The lab I work in uses equivalence. When we get a significant result from a two one-sided test, we conclude that true difference between mu1 and mu2 is likely inside the specified margin, which is taken to be the largest difference that is of no clinical significance. I'm interested in using Bayesian estimation instead of a Frequentist approach for this, but I'm having trouble parsing out what his objections with equivalence testing are.
Specifically, I have no idea what the bold part is trying to communicate:
ROPE method cannot be used to accept null value in NHST. Because an NHST confidence interval (CI) has some properties analogous to the Bayesian posterior HDI, it may be tempting to try to adopt the use of the ROPE in NHST. Thus, we might want to accept a null hypothesis in NHST if a 95% CI falls completely inside the ROPE. This approach goes by the name of equivalence testing in NHST (e.g., Rogers, Howard, & Vessey, 1993; Westlake, 1976, 1981). Unfortunately the approach fails because the meaning of the CI is not the same as the HDI. In a Bayesian approach, the 95% HDI actually includes the 95% of parameter values that are most credible. Therefore when the 95% HDI falls within the ROPE, we can conclude that 95% of the credible parameter values are practically equivalent to the null value. But a 95% CI from NHST says nothing directly about the credibility of parameter values. Crucially, even if a 95% CI falls within the ROPE, a change of intention will change the CI and the CI may no longer fall within the ROPE. For example, if the two groups being compared are intended to be compared to other groups, then the 95% CI is much wider and may no longer fall inside the ROPE.
Passage is from this article: http://www.indiana.edu/~kruschke/BEST/BEST.pdf
The way I see it, with equivalence testing I'm never actually accepting the null hypothesis of mu1 - mu2 = 0, I'm rejecting two separate null hypotheses show that the mean difference is less than some practically significant amount. I'm not sure how the BEST approach allows you to conclude that mu1 - mu2 = 0 any more than the Frequentist approach.