Distributional properties and variance-stabilizing transformations for measures of uncontrolled manifold effects

In the uncontrolled manifold (UCM) approach, variability in elemental variables (such as joint angles or muscle modes) is decomposed into goal-equivalent and non-goal-equivalent variability components (GEV, NGEV) with regard to a specified task variable. A UCM effect is present, when GEV exceeds NGEV, and different indices have been proposed to quantify the strength of UCM effects. We propose variance-stabilizing transformations for each of these measures. Our results indicate that the variability components should be log-transformed prior to statistical analysis, to reduce non-normality and inhomogeneity of variances. Moreover, it is formally shown that the UCM indices are identical after appropriate variance-stabilizing transformations. The theoretical analysis is illustrated by empirical and simulated data from a study on manual pointing.