 Equivalence test

An equivalence test asks whether an effect lies strictly inside a prespecified negligible-effect range, such as $(-\delta ,\delta )$ for $\delta >0$. Its null hypothesis is that the effect lies outside that range, including its boundary; its alternative hypothesis is $-\delta <\theta <\delta$. This differs from a test with a zero-effect null hypothesis: failing to reject zero does not establish that the effect is sufficiently small.

One construction performs two level-$\alpha$ one-sided tests, rejecting $\theta \le -\delta$ and rejecting $\theta \ge \delta$, and declares equivalence only if both reject. If the equivalence null hypothesis is true, at least one component null is true. The probability of rejecting both cannot exceed the probability of rejecting that true component, which is at most $\alpha$. Thus the combined procedure controls Type I error at level $\alpha$. In a continuous symmetric t-based location model with a common standard error, this is equivalent to putting the central $100(1-2\alpha )\%$ confidence interval wholly inside $(-\delta ,\delta )$. At $\alpha =0.05$ that is a 90% interval, not the 95% interval used for the usual two-sided zero-effect test. The margin must be chosen for substantive reasons rather than selected afterward to accommodate an observed interval.