive biomarker of MS progression and an early indicator of balance deficit.

Optimal sensor location and direction to accurately classify people with early-stage multiple sclerosis using gait stability.

BACKGROUND: The local divergence exponent (LDE) has been used to assess gait stability in people with multiple sclerosis (pwMS). Although previous studies have consistently found that stability is lower in pwMS, inconsistent methodologies have been used to assess patients with a broad range of disability levels. QUESTIONS: What sensor location and movement direction(s) are better able to classify pwMS at early stages of the disease? METHODS: 49 pwMS with EDSS ≤ 2.5 and 24 healthy controls walked overground for 5 min while 3D acceleration data was obtained from sensors placed at the sternum (STR) and lumbar (LUM) areas. Unidirectional (vertical [VT], mediolateral [ML], and anteroposterior [AP]) and 3-dimensional (3D) LDEs were calculated using STR and LUM data over 150 strides. ROC analyses were performed to assess classification models using single and combined LDEs, with and without velocity per lap (VELLAP) as a covariate. RESULTS: Four models performed equally well by using combinations of VELLAP, LUM3D, LUMVT, LUMML, LUMAP, STRML, and STRAP (AUC = 0.879). The best model using single sensor LDEs included VELLAP, STR3D, STRML, and STRAP (AUC = 0.878), whereas using VELLAP + STRVT (AUC = 0.869) or VELLAP + STR3D (AUC=0.858) performed best using a single LDE. SIGNIFICANCE: The LDE offers an alternative to currently insensitive tests of gait impairment in pwMS at early stages, when deterioration is not clinically evident. For clinical purposes, the implementation of this measure can be simplified using a single sensor at the sternum and a single LDE measure, but speed should be considered. Longitudinal studies to determine the predictive power and responsiveness of the LDE to MS progression are still needed.