Equivalence and Noninferiority Tests for Quality, Manufacturing and Test Engineers

93Reliability

Rt

et

h() Pr

T|

c

Ht

c

()

c

{}

==≥

−

= reliability function, comparator system

Rt eth() Pr T|

e

Ht

e

()

e

{}

==≥

−

= reliability function, evaluation system

T = time at which reliability of the evaluation system is to be compared to

that of the comparator system

δ = maximum allowable reduction in reliability of evaluation system com-

pared to the comparator system at time T

Hypotheses:

H

0

: R

e

(T) < (1 − δ) R

c

(T)

H

1

: R

e

(T) ≥ (1 − δ) R

c

(T)

Note that in this “nonparametric” case the hypotheses about R(t) will be

tested directly using estimates of R(t).

Data:

t

c1

, t

c2

, …, t

cn

= n sample order statistics for failure times, comparator system

t

e1

, t

e2

, …, t

em

= m sample order statistics for failure times, evaluation system

r(t

c1

), r(t

c2

), …, r(t

cn

) = sample reliabilities, comparator system

r(t

e1

), r(t

e2

), …, r(t

em

) = sample reliabilities, evaluation system

And the Kaplan-Meier estimators:

Rt

n

n

ni

ni

ˆ

()

1

11

1

ck

i

k

2

∏

()

=

−

−− −

−−

=

Rt

m

m

mi

mi

ˆ

()

1

11

1

ek

i

k

2

∏

()

=

−

−− −

−−

=

The standard errors of reliability at time T are:

SE RT RT

nini

ˆ

()

ˆ

()

1

1

cc

i

k

1

∑

()

()

()

≈

−−+

=

94 Equivalence and Noninferiority Tests

SE RT RT

mimi

ˆ

()

ˆ

()

1

1

ec

i

k

1

∑

()

()

()

≈

−−+

=

for k s.t. t

k

= T (If the life table method is used to calculate reliability estimates,

use Greenwood’s (1926) approximation formula for the standard error.)

Critical value(s):

Reject H

0

if:

RT RT tSERTSERT

ˆ

() (1 )

ˆ

() (

ˆ

())(

ˆ

()

)0

ec

ec

22

−−δ+

+≥

β

where t

β

= 100(1 − β) percentile of a (central) t distribution with m + n – 4

degrees of freedom.

Discussion:

The rationale for this test is identical to that of Test 2.3, the one-sided alter-

nate paradigm for comparing two independent means. Note that this test is

referred to here as “nonparametric” because no particular parametric forms

for the hazard functions or associated reliability functions are assumed.

Example:

Suppose at T = 0.5,

RT

ˆ

() 0.90

c

=

RT

ˆ

() 0.87

e

=

m = n = 100

δ = 0.025

SE RT

ˆ

( 0.5) 0.03

c

()

=≈

SE RT

ˆ

( 0.5)

0.03

e

()

=≈

.

Then

RT RT tSERTSERT

ˆ

() (1 )

ˆ

() (

ˆ

())(

ˆ

())

0.87 (1 0.025)0.90 1.653 0.03 0.03 0.063

0

ec

ec

22

22

−−δ+ +

=−−+ +≈≥

β

.

95Reliability

Therefore, the null hypothesis of inferiority, H

0

, is rejected in favor of nonin-

feriority, H

1

.

Condence interval formulation:

RT RT tSERTSERT

ˆ

() (1 )

ˆ

() (

ˆ

())(

ˆ

()

)

ec

ec

22

−−δ+ +

β

is an approximate 100(1 − β) percent upper condence limit on R

e

(T)

− (1 − δ)R

c

(T).

Computational considerations:

This test is conceptually identical to Test 2.3.

Test 7.3 Accelerated Life Test with Type I Censoring

Parameters:

λ

u

= unaccelerated failure rate

λ

a

= accelerated failure rate

f = acceleration parameter, that is, λ

a

= f λ

u

n = number of units tested

T

c

= censoring time, accelerated conditions

T

u

= f T

c

= time equivalent to T

c

censoring time under unaccelerated conditions

λ

u,0

= maximum tolerable failure rate, unaccelerated conditions

λ

a,0

= maximum tolerable failure rate, accelerated conditions

λ

a,0

= fλ

u,0

re e

fT T

0

uc

ac

,0 ,0

==

−λ −λ

= reliability at time T

c

(accelerated)

Hypotheses:

H

0

: λ

u

> λ

u,0

H

1

: λ

u

≤ λ

u,0

or equivalently

H

0

: λ

a

> f λ

u,0

H

1

: λ

a

≤ f λ

u,0

Note that from the parameter list:

r

T

ln

a

c

,0

0

λ=

−

.

96 Equivalence and Noninferiority Tests

Data:

K = number of units failing at or before time T

c

x

i

= time of ith failure, for the K units that failed at or before time T

c

T

x

K

nKT

K

()

i

i

K

c

1

∑

=+

−

=

.

If no units failed before time T

c

, then

TT

c

=

.

T

ˆ

1

a

λ=

Critical value(s):

The same critical values used in Test 4.1 would apply; namely, if time to

event, T, has an exponential distribution, then

Tn

T

ˆ

i

i

n

1

∑

γ= =

=

has a gamma distribution with parameters λ

a,0

and n. Often n is called the

“shape” parameter, and λ

a,0

is called the “scale” parameter. Thus, if

Gn

n

xe dx(

ˆ

|, )

()

a

a

n

nx

,0

,0

1

0

ˆ

0

∫

γλ=

λ

Γ

≥β

−

γ

−λ

then reject H

0

. That is, G(.) is the cumulative distribution function for a

gamma-distributed random variable with parameters λ

a,0

and n. Similarly,

nd the value, call it γ

c

, such that

Gn

n

xe dx(|,)

()

ca

a

n

n

x

,0

,0

1

0

c

a,0

∫

γλ=

λ

Γ

=β

−

γ

−λ

and reject H

0

if

ˆ

c

γ≥γ

.

One could also rely on the central limit theorem, and reject the null

hypothesis if:

Tt

T

n

1

a

1

,0

+≥

λ

−β

.