Equivalence and Noninferiority Tests for Quality, Manufacturing and Test Engineers

65

5

Capability Indices

Test 5.1 C

pk

Parameters:

L = lower specication limit

U = upper specication limit

μ = population mean

σ = population standard deviation

n = sample size

=

µ−

σ

C

L

3

pL

=

−µ

σ

C

U

3

pU

CC

Cmin,

pk pL pU

)

(

=

K

0

= minimum desirable value for C

pk

Hypotheses:

H

0

: C

pk

< K

0

H

1

: C

pk

≥ K

0

Data:

=X sample mean

S = sample standard deviation

=

−

C

XL

S

ˆ

3

pL

66 Equivalence and Noninferiority Tests

=

−

C

UX

S

ˆ

3

pU

CC

C

ˆ

min

ˆ

,

ˆ

pk pL pU

)

(

=

Critical value(s):

Both

C

ˆ

pL

and

C

ˆ

pU

have distributions that are proportional to a noncentral

T. Specically,

nC Tn nC3

ˆ

~1,3

pL pL

)

(

′

−δ=

.

Thus, to test the hypotheses:

H

0

: C

pL

< K

0

H

1

: C

pL

≥ K

0

reject the null hypothesis if

C

Tn nK

n

ˆ

1, 3

3

pL

0

)

(

≥

′

−δ=

β

where

Tn nK1, 3

0

)

(

′

−δ=

β

is the lower 100βth percentile of a noncentral t-distribution with n – 1 degrees

of freedom and noncentrality parameter

δ= nK3

0

. The same logic is used

for

C

ˆ

pU

; namely, reject the null hypothesis if

C

Tn nK

n

ˆ

1, 3

3

pU

0

)

(

≥

′

−δ=

β

.

Note that since

CC

C

ˆ

min

ˆ

,

ˆ

pk pL pU

)

(

= , the decision to accept or reject is made

based on only one of

C

ˆ

pL

or

C

ˆ

pU

. The rejection probability should actually

be based on the distribution of the minimum of two noncentral T random

variables. However, for practical purposes, the fact that the test statistic is

actually a function of two random variables will be ignored.

An approximation to the standard error of

C

ˆ

pk

is described by Bissell (1990):

SE

n

C

n

1

9

ˆ

2( 1)

pk

2

=+

−

⋅

67Capability Indices

As an alternative approach, a normal approximation could be used with

this expression for the standard error. The critical value would be similar to

the case of the single mean, namely:

Reject H

0

if:

+≥

−β

CzSE K

ˆ

pk

10

where z

1 − β

= 100*(1 − β) percentile of a standard normal distribution.

Discussion:

As in the case of all the hypothesis tests described in this work, we assume

that the “null” value stated for C

pk

is acceptable, so that we desire a high

probability of rejecting the null hypothesis if the true C

pk

is equal to the null

value. Thus, the critical value is the lower 100βth percentile of the appropri-

ate sampling distribution.

It should be noted that the estimates

C

ˆ

pL

and

C

ˆ

pU

are biased, in that their

expected values exceed the value of the parameter. The bias is a multiplier

described by:

=

−

Γ

−













Γ

−













bias

n

n

n

1

2

2

2

1

2

where Γ(.) is the gamma function. The bias is relatively small for values of

n≥100. At n = 100, the bias is approximately 1.008, or 0.8 percent too high.

To compensate for bias, multiply the critical value by the bias formula.

Alternatively, divide the usual estimator by the bias, and then use the “unad-

justed” critical value.

The power to reject the null is given by:

C

Tn nK

n

nKPr

ˆ

1, 3

3

|3

pL

aa

0

)

(

≥

′

−δ=

δ=





















β

and

Tn nK1, 3

0

)

(

′

−δ=

β

is the lower 100βth percentile of a noncentral t-distribution with n – 1 degrees

of freedom and noncentrality parameter

δ= nK3

0

. K

a

is the alternate value

of the population C

pL

.

68 Equivalence and Noninferiority Tests

Note that the test for C

pk

is identical in theory to the criteria used in ANSI

ASQ Z1.9 standard sampling plans (1993), or ISO 3951 (1989). That is, the

statistics:

=

−

C

XL

S

ˆ

3

pL

and

=

−

C

UX

S

ˆ

3

pU

are replaced in the standard sampling plans by

=

−

K

XL

S

L

and

=

−

K

UX

S

U

which only differ from the C

pk

statistics by the multiplicative constant 3.

Thus, power characteristics for the hypothesis tests concerning C

pk

are iden-

tical to those of the ANSI/ASQ Z1.9 or ISO 3951 plans.

Example:

Suppose that K

0

= 1.33. With a sample of n = 100, suppose that

=C

ˆ

1.30

pL

and

=C

ˆ

2.00

pU

. Then the decision to reject the null hypothesis will be based

entirely on

C

ˆ

pL

. For β = 0.05, the noncentrality is

δ= =nK3 39.9

0

and the critical value for

≈C

ˆ

1.18

pL

. Since

=C

ˆ

1.30

pL

, the null hypothesis is

rejected. The bias-corrected critical value is approximately 1.19. Figure5.1

shows a power curve for this test.

Condence interval formulation:

For the exact lower 100(1 − β) percent condence limit on C

pL

or C

pU

nd

Clow

ˆ

()

pL

such that

Tn nC lowCPr 1, 3

ˆ

(

ˆ

1

pL pL

{}

)

(

′

−δ

=≤=−

β

69Capability Indices

or

Cl

ow

ˆ

()

pU

such that

Tn nC lowCPr 1, 3

ˆ

()

ˆ

1

pU pU

{}

)

(

′

−δ

=≤=−β

.

See Kushler and Hurley (1992).

Using the normal approximation, the lower condence limit would be:

−+

−

−β

Cz

n

C

n

ˆ

1

9

ˆ

2( 1)

pL

pL

1

2

or

−+

−

−β

Cz

n

C

n

ˆ

1

9

ˆ

2( 1)

pU

pU

1

2

Computational considerations:

• SAS code

1

0.9

0.8

0.7

0.6

0.5

Power

0.4

0.3

0.2

0.1

0

0.9 0.9511.05 1.1

Cpk

1.15 1.2 1.25 1.3

FIGURE 5.1

Test 5.1, power curve for noninferiority test on C

pk

, K

0

= 1.33, n = 100.