 21Means
Example:
Suppose
=X 10.0
S = 2.0
n = 25
μ
L
= 8.0
μ
H
= 9.5
β = 0.05
Then:
Xt
S
n
10.0 1.708 *
2.0
25
10.68 8.0
1
+=
+≈
>
−β
Xt
S
n
10.0 1.708 *
2.0
25
9.32 9.5
1
−=
−≈
<
−β
Therefore, we reject H
0
in favor of H
1
: 8.0 ≤ μ ≤ 9.5, even though
=>X 10.0 9.5
.
Under TOST, the power calculations for the two-sided test for a single mean
are like those for the one-sided test. However, instead of simply calculating
Xt
S
n
n
nX
S
tn
Pr |,Pr
|,
1
LaL
L
aL1 1
()
+≥µµ
=
−µ
≥− µ=µ<µ
=−β
−β −β
we must also calculate
Xt
S
n
n
nX
S
tn
Pr |,Pr
|,
1.
HaH
H
aH
1 1
()
−≤µµ
=
−µ
≤+ µ=µ>µ
=−β
−β −β
As in the one-sided case, the quantity:
X
Sn
L
−µ
has a noncentral t-distribution with noncentrality parameter 22 Equivalence and Noninferiority Tests
n
L
aL
()
δ=
µ−µ
σ
.
Similarly, the random variable
X
Sn
H
−µ
has a noncentral t-distribution with noncentrality parameter
n
H
aH
()
δ=
µ−µ
σ
.
Note that as either
nn
or
LaLH
aH
δ
=
µ−µ
σ
δ
=
µ−µ
σ
goes to zero, the likelihood of rejecting the null hypothesis goes increasingly
to 1 − β.
Condence interval formulation:
The interval
Xt
S
n
Xt
S
n
,
11
−+
−β −β
is a 100(1 − 2β) percent condence interval for μ. Using the example data
yields the 90 percent condence interval (9.32, 10.68).
Computational considerations:
The code for this test is essentially the same as the code for Test 2.1, except
that both a lower and upper test must be performed. In light of the TOST
philosophy, both lower and upper test statistics should use a 100(1 − β) per-
centile from the appropriate t-distribution.
Test 2.5 A Special Case of Single Mean
(Two-Sided)Regression Slope
A special case of the single mean, two-sided test (Test 2.4) is for simple
linearregression slope parameters. A common hypothesis and its alternate are:
H
0
: β
1
< 1 Δ OR β
1
> 1 + Δ 23Means
H
1
: 1 − Δβ
1
≤ 1 + Δ
where β
1
is the slope parameter and Δ is a number between 0 and 1.
The standard error of the ordinary least squares (OLS) slope estimate is
SE
XX
ˆ
i
i
n
2
1
()
=
σ
=
where
ˆ
σ
is the root mean square error from the ANOVA for the regression.
The null hypothesis, H
0
, is rejected if
b
1
+ t
1−β
SE 1−Δ and b
1
t
1−β
SE ≤ 1 + Δ
where b
1
is the ordinary least squares (OLS) slope estimate, and t
1 − β
=
100*(1−β) percentile of a central t-distribution with n − 2 degrees of freedom.
(b
1
t
1−β
SE, b
1
+ t
1−β
SE)
is a 100(1 − 2β) percent condence interval for the slope parameter β
1
(do not
confuse the slope β
1
with the Type II error risk, β).
Computational considerations:
The computations for Test 2.5 are very similar to those for Test 2.4. The
principal differences are a special case of the hypotheses, and the computa-
tion of the standard error of the estimate.
SAS code
libname stuff 'H:\Personal Data\Equivalence & Noninferiority\
Programs & Output';
data calc;
set stuff.d20121105_test_2_5_example_data;
run;
proc means data = calc;
var X Y del0 beta;
output out = outmeans MEAN = xbar ybar delbar betabar N = nx
ny CSS = ssx ssy;
run;
proc print data = calc;
run;
proc print data = outmeans;
run;
24 Equivalence and Noninferiority Tests
proc reg data = calc outest = outreg1;
model Y = X;
run;
data outreg2;
set outreg1;
drop _TYPE_;
run;
data bigcalc;
set outmeans;
set outreg2;
seslope = _RMSE_/sqrt(ssx);
tval = tinv(1-betabar,nx-2);
lowlim = X + tval*seslope;
upplim = X - tval*seslope;
run;
proc print data = bigcalc;/*dataset bigcalc: var X is slope
estimate */
var nx X lowlim upplim;
run;
The output parameter “CSS” in proc reg is the corrected sums of squares.
So, “ssx” is the corrected sums of squares for the X variable, as X is the rst
column in the data le.
The SAS System 09:33 Monday, November 5, 2012 10
The MEANS Procedure
Variable Label N Mean Std Dev Minimum Maximum
ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ
X X 30 10.2108583 1.2373869 7.6748992 13.4843440
Y Y 30 15.4861972 1.3359807 12.6635617 18.7004035
del0 del0 30 0.0750000 0 0.0750000 0.0750000
beta beta 30 0.0500000 0 0.0500000 0.0500000
ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ
The SAS System 09:33 Monday, November 5, 2012 11
Obs X Y del0 beta
1 10.05 15.12 0.075 0.05
2 8.97 14.55 0.075 0.05
3 9.78 15.62 0.075 0.05
4 7.67 12.66 0.075 0.05

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