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Equivalence is basic to mathematics, e.g. https://en.wikipedia.org/wiki/Equivalence_relation.

"Equivalence tests" are common in statistics, e.g. https://en.wikipedia.org/wiki/Equivalence_test.

Is the name "statistical equivalence" used intentionally to reflect a particular set-theoretic equivalence (and where is this best explained)?

Any views/references welcome. Thanks!

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  • $\begingroup$ It is a recognized pedagogical problem in statistics that many basic statistical terms like "Normal," "conditional," "independent," and so on differ so much from their colloquial meanings or their definitions in other fields (like mathematics) that it is erroneous to apply those "alien" concepts to help understand their statistical cognates. "Equivalence," although not so basic, suffers from the same problem. $\endgroup$
    – whuber
    Commented Aug 10, 2022 at 14:51

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I do not think statistical equivalence would count as an equivalence relation, as it seems to lacks transitivity.

Let’s say that equivalence is the means being within $2$ of each other.

$$ \bar x_A =0.0\\ \bar x_B =1.5\\ \bar x_C =3.0 $$

$A$ is equivalent to $B$, and $B$ is equivalent to $C$, but $A$ is not equivalent to $C$.

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    $\begingroup$ Transitivity was my first thought. My second was that in set theory you would expect two things to remain equivalent every time you looked at them, but with hypothesis testing you could (due to different samples) not always get the same result $\endgroup$
    – Henry
    Commented Aug 10, 2022 at 13:29

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