Zero- vs. one-dimensional, parametric vs. non-parametric, and confidence interval vs. hypothesis testing procedures in one-dimensional biomechanical trajectory analysis

Abstract

Biomechanical processes are often manifested as one-dimensional (1D) trajectories. It has been shown that 1D confidence intervals (CIs) are biased when based on 0D statistical procedures, and the nonparametric 1D bootstrap CI has emerged in the Biomechanics literature as a viable solution. The primary purpose of this paper was to clarify that, for 1D biomechanics datasets, the distinction between 0D and 1D methods is much more important than the distinction between parametric and non-parametric procedures. A secondary purpose was to demonstrate that a parametric equivalent to the 1D bootstrap exists in the form of a random field theory (RFT) correction for multiple comparisons. To emphasize these points we analyzed six datasets consisting of force and kinematic trajectories in one-sample, paired, two-sample and regression designs. Results showed, first, that the 1D bootstrap and other 1D nonparametric CIs were qualitatively identical to RFT CIs, and all were very di↵erent from 0D CIs. Second, 1D parametric and 1D non-parametric hypothesis testing results were qualitatively identical for all six datasets. Last, we highlight the limitations of 1D CIs by demonstrating that they are complex, designdependent, and thus non-generalizable. These results suggest that (i) analyses of 1D data based on 0D models of randomness are generally biased unless one has explicitly specified an a priori 0D variable, and (ii) parametric and non-parametric 1D hypothesis testing provide an unambiguous framework for analysis when one's hypothesis explicitly or implicitly pertains to whole 1D trajectories.

Toward goal-oriented robotic gait training: The effect of gait speed and stride length on lower extremity joint torques

“…Since torques during swing are smaller in amplitude than those measured during stance, analysis methods such as PCA and torque pulse approximation are not sensitive enough to capture the effect of any factor on joint torques measured during swing. Continuum analysis methods, such as the one recently developed in [19], could instead test the statistical significance of the effect of SL on the hip joint torque at early swing, as appears from visual inspection of (Fig. 3, top row central column).…”

The probability of false positives in zero-dimensional analyses of one-dimensional kinematic, force and EMG trajectories

“…The main result of this study was that smooth, random 1D trajectories generally produce false positives in 0D analyses with a probability much higher than ↵. Even for the best case -maximum smoothness (FWHM=67.0) and one scalar trajectory -false positive rates were nearly three times greater than ↵ (p=0.145, (Lenho↵ et al, 1999;Pataky et al, 2015;Robinson et al, 2015) but to our knowledge have not been previously quantified.…”

“…One could alternatively analyze the data using 1D methods (Lenho↵ et al, 1999;Pataky et al, 2015) (Fig.1, right panels). Analogous to the 0D procedure, the 1D residuals ( Fig.1d) embody the variance about mean trajectories, and the null hypothesis is the null di↵erence trajectory:…”

Pooling sexes when assessing ground reaction forces during walking: Statistical Parametric Mapping versus traditional approach

“…The discrepancy is caused by the randomness model from which p values and critical thresholds are computed. SPM uses a 1D model of randomness and the traditional approach uses a 0D model, and it has recently been shown that a 0D model of randomness is inappropriate for making probabilistic conclusions regarding 1D data (Pataky et al, 2015). Equivalently, if one's hypothesis explicitly or implicitly pertains to the entire threecomponent 1D GRF waveform, then we would argue that one is obliged to analyze the entire three-component waveform using SPM or another method which employs a multi-component 1D model of randomness.…”

“…Just like SPM, the traditional approach uses α to determine a critical test statistic threshold (t*) and if the observed t value exceeds t* then the null hypothesis is rejected. The only difference is that SPM uses a model of smooth 1D waveform randomness, and the traditional approach uses a model of 0D scalar randomness (Pataky et al, 2015).…”