As for ANOVA, the partial eta-squared η2 can be used as a measure of effect size for MANOVA.
This statistic is calculated by
partial η2 =
which is equivalent to the following, where b and s are as in Property 4 and 5 of Manova Basic Concepts.
Wilks Lambda: 1 – Λ1/b
Pillai-Bartlett Trace: V/s