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In frequentist hypothesis testing, the $p$-value is the probability of a result as extreme (or more) than the observed result, under the assumption that the null hypothesis is true.

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Why Do Different MANOVA Test Statistics Produce the Same Approximate F and P Values in R?

I am using the manova() function in R to perform multivariate analysis of variance on my dataset. I have tried using different test statistics such as Pillai's trace, Wilks' lambda, Hoteling's trace, …