Equivalence and Noninferiority Tests for Quality, Manufacturing and Test Engineers

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=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=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 − Δ ≤ β

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