# Equivalence test for non-normal distribution: how to determine equivalence bounds?

I am trying to conduct a Mann-Whitney equivalence test (as described in Wellek's book Testing Statistical Hypotheses of Equivalence and Noninferiority) with the following code in R:

mawi(alpha,m,n,eps1_,eps2_,x,y)
[alpha-    significance level;
m- size of Sample 1;
n- size of Sample 2;
eps1_- absolute value of the left-hand limit of the hypothetical equivalence range for
π_+ - 1/2;
eps2_- right-hand limit of the hypothetical equivalence range for π_+ - 1/2;
x- row vector with the m observations making up Sample1 as components;
y- row vector with the n observations making up Sample2 as components]


Where eps1 and eps2 represent the equivalence bounds I want to test for. The equivalence bounds need to be specified as the absolute value of the left-hand and right-hand limit of the hypothetical equivalence range for $$\pi_{+} - 1/2$$. However, I don't know how to determine this value.

In case I want to test for an equivalence bound corresponding to Cohen's $$d=0.57$$ (for two samples of $$N = 25$$; $$\alpha = 0.5$$) how do I compute eps1 and eps2?

• Welcome to CV, DOMINIKA SLUSNA. :) – Alexis Mar 24 at 20:11
• It is bizarre to think of what "equivalence" means when performing a Mann-Whitney test. If the means are different because the distributions are skewed, then what is considered to be the same about the two distributions if you conclude they're "equivalent"? – AdamO Mar 24 at 21:04
• @AdamO Bizarre or not, these tests appear in the biostats lit. :) If we recall that the positivist rank sum test can be expressed as $\text{H}_{0}^{+}\text{: }P(X\ge Y) = \frac{1}{2}$, then the general (symmetric) negativist rank sum null is $\text{H}_{0}^{-}\text{: }|P(X\ge Y)| \ge \frac{1}{2}+\varepsilon$, with the specific one-side nulls $\text{H}_{01}^{-}\text{: }P(X\ge Y) \ge \frac{1}{2}+\varepsilon$ and $\text{H}_{02}^{-}\text{: }P(X\ge Y) \leq \frac{1}{2}-\varepsilon$. – Alexis Mar 24 at 22:15

You do not compute eps1 and eps2 because these are researcher choices.
Equivalence testing (both the lower powered Two One-Sided Tests, or TOST framework, and the uniformly most powerful, or UMP test framework professed by Wellek) rely on a notion of equivalence threshold, or the minimum effect size that you the researcher find to be relevant (possibly informed by regulatory guidelines, as is the case with the FDA some some applications of bioequivalence in the United States). Put another way, the researcher effectively states a priori "I do not care about effect sizes smaller the equivalence threshold." I do not know what your data are, but, for example, when studying whether a policy changes national incidence rates of, say, HIV a researcher might say a priori "I do not care about effect sizes smaller than $$\pm$$1 new case of HIV per 100,000 people per year."
Some equivalence thresholds are expressed symmetrically on an absolute or relative scale. Some are expressed asymmetrically. The values of esp1 and esp2 are non-negative values that express what you find to be the appropriate boundary between relevant and equivalent, and correspond to your choices for the values of $$\varepsilon_1^{\prime}$$ and $$\varepsilon_2^{\prime}$$ in expression 6.10 on page 126 of Wellek (Second ed.).