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I very much like the idea of the ROPE (region of practical equivalence) (e.g. see here), where you compute the posterior probability that a given parameter is in a previous range that counts as "small" in subject-area terms. (This is not a substitute-for-a-hypothesis-rejection-procedure question; in the particular context I'm working on, the authors want specifically to test for equivalence.)

As I have seen it, the ROPE is a univariate procedure, i.e. testing the marginal posterior distribution of some particular parameter. Does any discussion of a multivariate version exist? I can think of some crude approaches (e.g. test for the simultaneous satisfaction of a bunch of univariate ROPEs — probability of being inside a specified 'rectangular' region — or some kind of sum-of-squares/Mahalanobis variant) but am hoping not to reinvent any wheels.

(I do see that there are some papers on frequentist multivariate equivalence testing, e.g. Hoffelder et al 2015, perhaps these could be adapted ...)

Hoffelder, Thomas, Rüdiger Gössl, and Stefan Wellek. 2015. “Multivariate Equivalence Tests for Use in Pharmaceutical Development.” Journal of Biopharmaceutical Statistics 25 (3): 417–37. https://doi.org/10.1080/10543406.2014.920344.

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If you use a rectangular multivariate ROPE, i.e., have a series of assertions in the form of inequalities about an array of effects on different outcomes, and you have the dependency between outcomes modeled (e.g. using a multivariate outcome distribution or a copula connecting the various univariate outcomes), the computation of the posterior probability of being int he ROPE rectangle is straightforward using multivariate posterior samples.

If you don’t have a multivariate outcome model but model the outcomes as independent, the multivariate posterior draws will lead to bounds on the true dependent posterior samples. Sometimes the bounds may be tight enough that you don’t need to have a full multivariate model. For example, if Pr(A & B) is very high assuming A and B are independent, this probability will only be higher if there is a positive dependence between A and B.

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