A randomization method for the calculation of covariation in multiple nonlinear relations: illustrated with the example of goal-directed movements

A randomization method is developed for the calculation of covariation between multiple variables that are linked nonlinearly to a dependent variable. Covariation is a phenomenon often invoked in the study of movement coordination to capture the fact that in coordinated movement the outcome shows greater than expected consistency from the variability in the component processes. However, in most cases, the problem is that more than two variables covary in a nonlinear fashion, which makes quantification with the bivariate linear covariation and correlation coefficient inapplicable. This paper presents a generalization of the calculation of linear bivariate covariance using a variant of a randomization method that is based on the comparison between the empirically measured variability in the outcome and a covariation-free variability. The latter can be estimated by permuting data sets. A generalized correlation coefficient is derived, and it is shown how errors of estimation can be quantified. The permutation method can also quantify partial multiple nonlinear covariation. The calculations are illustrated in a numerical example of an arm-reaching task. However, the method is applicable to all cases where the internal organization of a nonlinear system of multiple variables needs to be quantified. The relation and applicability of the permutation method compared to other methods using regression and principal component analysis are discussed and illustrated with a numerical example.