# Why Hotelling's $T^2$ is taught in the book but it is not used anywhere so far?

I have seen various multivariate statistics book covering Hotelling $$T^2$$ statistics. However, when reading medical journals and papers, it does not show up anywhere.

Q1:Is this test of any practical value? All I can tell is that it detects multivariate mean of multi-groups equal to something. I could also use ANOVA contrast matrices instead. Hotelling will require asymptotic result. So I would guess it has lower power compared to ANOVA contrasts as ANOVA may not be sensitive to deviation from normality in small sample case.

Q2:Why is it still covered in multivariate statistics book, even if it is not seen a lot in journals and papers?

• For your Q2, I might turn the question around to ask why practitioners are still using univariate tests when multivariate tests are well-established.
– Dave
Aug 25, 2022 at 18:05
• @Dave Simple univariate test is easy and multiple adjustment is easy. I guess Hotelling's $T^2$'s rejection region might be more stringent compare to those of adjusted univariate testing. Multivariate testing for point should really use contrasting from my perspective. Aug 25, 2022 at 18:07
• The advantage of the multivariate test is that it takes advantage of the correlations between the marginal distributions. If you’re unclear about how this happens, you might consider posting a question. I have in mind a plot I can make that will illustrate this. (I had the same question in my multivariate class, why we didn’t just test the margins. The example I would post mimics what my professor said that I found so helpful.)
– Dave
Aug 25, 2022 at 18:13
• @Dave I would be interested in that as I am not aware how it take advantage of correlation of marginalized variables. I will edit my question. Aug 25, 2022 at 18:18
• One reason is that $T^2$ can detect a difference that is not of interest. For example on might test whether a drug lowers either systolic or diastolic blood pressure. If the drug changed only the pulse pressure (SBP - DBP) $T^2$ may detect that but it may not be of clinical interest. Aug 25, 2022 at 18:43