= Equivalence test
{title2=$H_0:\theta\le-\delta\text{ or }\theta\ge\delta$}
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= Equivalence testing
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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.
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