Statistics for Environmental Engineers

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One statistical definition of equivalence is the classical null hypothesis H0: n — Ц2 = 0 versus the alternate hypothesis H1: n -n2 ^ 0. If we use this problem formulation to determine the sample size for a two-sided test of no difference, as shown in the previous section, the answer is likely to be a sample size that is impracticably large when A is very small.


Stein and Dogansky (1999) present an alternate formulation of this classical problem that is often used in bioequivalence studies. Here the hypothesis is formed to demonstrate a difference rather than equivalence. This is sometimes called the interval testing approach. The interval hypothesis (H1) requires the difference between two means to lie with an equivalence interval [0L, 6У so that the rejection of the null hypothesis, H0 at a nominal level of significance (a), is a declaration of equivalence. The interval determines how close we require the two means to be to declare them equivalent as a practical matter:


H0: П1 — П2 ^6l    or П1 — П2 — в,2


versus


H1: 6l <П1 — П2 <Vv


This is decomposed into two one-sided hypotheses:


H01: n1 — n2 — @L    and    H02 : n1 — n2 —


H 11: П1 ^2 > ®L    H12: П1 ^2 < @U


where each test is conducted at a nominal level of significance, a. If H01 and H02 are both rejected, we conclude that 6L <p1 — p2 < в и and declare that the two treatments are equivalent.

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