The Mahalanobis distance squared D2 can be used as a measure of effect size, where
D2 = T2/n
For Example 1 of One-sample Hotelling’s T-square Test
D2 = T2/n = 52.6724/25 = 2.107
Rough guidelines for D2 are .25 is a small effect, .50 is a medium effect and 1.00 is a large effect.
As for ANOVA, the partial eta-square is another measure of effect size, where
For Example 1 of One-sample Hotelling’s T-square Test
Generally, .01 is considered to be a small effect, .06 is a medium effect and .14 is a large effect.
The calculation of the power of the one-sample Hotelling’s T-square test is based on using the Mahalanobis measure of effect size, namely
where μ0 is the mean vector based on the null hypothesis and μ1 is the mean vector based on the alternative hypothesis. We use λ = D2 as the noncentrality parameter. The power of such a test is equal to 1 – β where
where k = the number of dependent variables and n = the sample size.
See Hotelling T-square Power for more information about the statistical power of one-sample Hotelling’s T-square test and the minimum sample size.
References
Schumacker, R. E. (2015) Hotelling’s T-square. Using R with multivariate statistics. SAGE.
https://us.sagepub.com/sites/default/files/upm-assets/70364_book_item_70364.pdf
Penn State University (2013) Hotelling’s T-square. STAT 505: Applied multivariate statistical analysis (course notes)
https://online.stat.psu.edu/stat505/lesson/7/7.1/7.1.3
Sapp, M., Obiakor, F. E., Gregas, A. J., Scholze, S. (2007) Mahalanobis distance: A multivariate measure of effect in hypnosis research
https://psycnet.apa.org/record/2008-04630-004