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Also, if the experiment was intended primarily to add to the body of published literature, it should be
acceptable to increase α to 0.05 or 0.10.
Having specified α, the investigator needs to specify β, or 1 − β. Cohen (1969) suggests that in the
context of medical treatments, a type I error is roughly four times as serious as a type II error. This
implies that one should use approximately β = 4α so that the power of the test is 1 − β = 1 − 4α. Thus,
when α = 0.05, set 1 − β = 0.80, or perhaps less.

Sample Size for Assessing the Equivalence of Two Means

The previous sections dealt with selecting a sample size that is large enough to detect a difference between
two processes. In some cases we wish to establish that two processes are not different, or at least are close
enough to be considered equivalent. Showing a difference and showing equivalence are not the same
problem.
One statistical definition of equivalence is the classical null hypothesis H 0 : η 1 − η 2 = 0 versus the
alternate hypothesis H 1 : η 1 − η 2 ≠ 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 ∆ 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 (H 1 ) requires
the difference between two means to lie with an equivalence interval [θ L , θ U ] so that the rejection of
the null hypothesis, H 0 at a nominal level of significance (α), 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:

–
H 0 : η 1 η 2 ≤– θ L or η 1 η 2 ≥ θ U
versus

H 1 : θ L < η 1 η 2 < θ U
–
This is decomposed into two one-sided hypotheses:

–
H 01 : η 1 η 2 ≤– θ L and H 02 : η 1 η 2 ≥ θ U
–
H 11 : η 1 η 2 >– θ L H 12 : η 1 η 2 < θ U
where each test is conducted at a nominal level of significance, α. If H 01 and H 02 are both rejected, we
conclude that θ L < η 1 η 2 < θ U and declare that the two treatments are equivalent.
–
2
We can specify the equivalence interval such that θ = θ U = −θ L . When the common variance σ is
known, the rule is to reject H 0 in favor of H 1 if:
– θ + z α σ y 1 −y 2 ≤ y 1 − y 2 ≤ θ – z α σ y 1 −y 2

The approximate sample size for the case where n 1 = n 2 = n is:
2σ z α +( z β ) 2
2
n = ------------------------------- + 1
( θ ∆) 2
–
θ defines (a priori) the practical equivalence limits, or how close the true treatment means are required
to be before they are declared equivalent. ∆ is the true difference between the two treatment means under
which the comparison is made.
© 2002 By CRC Press LLC