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Statistical power analysis is fundamental for valid inference and sample size determination, but its application to one-dimensional data introduces additional complexity. In contrast to zero-dimensional data, statistical power for one-dimensional data does not have a unique definition. The statistical power analysis for two-sample hypothesis tests when the data is one-dimensional is further complicated due to the data characteristics, such as smoothness and differences between group mean trajectories, referred to as the effect trajectory. This study investigates how noise smoothness and effect trajectory jointly influence statistical power in the analysis of one-dimensional data, using two different definitions of statistical power: omnibus power and sensitivity. Analysis of six biomechanical datasets confirmed that diverse characteristics of the effect trajectory exist in practice and influence omnibus power trends. Using simulation experiments with different characteristics of the effect trajectories under varying smoothness levels, we assessed the statistical power for both statistical parametric mapping and its nonparametric version. The results show that when non-zero effects cover large portions of the domain, increasing smoothness decreases omnibus power, whereas for effects on smaller portions, smoother data enhances omnibus power. In contrast, sensitivity consistently increases with smoothness, independent of the effect's characteristics. These findings highlight that the relationship between smoothness and statistical power is not universal, but instead, depends on the definition of power and the underlying effect structure.

Background  Recent medical studies have shown an increasing interest in inferential methods for analysing functional data, while statistical power analysis for sample size planning for such data is less explored. As a result, researchers often rely on classical scalar approaches to estimate sample size, despite working with functional data. This can substantially underestimate the required sample sizes. Moreover, there are no guidelines to assist researchers in planning, conducting, and reporting sample size estimation for studies analysing functional data.

Methods  Two functional data sets from medical sciences are used in a simulation study to explore a functional approach for sample size planning. These data represent two distinct patterns in mean function differences. Six wellknown local inferential methods are evaluated for two-population comparisons of functional data. The evaluation focuses on the sample sizes required to achieve the target statistical power, under different data characteristics and assuming equal group sizes and stationary noise in the data generation process. We have also developed an interactive web-based application that helps researchers in performing a priori power analysis by allowing them to explore how changes in data characteristics affect statistical power, and consequently, the required sample size.

Results  Our comparison revealed distinct patterns in the estimated sample sizes for different data characteristics and inferential methods. Even when based on the same baseline data, the required sample sizes to achieve a target statistical power of 0.80 differed noticeably, ranging from very small to moderately large sample sizes, depending on the mean function pattern, underlying noise characteristics, and inferential approach.

Conclusions  Overall, our results emphasise the importance of appropriate sample size planning and inferential method selection for valid inference in medical studies that include functional data analysis. Based on these findings, we provide guidance for researchers to follow, from study design conception through to reporting.

Long-term balance impairments are prevalent after anterior cruciate ligament (ACL) injury and are possibly linked to an overreliance on visual information and related cortical processing. We therefore aimed to explore characteristics of functional brain networks related to postural control with and without vision following ACL reconstruction (ACLR). Twenty-seven individuals after ACLR and 24 non-injured controls performed single-leg balance tasks under eyes-open/eyes-closed conditions. Graph-theoretical measures of functional network segregation (clustering coefficient, CC) and integration (path length, PL) were derived from mobile electroencephalography. Sway characteristics were calculated based on centre of pressure (CoP; area and velocity) and the mean distance between CoP and centre of mass (CoM). Knee antero-posterior kinematics were also explored. Group effects were analysed using permutation-based ANCOVA. During eyes-open only, the ACLR group exhibited greater cortical network segregation (higher CC; p = 0.025) in the alpha-1 band (8–10 Hz). While sway characteristics were similar between groups, the ACLR leg demonstrated greater knee flexion compared to their contralateral leg (p = 0.036). Individuals post-ACLR showed more efficient functional brain connectivity during eyes-open, combined with kinematic adaptations in their injured leg. These findings suggest post-ACLR neural adaptations of postural control mechanisms, particularly when visual information is available.

Jaw-head movement coordination develops during adolescence. However, functional adjustments during this period remain poorly understood. This study aimed to characterize jaw and head movement adjustments in early adolescents and compare this to adults. Three-dimensional optical cameras captured jaw and head movements during maximum jaw opening-closing and chewing. Twenty (8 females, 12 males) adolescents (mean 13.5 yr, standard deviation [SD] 8 months) and 20 (9 females, 11 males) adults (mean 28.2 yr, SD 80 months) participated. Outcomes included jaw and head movement magnitudes, movement cycle time, time to first peak value, and initial phase. Functional data analysis and Wilcoxon rank-sum tests were employed. Adolescents showed larger head magnitude in jaw opening-closing and smaller jaw magnitude than did adults during chewing in the first movement cycle. Adolescents exhibited longer time to peak and time of first movement cycle during jaw opening-closing. During chewing, adolescents showed a longer initial phase, time to peak for consecutive cycles, and movement cycle time. For both age groups, the first cycle differed from consecutive cycles in jaw and head movement magnitudes and cycle times. Compared to adults, adolescents displayed pronounced spatiotemporal initial jaw-head movement adjustments during jaw function, particularly in the first movement cycle. Jaw-head coordination refines from early adolescence into adulthood.

This study provides guidance for researchers who work with functional (curve) data and aim to perform a prior sample size estimation. Focusing on the two-population framework, we test mean differences between two populations — a scenario common in fields such as human movement science. Through simulations, we examine how standard deviation and smoothness influence sample size requirements to achieve 0.80 statistical power, using four methods with control the family-wise error rate: interval-wise testing (IWT), threshold-wise testing (TWT), F-max, and Extreme Rank Length (ERL) global envelope. For instance, increasing the standard deviation from 5 to 10 can raise the sample size from approximately 10 to over 30. Adjusting the smoothness parameter from 5 to 45 can lead to varied outcomes: the required sample size may increase to over 50, remain near 10, or even decrease, depending on the method and data characteristics. Three key findings are: (1) higher noise levels require larger sample sizes, (2) smoother data necessitate more samples when mean differences span larger domains, and (3) TWT and IWT are more efficient for large-domain differences, while ERL and F-max performbetter for differences on narrower domains.

Musculoskeletal trauma often leads to lasting psychological impacts stemming from concerns of future injuries. Often referred to as kinesiophobia or re-injury anxiety, such concerns have been shown to hinder return to physical activity and are believed to increase the risk for secondary injuries. Screening for re-injury anxiety is currently restricted to subjective questionnaires, which are prone to self-report bias. We introduce a novel approach to objectively identify electrocortical activity associated with the threat of destabilising perturbations. We aimed to explore its feasibility among non-injured persons, with potential future implementation for screening of re-injury anxiety. Twenty-three participants stood blindfolded on a translational balance perturbation platform. Consecutive auditory stimuli were provided as low (neutral stimulus [CS^–]) or high (conditioned stimulus [CS⁺]) tones. For the main experimental protocol (Protocol I), half of the high tones were followed by a perturbation in one of eight unpredictable directions. A separate validation protocol (Protocol II) requiring voluntary squatting without perturbations was performed with 12 participants. Event-related potentials (ERP) were computed from electroencephalography recordings and significant time-domain components were detected using an interval-wise testing procedure. High-amplitude early contingent negative variation (CNV) waves were significantly greater for CS⁺ compared with CS– trials in all channels for Protocol I (> 521-800ms), most prominently over frontal and central midline locations (P ≤ 0.001). For Protocol II, shorter frontal ERP components were observed (541-609ms). Our test paradigm revealed electrocortical activation possibly associated with movement-related fear. Exploring the discriminative validity of the paradigm among individuals with and without self-reported re-injury anxiety is warranted.

Nonparametric bagging clustering methods are studied and compared to identify latent structures from a sequence of dependent categorical data observed along a one-dimensional (discrete) time domain. The frequency of the observed categories is assumed to be generated by a (slowly varying) latent signal, according to latent state-specific probability distributions. The bagging clustering methods use random tessellations (partitions) of the time domain and clustering of the category frequencies of the observed data in the tessellation cells to recover the latent signal, within a bagging framework. New and existing ways of generating the tessellations and clustering are discussed and combined into different bagging clustering methods. Edge tessellations and adaptive tessellations are the new proposed ways of forming partitions. Composite methods are also introduced, that are using (automated) decision rules based on entropy measures to choose among the proposed bagging clustering methods. The performance of all the methods is compared in a simulation study. From the simulation study it can be concluded that local and global entropy measures are powerful tools in improving the recovery of the latent signal, both via the adaptive tessellation strategies (local entropy) and in designing composite methods (global entropy). The composite methods are robust and overall improve performance, in particular the composite method using adaptive (edge) tessellations.

Background: Psychophysiological consequences often persist following musculoskeletal trauma and can result in vastly decreased quality of life. Re-injury anxiety is particularly common among individuals following anterior cruciate ligament (ACL) injury. Existing assessments of re-injury anxiety are, however, restricted to subjective suboptimal questionnaires, which may result in under-reporting and thus poorer injury management. We propose a novel approach to objectively quantify arousal response to movement-related anxiety. A new experimental paradigm was implemented to induce and record a conditioned electrophysiological response to a sudden perturbation, experienced to be potentially injurious.

Objective: To explore the feasibility of detecting anxiety-associated electrocortical response and to evaluate its discriminative ability between asymptomatic individuals and those who had experienced an ACL injury.

Methods: Physically-active asymptomatic persons and individuals post-ACL reconstruction stood blindfolded on a perturbation platform capable of generating high-acceleration translations (1.5 m/s2). Auditory stimuli were repeatedly presented in four-second intervals, as either low- or high-frequency tones. Half of the high-frequency tones were followed 1.5 seconds later by a destabilizing perturbation in one of eight randomized directions. The two tone conditions were thus termed ‘Neutral’ and ‘Anxiety’, as the high-frequency tone was intended to invoke an arousal response in anticipation of a potential perturbation. Event-related potentials (ERP) were computed for nine electrodes by averaging 100 Neutral and 100 Anxiety trials. Significant ERP components were identified using functional data analysis. Paired difference-waves’ amplitudes (Neutral - Anxiety) were compared between groups.

Results: ERP correlates of anxiety were detected for both groups in frontal and central midline locations, with an observable contingent negative variation (CNV) from 500 ms post-stimulus in Anxiety compared with Neutral trials. This ERP component is reflective of a threat-induced arousal response, associated with attention and expectancy of an anxiety-relevant event. Preliminary data indicate no group differences in CNV amplitudes.

Conclusions: Objective evaluation of an arousal response to movement-related anxiety was found to be feasible, resulting in a threat-induced CNV. Further investigation will elucidate the discriminative power of such an approach to differentiate between individuals with high and low re-injury anxiety, as well as potential associations with existing patient-reported outcome measures.

Here we present and compare functional and spatiotemporal (Sp.T.) kriging approaches to predict spatial functional random processes, which can also be viewed as Sp.T. random processes. Comparisons are focused on Sp.T. kriging versus ordinary kriging for functional data (OKFD), since more flexible functional kriging approaches like pointwise functional kriging and functional kriging total model coincide with OKFD in several situations. Prediction performance is evaluated via functional cross-validation on simulated data as well as on a Canadian weather data set. The two kriging approaches perform in many cases rather equal for stationary Sp.T. processes. For nonstationary Sp.T. processes, OKFD performs better than Sp.T. kriging. The computational time for OKFD is considerably lower compared to those for the Sp.T. kriging methods.

Three-dimensional human motion analysis provides in-depth understanding in order to optimize sports performance or rehabilitation following disease or injury. Recent developments of statistical methods for functional data allow for novel ways to analyze often complex biomechanical data. Even so, for such methods as well as for traditional well-established statistical methods, the interpretations of the results may be influenced by analysis choices made prior to the analysis. We evaluated the consequences of three such choices when comparing one-leg vertical hop (OLVH) performance in individuals who had ruptured their anterior cruciate ligament (ACL), to that of asymptomatic controls, and also athletes. Kinematic data were analyzed using a statistical approach for functional data, targeting entire curve data. This was done not only for one joint at a time but also for multiple lower limb joints and movement planes simultaneously using a multi-aspect methodology, testing for group differences while also accounting for covariates. We present the results of when an individual representative curve out of three available was either: (1) a mean curve (Mean), (2) a curve from the highest hop (Max), or (3) a curve describing the variability (Var), as a representation of performance stability. We also evaluated choice of sample leg comparison; e.g., ACL-injured leg compared to either the dominant or non-dominant leg of asymptomatic groups. Finally, we explored potential outcome effects of different combinations of included joints. There were slightly more pronounced group differences when using Mean compared to Max, while the specifics of the observed differences depended on the outcome variable. For Var there were less significant group differences. Generally, there were more disparities throughout the hop movement when comparing the injured leg to the dominant leg of controls, resulting in e.g., group differences for trunk and ankle kinematics, for both Mean and Max. When the injured leg was instead compared to the non-dominant leg of controls, there were trunk, hip and knee joint differences. For a more stringent comparison, we suggest considering to compare the injured leg to the non-dominant leg. Finally, the multiple-joint analyses were coherent with the single-joint analyses. The direct effects of analysis choices can be explored interactively by the reader in the Supplementary Material. To summarize, the choices definitively have an impact on the interpretation of a hop test results commonly used in rehabilitation following knee injuries. We therefore strongly recommend well-documented methodological analysis choices with regards to comparisons and representative values of the measures of interests.