Fig 1.
Participants sat with the trunk secured to the backrest to limit torso motion. Head motion was recorded using eight markers attached to a rigid helmet. The local reference frame was centered at the midpoint between both tragus points. The X-axis (lateral flexion) pointed anteriorly through the line connecting the infraorbital landmarks, the Z-axis (flexion extension) connected the tragus points toward the right side, and the Y-axis (axial rotation) completed the right-handed triad. Additional details on the determination of anatomical axes are described in Venegas et al. [27].
Table 1.
Mean and standard deviation of the variables NDI, age (years), weight (kg), height (cm), neck length (cm), head mass (kg), and neck mass (kg) by gender.
Fig 2.
Angular position of (a) flexion-extension; (b) lateral flexion; (c) axial rotation. Angular velocity of: (d) flexion-extension; (e) lateral flexion; (f) axial rotation. The grey lines are the curves for each subject, with the functional mean (black) plus and minus one functional standard deviation (dotted black).
Table 2.
Mean and standard deviation of the range of angular motion (RoM) and range of angular velocity (RoV) for flexion-extension, lateral flexion and axial rotation.
Table 3.
Multiple linear correlation coefficient value r, F-statistic, and p-value of the scalar models for the three movements: flexion-extension, lateral flexion, and axial rotation.
In all models, the predictors used were range of motion (RoM), range of velocity (RoV), and their interactions.
Fig 3.
Selection of the optimal number of principal components.
The plot shows the multiple linear correlation coefficient (x-axis) and the AIC and BIC (y-axis) according to different numbers of principal components, varying from four (first point) to 20 (last point). Black and blue represent the AIC and BIC values, respectively. The red point corresponds to the optimal number of principal components chosen (11 for both models). (a) The plot on the left corresponds to the functional regression model, whose predictor is the lateral flexion velocity. (b) The one on the right is for the principal components model concatenating the flexion-extension and lateral flexion velocity curves.
Table 4.
Optimal number of functional principal components, proportion of variance explained, Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), multiple correlation coefficient (r), and F statistic with associated p-value for each model.
Table 5.
AIC, BIC, multiple linear correlation coefficient r, F statistic, and p-value of the linear regression models.
Different scalar predictors were added to the base model (only functional) to analyze whether their introduction improved the initial functional model.
Fig 4.
Coefficient and product functions of the functional regression model.
This plot shows the coefficient and product functions of the chosen bivariate flexion-extension (top) and lateral flexion (bottom) principal component models for 11 principal components. The flexion-extension velocity curves are plotted for a participant with a low NDI of 4 (a) versus one with a high NDI of 15 (b); analogously for lateral flexion (c) and (d). In each plot, the velocity of the subject (dashed black), coefficient function (blue), and product of the velocity by the coefficient function (black) are shown. The negative and positive areas under the product function are shaded in orange and green, respectively. It is important to note that although the coefficient function is continuous from the start of the flexion extension velocity to the end of the lateral flexion velocity, as it is for the bPCA model, it has been separated for each velocity for a better interpretation of the results. The coefficient function is multiplied by 25 for better visualization in the same plot. The vertical scales have been adjusted to highlight the relative differences between patients.