6 Equivalence and Noninferiority Tests
Test 2.2 Comparison of Two Means—Two Independent
Samples, Fixed Δ Paradigm (One-Sided)
Parameters:
μ
1
= population mean, “group” 1
μ
2
= population mean, “group” 2
σ
1
= population standard deviation, “group” 1
σ
2
= population standard deviation, “group” 2
Δ
0
= maximum allowable difference between μ
1
and μ
2
Hypotheses:
H
0
: μ
1
< μ
2
− Δ
0
H
1
: μ
1
≥ μ
2
− Δ
0
Data:
=X sample mean
1
, “group” 1
FIGURE 2.2
Test 2.1, JMP screen.
7Means
S
1
= sample standard deviation, “group” 1
n
1
= sample size, “group” 1
=X sample me
an
2
, “group” 2
S
2
= sample standard deviation, “group” 2
n
2
= sample size, “group” 2
Critical value(s):
Reject H
0
if:
XXtSE
12
10
−+ ≥−∆
−β
where t
1 − β
= 100*(1 − β) percentile of a central t-distribution with n
1
+ n
2
− 2
degrees of freedom and SE is the standard error for the difference of two
means:
SE
S
n
S
n
1
2
1
2
2
2
=+
.
Discussion:
If μ
1
= μ
2
– Δ
0
exactly, then we would expect that
XX
120
>−∆
about as fre-
quently as
XX
120
<−∆
. Since μ
1
= μ
2
– Δ
0
would be minimally acceptable,
we would want to avoid failing to conclude that μ
1
≥ μ
2
– Δ
0
just because
XX
120
<−∆
. That is, we would want to conclude that μ
1
< μ
2
– Δ
0
only when
X
1
was sufciently less than
X
2
. In other words, we are willing to believe that
μ
1
≥ μ
2
– Δ
0
(i.e., the alternate hypothesis) as long as
XX
tSE
1201
≥−∆−
−β
.
As in the case of the single mean, the test statistic under various alternate
hypotheses has a noncentral t-distribution with n
1
+ n
2
– 2 degrees of free-
dom and noncentrality:
nn
a 0
1
2
12
2
2
δ=
∆−∆
σ+σ
.
Welch (1947) provided an alternative calculation for the degrees of free-
dom of the two-sample t-test, when it is assumed that the variances for the
two populations or systems are not equal.
8 Equivalence and Noninferiority Tests
Let:
()
=
+
−
W
Sn
Sn Sn
n 1
1
1
2
1
1
2
12
2
2
2
1
and
()
=
+
−
W
Sn
Sn Sn
n 1
2
2
2
2
1
2
12
2
2
2
2
.
Then the degrees of freedom for Welch’s t-test are
df
WW
..
1
12
=
+
For simplicity, the conventional n
1
+ n
2
– 2 degrees of freedom will be used for
the examples presented. In actual practice, Welch’s formula is recommended.
Power calculations are made as a function of the noncentrality parameter,
and particularly as a function of Δ
a
.
Example:
Suppose we hypothesize that the mean of “group” 1 is no more than
Δ
0
=5.0 units less than the mean of “group” 2. The data are
=X 96.0
1
S
1
= 2.40
n
1
= 12
=X 101.5
2
S
2
= 2.20
n
2
= 14
We choose 1 – β = 0.95, so t
1 − β
= 1.711 (12 + 14 – 2 = 24 degrees of freedom).
We compute
=+=+≈SE
S
n
S
n
2.40
12
2.20
14
0.909
1
2
1
2
2
2
22
.
The critical value is
XXtSE 96 101.5 1.711* 0.909 3.94
5.0
121
−+ =−
+≈−≥
−
−β
.
9Means
Therefore, we reject the null hypothesis, H
0
in favor of the alternate, H
1
.
If μ
1
= μ
2
– 5.0, then
t
XX
S
n
S
n
5.0
12
1
2
1
2
2
2
=
−+
+
has a central t-distribution with n
1
+ n
2
– 2 degrees of freedom.
Thus, the probability of rejecting the null hypothesis when μ
1
= μ
2
– 5.0 is
XXtSEPr 5.0 1
121
{}
−+ ≥−
=−β
−β
.
Under some specic alternate hypothesis, such as μ
1
= μ
2
– Δ
a
, where
Δ
a
>5.0, then
t
XX
S
n
S
n
5.0
12
1
2
1
2
2
2
=
−+
+
has a noncentral t-distribution with n
1
+ n
2
– 2 degrees of freedom and non-
centrality parameter
nn
5.0
a
1
2
12
2
2
δ=
∆−
σ+σ
.
To calculate a power curve, we will make the simplifying assumptions that
σ=σ=σ==
+
=nn
nn
and
2
13
12 12
12
.
Thus, the noncentrality parameter simplies to
n 5.0
2
()
δ=
∆−
σ
.
Expressing
5.0
a
∆−
σ
as a proportion (i.e., the difference in σ units) is usually easier than obtaininga
reasonable estimate of σ. Thus, the power curve will be expressed as a function of
5.0
a
γ=
∆−
σ
10 Equivalence and Noninferiority Tests
for γ = 0, . . . , 2.0 (σ units). Figure2.3 shows the power curve for this test.
Condence interval formulation:
The expression
XX
tS
E
121
−+
−β
is a one-sided 100(1 − β) percent upper condence limit for μ
1
– μ
2
. From the
example, the upper 95 percent condence limit for μ
1
– μ
2
is
XXtSE 96 101.5 1.711* 0.909
3.94
121
−+ =−
+≈
−
−β
.
Computational considerations:
For this test, one could use a condence limit computed by various pro-
grams for the difference between two means. The upper condence limit
should be compared to the lower “acceptable” bound on the difference, and a
one-sided 100(1 − β) percent [or a two-sided 100(1 − 2β) percent] limit should
be computed, in concert with the two one-sided test (TOST) philosophy.
1
0.9
0.8
0.7
0.6
Power
0.5
0.4
0.3
0.2
0.1
0
–1.5 –1.3 –1.1 –0.9 –0.7
Delsig
–0.5 –0.3 –0.1 0 0.1 0.2
FIGURE 2.3
Test 2.2, power curve for equivalence of two means.

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