  The multivariate coefficient of variation (MCV) is an attractive and
easy-to-interpret effect size for the dispersion in multivariate data.
Recently, the first inference methods for the MCV were proposed by Ditzhaus and
Smaga (2022) for general factorial designs covering k-sample settings but also
complex higher-way layouts. However, two questions are still pending: (1) The
theory on inference methods for MCV is primarily derived for one special MCV
variant while there are several reasonable proposals. (2) When rejecting a
global null hypothesis in factorial designs, a more in-depth analysis is
typically of high interest to find the specific contrasts of MCV leading to the
aforementioned rejection. In this paper, we tackle both by, first, extending
the aforementioned nonparametric permutation procedure to the other MCV
variants and, second, by proposing a max-type test for post hoc analysis. To
improve the small sample performance of the latter, we suggest a novel
studentized bootstrap strategy and prove its asymptotic validity. The actual
performance of all proposed tests and post hoc procedures are compared in an
extensive simulation study and illustrated by a real data analysis.
