83
7
Reliability
Test 7.1 Comparison of Reliability Using Parametric Models
Parameters:
T = time to failure
h
c
(t) = α
0
+ 2α
1
t = hazard rate function, comparator system
h
e
(t) = β
0
+ 2β
1
t = hazard rate function, evaluation system
(these hazard rate functions are a slight generalization of an exponential
hazard rate, where α
1
= β
1
= 0)
Ht hd tt() ()
=cumulativehazardfunction, comparator system
cc
t
01
2
0
∫
=ττ=α+α
Ht hd tt() ()
=cumulativehazardfunction, evaluation system
ee
t
01
2
0
∫
=ττ=β+β
Rt eth() Pr T| reliability function,comparatorsystem
c
Ht
c
()
c
{}
==≥=
−
Rt eth() Pr T| reliabilityfunction, evaluation system
e
Ht
e
()
e
{}
==≥=
−
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
compared 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)
84 Equivalence and Noninferiority Tests
Alternatively, the hypotheses can be stated as:
H
0
: −H
e
(T) + H
c
(T) < ln(1 − δ)
H
1
: −H
e
(T) + H
c
(T) ≥ ln(1 − δ).
Based on the parametric model, these hypotheses can then be expressed as:
H
0
: (α
0
− β
0
)T + (α
1
− β
1
)T
2
< ln(1 − δ)
H
1
: (α
0
− β
0
)T + (α
1
− β
1
)T
2
≥ ln(1 − δ)
Or
H
0
: D
0
T + D
1
T
2
< ln(1 − δ)
H
1
: D
0
T + D
1
T
2
≥ ln(1 − δ)
D
k
= α
k
− β
k
, k = 0, 1, or nally:
+− −δ <HD
T
D
T
:
11
ln(1
)0
01 0
2
+− −δ ≥HD
T
D
T
:
11
ln(1
)0
11 0
2
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
rt
n
n
ni
ni
()
1
11
1
(Kaplan-Meierestimator,see Lee, 1992)
k
i
k
2
i
1
∏
()
=
−
−− −
−−
⋅
δ
δ
=
with
δ=
iobs uncensored
iobs censored
1.
0.
i
th
th
85Reliability
Ht tt
ˆ
()
ˆˆ
c 01
2
=α +α
= least squares t of cumulative hazard, comparator
system, with response variable y = −ln r(t
i
).
Ht tt
ˆ
()
ˆˆ
e 01
2
=β +β
= least squares t of cumulative hazard, evaluation system.
Dk
ˆ
ˆ
ˆ
,0,1
kkk
=α −β =
.
Critical value(s):
Reject H
0
if
D
T
D
T
tSE
ˆ
1
ˆ
1
ln(1
)0
reg10
2
+− −δ
+≥
β
where t
β
= 100(1 − β) percentile of a (central) t-distribution with m + n – 4
degrees of freedom, and
SE SE D
T
SE D(
ˆ
)
1
(
ˆ
)
reg
2
1
2
2
0
=+
where
SE DSESEk SE SE(
ˆ
)(
ˆ
)(
ˆ
)0,1 and(
ˆ
), (
ˆ
)
kk
kk
k
222
=α+β
=α
β
are standard errors for the regression estimates of the corresponding
parameters.
Discussion:
The parametric models for the hazard functions,
h
c
(t) = α
0
+ 2α
1
t
and
h
e
(t) = β
0
+ 2β
1
t
could be somewhat generalized to a pth-order polynomial in t:
ht
kt
() (1)
k
k
k
p
0
∑
=+
γ
=
.
86 Equivalence and Noninferiority Tests
Thus, the cumulative hazard function would have the form:
Ht hd t() ()
k
k
k
p
t
1
0
0
∑
∫
=ττ= γ
+
=
.
Example:
Suppose from a sample of m = n = 100 times to failure for two systems, least
squares estimates of the parameters are
ˆ
0.60902
0
α=
ˆ
0.01696
1
α=
ˆ
0.74436
0
β=
ˆ
0.17178
1
β=
and the standard errors are
SE(
ˆ
) 0.014285
0
α≈
SE(
ˆ
) 0.002053
1
α≈
SE(
ˆ
) 0.008214
0
β≈
SE(
ˆ
) 0.002445
1
β≈
Figure7.1 shows the sample reliability curves, R
c
(t) and R
e
(t).
Thus,
D
ˆ
ˆ
ˆ
0.60902 0.74436 0.13534
000
=α −β =−≈−
D
ˆ
ˆ
ˆ
0.01696 0.17178 0.15482
111
=α −β =−≈−
At T = 0.5, with δ = 0.10,
SE D(
ˆ
) 0.016478
0
≈
87Reliability
SE D(
ˆ
) 0.003193
1
≈
SE
reg
≈ 0.03311
With 100 + 100 – 4 = 196 degrees of freedom, the statistic is
D
T
D
T
tSE
ˆ
1
ˆ
1
ln(1 ) 0.000108
0
reg10
2
+− −δ +≈ ≥
β
.
Therefore, reject the null, and conclude that the evaluation system is equiv-
alent to the comparator in reliability at T = 0.5.
Condence interval formulation:
The one-sided upper 100(1−β) percent condence limit on
D
ˆ
ˆ
ˆ
,
kkk
=α −β
k = 0, 1 is
DtSE DtSE SE
ˆ
(
ˆ
)
ˆ
ˆ
(
ˆ
)(
ˆ
)
kkkk
kk
22
+=α−β+ α+ β
ββ
where t
β
= 100(1 − β) percentile of a (central) t-distribution with m + n – 4
degrees of freedom.
0 0.4 0.8 1.4 1.8
Time
2.633.4 3.8 4.2 4.6 52.21
Y Rc(t) Re(t)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
R(t)
FIGURE 7.1
Sample reliability curves.

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