An equivalence study is designed to test whether one treatment is nearly equal, or 'equivalent' to another.
Equivalence studies are common in, but not limited to, drug trials. They amount to an inversion of the typical Neyman-Pearson hypothesis testing approach, in that the researcher is trying to provide evidence against a null hypothesis that two groups differ by at least some amount:
H$^{-}_0\text{: }|\theta|\ge\Delta$
H$^{-}_\text{A}\text{: }-\Delta< \theta < \Delta$
Such "negativist" null hypotheses can also easily extend to asymmetric equivalence intervals:
H$^{-}_0\text{: }\theta\leq\Delta_2$ OR $\theta \ge \Delta_{1}$
H$^{-}_\text{A}\text{: }\Delta_{2}< \theta < \Delta_{1}$
Because it is impossible to prove that an estimated parameter is equal to any point value, it can only be shown that the parameter appears to fall within an acceptably narrow range that covers perfect equality. A frequent approch to testing for equivalence is by using two one-sided tests.
