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Let's say I have a dataset with two groups (male and female), a target variable ($y$) and multiple features ($X_1$, $X_2$ and $X_3$).

gender y X1 X2 X3
m 100 39 150 12
m 120 44 180 16
f 100 22 140 14
f 150 40 169 20

I can test the hypothesis that the population means of $X_1$ are equal between men and women with a simple $t$ test.

$H_0$: $\mu_{X_1}^{m} = \mu_{X_2}^{f}$

$H_1$: $\mu_{X_1}^{m} \neq \mu_{X_2}^{f}$

I can carry out the same test for $X_2$ and $X_3$.

If I do this for every feature in my dataset, I will test each hypothesis individually. However, how can I test if all the population means are different between men and women simultaneously?

$H_0$: \begin{bmatrix} \mu_{X_1}^m - \mu_{X_1}^f = 0\\ \mu_{X_2}^m - \mu_{X_2}^f = 0\\ \mu_{X_3}^m - \mu_{X_3}^f = 0 \end{bmatrix}

$H_1$: Any other case (at least one pair of means does not hold)

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    $\begingroup$ It sounds like you are looking for the multivariate T test. $\endgroup$
    – whuber
    Oct 24, 2021 at 15:10
  • $\begingroup$ In your first null hypothesis you compare male mean of $X_1$ to female mean of $X_2$? Is that what you want? $\endgroup$ Oct 25, 2021 at 15:06
  • $\begingroup$ Yes, but also the male mean of $X_2$ vs the female mean of $X_2$. So basically, using a multivariate distribution to simultaneously test those two hypotheses. $\endgroup$
    – Arturo Sbr
    Oct 25, 2021 at 16:05
  • $\begingroup$ It sounds like Hotelling's $T^2$ test is exactly what you want, though I don't see how the target variable $y$ comes up. $\endgroup$
    – Dave
    Oct 25, 2021 at 16:16
  • $\begingroup$ Oh, it doesn't. I just added it to make the example more relatable. $\endgroup$
    – Arturo Sbr
    Oct 25, 2021 at 16:17

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