Equivalence and Noninferiority Tests for Quality, Manufacturing and Test Engineers

114 Equivalence and Noninferiority Tests

Hypotheses:

H

tt

t:

er

er er

0

22

0

0

22

))

((

µ−µ

σ+σ

>γ =

δ

σ+σ

∀

H

tt

t:

er

er er

1

22

0

0

22

))

((

µ−µ

σ+σ

≤γ =

δ

σ+σ

∀

Data:

t

i

= the ith value of a sequencing variable (often time), for i = 1, n

f

e

(t

i

) = observation of one of two sequences sampled at t

i

f

r

(t

i

) = observation of the other of two sequences sampled at t

i

ft ft()

i

n

ei ri

2

1

2

∑

()

()

ψ= −

=

= the sum of squared differences between the

two sequences

ˆˆ

er

22

σ+σ=

an estimate of the variance of paired differences between the

two sequences

Critical value(s):

Reject H

0

if:

ψ

σ+σ

≤

′

λ=

δ

σ+σ













−β

Fnn

n

ˆˆ

,,

er

er

2

22

10

0

2

22

where

′

λ=

δ

σ+σ













−β

Fnn

n

,,

er

10

0

2

22

is the 100(1 – β) percent percentile of a noncentral F-distribution with n degrees

of freedom in the numerator and denominator, and numerator noncentrality

n

n

er

0

0

2

22

0

2

λ=

δ

σ+σ

=γ

.

In other words, since variability is unknown, the noncentrality parameter

could be specied in standard deviation (of the difference) units.

115Comparing Sequences of Points

Discussion:

If

()

()

() () ()

−= =ε−η =σ +σVarf tftVar DVar

ei ri

ie

r

22

then the maximum likelihood estimate of the variance of the difference

f

e

(t

i

) − f

r

(t

i

) is

n

DD

ˆˆ

1

er

i

n

i

22

1

2

∑

()

σ+σ= −

=

and

D

n

D

1

i

n

i

1

∑

=

=

.

The test statistic

ˆˆ

er

2

22

ψ

σ+σ

has a noncentral F-distribution with n degrees of freedom in both numerator

and denominator, and noncentraility:

n

n

er

0

0

2

22

0

2

λ=

δ

σ+σ

=γ

(Johnson, Kotz, and Balakrishnan, 1995).

Example:

From the example for Test 9.1,

ft ft 280.6508

i

n

ei ri

2

1

2

∑

()

() ()

ψ= −≈

=

.

The estimate of noise variance is

n

DD

ˆˆ

1

4.96

er

i

n

i

22

1

2

∑

()

σ+σ= −≈

=

.

The test statistic is

ˆˆ

280.6508

4.9582

56.6034

er

2

22

ψ

σ+σ

≈≈

.

116 Equivalence and Noninferiority Tests

With:

0.33

3.33

0.1808

er

0

0

22

γ=

δ

σ+σ

=≈

and n = 60, the “null” noncentrality is

n

n 60 .1808

1.9622

er

0

0

2

22

0

22

()

λ=

δ

σ+σ

=γ

≈≈

.

The critical value from the noncentral F-distribution with n = 60 degrees of

freedom in both numerator and denominator is approximately 1.5843. Since

56.6034 > 1.5843, the null is not rejected. The power curve for this example is

given in Figure9.4.

Condence interval formulation:

An upper 100(1 – β) percent condence limit on

sup

tt()

er

er

22

()

γ=

µ−µ

σ+σ

0.1

0

1

0.9

0.8

0.7

0.6

0.5

Power

0.4

0.3

0.2

0.1

0.2 0.3 0.4 0.5

Gamma alt.

0.6 0.7 0.8 0.9

1

FIGURE 9.4

Test 9.2, power curve for comparing two sequences, variability unknown.

117Comparing Sequences of Points

can be derived as follows. First, nd the value, λ

U

, the largest value of the

noncentrality parameter such that

Fnn

ˆˆ

|,

,1

er

U

2

22

ψ

σ+σ

λ













=−

β

where F(.) is the cumulative distribution function of the noncentral F with

n degrees of freedom in both numerator and denominator. Then compute:

n

U

U

1

2

γ=

λ













which is an upper 100(1 – β) percent condence limit for γ.

Computational considerations:

• SAS code

libname stuff 'H:\Personal Data\Equivalence & Noninferiority\

Programs & Output';

data calc;

set stuff.d20121116_test_9_2_example_data;

diff = fe - fc;

diffsq = diff**2;

run;

proc print data = calc;

run;

proc means data = calc;

var diff diffsq;

output out = onemean SUM = sdiff sdiffsq VAR = vdiff N =

ndiff;

run;

data outcalc;

set onemean;

delsig = 5;

nc0 = ndiff*delsig**2;

beta = 0.05;

psisq = sdiffsq/vdiff;

f_crit = finv(1-beta,ndiff,ndiff,nc0);

run;

118 Equivalence and Noninferiority Tests

proc print data = outcalc;/* has vars sdiff sdiffsq vdiff

ndiff delsig nc0 beta psisq f_crit */

run;

The SAS System 11:02 Friday, November 16, 2012 3

Obs time fc fe diff diffsq

1 1 10.0000 11.0749 1.07488 1.15536

2 2 5.9697 5.0135 -0.95619 0.91431

3 3 1.5610 1.6425 0.08156 0.00665

4 4 5.7207 5.8904 0.16973 0.02881

5 5 8.5700 8.4026 -0.16740 0.02802

6 6 9.8114 9.3124 -0.49895 0.24895

7 7 13.4625 13.9105 0.44803 0.20073

8 8 17.4663 16.6408 -0.82550 0.68145

9 9 19.5568 20.0469 0.49017 0.24027

10 10 18.9278 20.6939 1.76616 3.11933

11 11 22.5127 23.0164 0.50366 0.25367

12 12 22.4253 23.7516 1.32634 1.75918

13 13 14.7594 13.0602 -1.69920 2.88727

14 14 10.6493 11.0431 0.39383 0.15510

15 15 11.3332 10.6330 -0.70021 0.49029

16 16 8.2396 7.9960 -0.24355 0.05932

17 17 9.5967 8.8327 -0.76401 0.58371

18 18 10.6682 12.3575 1.68934 2.85388

19 19 12.2426 12.3369 0.09434 0.00890

20 20 17.9369 19.3441 1.40723 1.98031

21 21 12.4793 12.8372 0.35789 0.12808

22 22 11.5343 12.3369 0.80261 0.64418

23 23 16.0164 14.3267 -1.68972 2.85514

24 24 18.7508 18.9056 0.15478 0.02396

25 25 19.8442 19.3896 -0.45452 0.20659

26 26 18.9532 19.5501 0.59686 0.35624

27 27 12.0322 12.8241 0.79197 0.62721

28 28 16.4105 17.6849 1.27436 1.62401

29 29 8.5825 7.1428 -1.43971 2.07277

30 30 2.1924 2.2532 0.06076 0.00369

31 31 -1.5464 -3.1166 -1.57019 2.46550

32 32 -4.9271 -4.3207 0.60640 0.36772

33 33 -4.8922 -5.6441 -0.75185 0.56528

34 34 -7.9586 -9.0416 -1.08294 1.17276

35 35 -8.2935 -7.9490 0.34451 0.11868

36 36 -1.0380 -0.6618 0.37619 0.14152

37 37 -3.5019 -1.9691 1.53274 2.34928

38 38 -9.2971 -8.9091 0.38805 0.15058

39 39 -8.2664 -9.5914 -1.32503 1.75570

40 40 -10.0305 -9.7129 0.31761 0.10088