Chapter 24
Planning Accelerated Destructive Degradation Tests
William Q. Meeker and Luis A. Escobar
Iowa State University and Louisiana State University
Copyright 2002-2003 W. Q. Meeker and L. A. Escobar.
Complements to the authors’ text Statistical Methods for
Reliability Data, John Wiley & Sons Inc., 1998.
Work done jointly with Danny L. Kugler and Laura L. Kramer (Imaging
& Printing Group, Hewlett-Packard)
January 13, 2014
3h 42min
24 - 1
Planning Accelerated Destructive
Degradation Tests (ADDT)
Chapter 24 Objectives
• Outline practical issues in planning ADDTs.
• Describe criteria for ADDT planning.
• Illustrate how to evaluate the properties of a ADDTs.
• Describe methods of constructing and choosing among ADDT
plans.
• Present guidelines for developing practical ADDT plans with
good statistical properties.
24 - 2
Reasons for Conducting an Accelerated Test
Accelerated destructive degradation tests (ADDTs) are used
for different purposes. These include:
• To identify failure modes and other weaknesses in product
design.
• Improve reliability.
• Assess the durability of materials and components.
• Monitor and audit a production process to identify changes
in design or process that might have a seriously negative
effect on product reliability.
24 - 3
Motivation/Example
Reliability assessment of an adhesive bond (AdhesiveBondB)
• Need: Estimate of the B05 and the proportion failing after
3 years (156 weeks) and 5 years (260 weeks) operating at
room temperature of 25◦C.
• Destructive measurements of strength in units of pounds.
• Constraints
◮ 88 test units.
◮ 16 weeks for testing.
• Test at 25◦C expected to yield little useful data.
24 - 4
A Class of Linear Degradation Models
• These models are of the form
yijk = µij + ǫijk
= β0 + β1 exp(β ′2xj )τi + ǫijk
where yijk , τi, and xj may be transformations of the measured degradation, ti, and the accelerating variable(s) AccVarj ,
respectively.
• The model is linear in the sense that for specified AccVar
vector xj , the degradation is linear in τi.
◮ For a scalar xj , β ′2xj = β2xj .
◮ For multiple AccVar, xj = (xj1, . . . , xjp)′, β ′2 = (β2, β3, . . . ),
and β ′2xj is a linear combination of the AccVar, i.e.,
β ′2xj =
p
X
i=2
βixji = β2xj2 + β3xj3 + · · · + βpxjp.
24 - 5
Degradation Model for the AdhesiveBondB ADDT
• The degradation model for the planning and data analysis
is
yijk = β0 + β1 exp(β2xj )τi + ǫijk
where
yijk = log(Newtonsijk )
p
√
τi =
ti = Weeksi
11605
xj = − ◦
Cj + 273.15
(ǫijk /σ) ∼ Φnor (z).
• Thus transformed time is square–root of Weeks and transformed AccVar is Arrhenius-transformed temperature.
24 - 6
Required Planning Information
The ADDT planning requires some information which includes:
• Purpose of the study.
• Planning values for the parameters β that characterize degradation model for Dij .
• A planning value for the parameter σ.
• A plausible distribution to model the variability ǫ.
• An specification of the critical degradation level Df .
• The range of AccVar conditions available for experimentation.
• The maximum length of the experiment.
24 - 7
Assumed Planning Information for the
AdhesiveBondB Experiment
The objective is to find a test plan to estimate with good
precision the B05 and the proportion failing at use conditions
(25◦C) after 3 and 5 years of operation.
• A Normal failure-time distribution linear degradation model
with constant regression parameters (β0, β1, β2) and same
σ at each level of temperature.
• Planning values for the parameters θ = (β0, β1, β2, σ)′ are
β0✷ = 4.471, β1✷ = −864064160, β2✷ = 0.6364, σ ✷ = 0.1580
• Critical degradation level is specified as: Df = 40 pounds.
Result: The planning information defines the degradation
curves at all levels of temperature. It also defines the failure
time distribution at a given temperature.
24 - 8
Alternative Method for Specifying
the Planning Model Parameters
• Specify planning values for β0, β2, σ, say
β0✷ = 4.471
β2✷ = 0.6364
σ ✷ = 0.1580
• Specify the degradation rate (slope of the line), υ ✷ of µ(τ, x, β )
for a given temperature, say υ ✷ = −0.1025 at 50◦C.
Because υ ✷ = β1✷ exp(β2✷x), with x = −11605/(50◦C +
273.15) = −35.912, then
β1✷ = −0.1025 exp(0.6364 × 35.912)
= −863499883
• The difference in the values for β1✷ obtained from the two
procedures is due to rounding in the specification of υ ✷ and
β2✷.
24 - 9
Degradation Paths from Planning Information
D(τ, x, β ✷) = exp[β0✷ + β1✷ exp(β2✷x)τ ]
√
τ = Weeks, x is Arrhenius-Transformed Temp
Square Root–Log Axes
100
Newtons
50
20
70 DegreesC
60 DegreesC
50 DegreesC
10
0
10
20
30
40
50
60
Weeks
24 - 10
Original Test Plan
for the ADDT AdhesiveBondB Data
Test plan: 8 units with no ageing were measured at the
start of the experiment. A total of 80 additional units were
aged and measured according to the following temperatures
and time schedule.
Temp
◦C
—
50
60
70
Totals
0
Weeks Aged
2 4
6 12
Totals
16
8
8
8
6
6
20
0
0
6
6
8
6
4
18
8
6
9
23
7
6
0
13
8
31
24
25
88
24 - 11
DegreesC
Original AdhesiveBondB ADDT Plan
70
6
60
50
25
6
4
9
6
6
6
6
8
8
8
7
6
12
16
8
0
2
4
Weeks
24 - 12
Comments/Questions on
the Original AdhesiveBondB ADDT Plan
• The 8 observations at ti = 0 were not aged, i.e., there were
never put into an oven.
• Too much extrapolation in temperature?
• Reasonable to extrapolate accurately to 25◦C?
24 - 13
Want a Plan That
• Meets practical constraints and is intuitively appealing.
• Is robust to deviations from assumed inputs.
• Has reasonably good statistical properties.
24 - 14
Some Practical Guidelines for
One AccVar ADDT Plans
• Statistically optimum plans are a research topic. They are
not practical plans but provide insight into the problem.
• Use three or four levels of AccVar and spread them out as
much as possible, subject to model adequacy.
• Limit high level of the accelerating variable to a maximum
reasonable condition.
• Reduce lowest level of the accelerating variable (to minimize
extrapolation)-subject to seeing some action.
• Measure test units at three or four different times for each
level of the AccVar.
• Choose initial sample size using large-sample approximations.
• Use simulation to compare alternative plans.
24 - 15
Evaluating Test Plan Properties
Suppose inferences are needed on a function g(θ ) (one-toone and all the first derivatives with respect to the elements
of θ exist, and are continuous).
• Properties depend on test plan, model and (unknown) parameter values. Need planning values.
b)
• Large–sample approximate standard error of g(θ
v"
"
#
#
u
u ∂g(θ ) ′
∂g(θ )
t
b
Ase[g(θ)] =
Σb
.
θ
∂θ
∂θ
where Σb is the inverse of the Fisher information matrix Iθ .
θ
For the θ = (β1, β 2, β3, σ)′ parameterization, the computation of Σb can be numerically unstable. Transform to stable
θ
parameters.
• Monte Carlo simulation is a powerful tool.
24 - 16
Evaluating Test Plans
• Large–sample approximations
◮ Fast.
◮ Provides some general insights.
◮ Provides simple sample size rules.
• Simulation
◮ Reflects actual variability (no approximations).
◮ Provides visualization of sampling variability.
◮ Requires up-front computational effort.
• Use large–sample approximations to get an initial indication
of sample size needs and to suggest potential designs (e.g.,
use of optimization techniques). Use simulation to study
the properties of particular designs.
24 - 17
Simulation of Original Test Plan
Degradation 0.05 Quantile Versus Time
Accelerated destructive degradation test simulation based on
originalplan.ADDTplan AdhesiveBondB.Normal.ADDTpv
0.05 quantile of degradation versus Weeks at 25. DegreesC
Resp:Log,Time:Square root,x:Arrhenius, Dist:Normal
72
68
Newtons
64
60
56
54
52
50
48
46
44
0
50
100
150
200
300
Weeks
24 - 18
Simulation of Original Test Plan
FT (t) Estimate at 25◦C with Df = 40 Newtons
Accelerated destructive degradation test simulation based on
originalplan.ADDTplan AdhesiveBondB.Normal.ADDTpv
Fraction failing versus Weeks for 40 Newtons at 25. DegreesC
Resp:Log,Time:Square root,x:Arrhenius, Dist:Normal
Lognormal Probability Plot
.01
.005
.002
.001
Fraction Failing
.0005
.0001
.00005
.00001
.000003
.000001
.0000003
.0000001
.00000003
.00000001
40
60
80
100
120
140
160 180
220
260
Weeks
24 - 19
Simulation of Original Test Plan
Failure-Time 0.05 Quantile Estimates Versus
Temperature
Accelerated destructive degradation test simulation based on
originalplan.ADDTplan AdhesiveBondB.Normal.ADDTpv
Failure time 0.05 quantile vs DegreesC for failure definition 40 Newtons
Resp:Log,Time:Square root,x:Arrhenius , Dist:Normal
10000
Weeks
1000
100
10
1
20
30
40
50
60
70
DegreesC
24 - 20
Simulation of Original Test Plan
Joint Distribution of tb.05 and βb2 Estimates
0
5
15
0.80
0.75
0.70
0.65
0.60
0.55
0.50
0
1000
2000
3000
4000
5000
8 12
t_0.05 at 25 DegreesC for a failure def of 40 Newtons
0.50
0.60
0.70
0.80
beta2
Mean of the simulation estimates
4
Focus quantity 1: 1588
beta2: 0.6509
0
beta2
Accelerated destructive degradation test simulation based on
originalplan.ADDTplan AdhesiveBondB.Normal.ADDTpv
Failure time 0.05 quantile vs DegreesC for failure definition 40 Newtons
Resp:Log,Time:Square root,x:Arrhenius , Dist:Normal
0
1000
3000
t_0.05 at 25 DegreesC for a failure def of 40 Newtons
24 - 21
ADDT Sample Size Determination for a Quantile tp
• Approximate 100(1 − α)% confidence interval for log(tp):
"
!
#
r
b
˜ p) = log(tbp)± √1 z
= log(tbp)±log(R
log(tp), log(t
(1−α/2) Vlog(b
t
)
p
n
e
Exponentiation yields a confidence interval for tp
[tbp/R, tbpR]
where
√
r
b
R = exp (1/ n)z(1−α/2) V
=
log(b
tp )
s
t˜p/tp.
e
b
d /tb2 with V✷
• Replace V
= nVar
, R with RT , and
b
log(b
tp )
tp p
log(b
tp )
solve for n to get
n=
2
z(1−α/2)
V✷
log(b
tp )
.
2
[log(RT )]
24 - 22
Statistical Properties of a Test Plan
With a Given Sample Size n
Test plans with a sample size of
n=
2
z(1−α/2)
V✷
log(b
tp )
[log(RT )]2
provides confidence intervals for tp having the following characteristics:
• In repeated samples approximately 100(1 − α)% of the intervals will contain tp.
b
b
• In repeated samples V
is
random
and
if
V
>
b
log(tr
log(b
tp )
p)
V✷ b then the ratio R = t˜p/tp will be greater than RT .
log(tp )
• The ratio R =
r
e
t˜p/tp will be greater than RT with a probe
ability approximately equal to 0.5.
24 - 23
Sample Size to Estimate
the AdhesiveBondB t0.05 Quantile
at 25◦C when Df = 40 Newtons
Needed sample size giving approximately a 50% chance of having
a confidence interval factor for the 0.05 quantile that is less than R
use condition= 25. DegreesC and a failure definition= 40
originalplan.ADDTplan AdhesiveBondB.Normal.ADDTpv
10000
5000
Sample Size
2000
1000
500
200
100
50
99%
20
95%
90%
80%
10
1.0
1.5
2.0
2.5
3.0
Confidence Interval Precision Factor R
24 - 24
A Two-Variables Linear Degradation Model
With Temperature and Relative Humidity as AccVar
• AdhesiveBondD model
yijk = β0 + β1 exp(β ′2xj )τi + ǫijk
= β0 + β1 exp(β2xj2 + β3xj3)τi + ǫijk
where
yijk = log(Newtonsijk )
p
√
ti = Weeksi
τi =
(ǫijk /σ) ∼ Φnor (z).
• The AccVar transformations are Arrhenius for Temp (Temperature) and logit for RH (Relative Humidity), i.e.,
11605
xj2 = − ◦
Cj + 273.15
!
RHj
xj3 = log
.
1 − RHj
24 - 25
Original Test Plan
for the AdhesiveBondD ADDT Data
• A 8 × 3 × 2 full factorial arrangement in Weeks, Temp, and
RH, with 6 units at each factor combination.
RH
%
20
Temp
◦C
50
60
70
80
50
60
70
Totals
0
1
6
6
6
6
6
6
36
6
6
6
6
6
6
36
Weeks Aged
2
4
6
8
6
6
6
6
6
6
36
6
6
6
6
6
6
36
6
6
6
6
6
6
36
6
6
6
6
6
6
36
Totals
12
16
6
6
6
6
6
6
36
6
6
6
6
6
6
36
48
48
48
48
48
48
288
• Might want to re-allocate units assigned to the corners
of the experimental region, i.e., highest (lowest) levels of
(Weeks, RH, Temp). Also, might want to decrease the
number of pens allocated to each factor level combination
to reduce size of the experiment.
24 - 26
Some Practical Guidelines for
Two or More AccVar ADDT Plans
• Moderate increases in two accelerating variables may be
safer than using a large amount of a single accelerating
variable.
• There may be interest in assessing the effect of nonaccelerating variables, i.e., factor Weeks in the AdhesiveBondD
example.
• There may be interest in assessing joint effects of two or
more accelerating variables, i.e., (RH, Temp) in the AdhesiveBondD example.
• Traditional factorial experiments are good as starting plans.
• When possible, a fractional factorial could provide economy
in testing.
• Choose initial sample size using large-sample approximations.
• Use simulation to compare alternative plans.
24 - 27
Choosing Experimental Variable Definition
to Minimize Interaction Effects
• Care should be used in defining experimental variables.
• Guidance on variable definition and possible transformation
of the response and the experimental models should, as
much as possible, be taken from mechanistic models.
• Proper choice can reduce the occurrence or importance of
statistical interactions.
• Models without statistical interactions simplify modeling,
interpretation, explanation, and experimental design.
• Knowledge from mechanistic models is also useful for planning experiments.
24 - 28
Examples of Choosing Experimental Variable
Definition to Minimize Interaction Effects
• For accelerated testing of dielectrics, use size and volts
stress (e.g., mm and volts/mm instead of mm and volts).
• For light exposure, use aperture and total light energy
(not aperture and exposure time).
• For accelerated humidity testing with a corrosion mechanism, use RH and temperature (not vapor pressure and
temperature).
24 - 29
•
•
•
•
•
•
•
100
150
200
3.0
•
•
•
2.0
•
•
•
50
Voltage Stress (Volts/mm)
1.0
1.5
Size (mm)
2.0
1.5
1.0
Size (mm)
•
2.5
•
2.5
3.0
Comparison of Experimental Layout with
Volts/mm Versus Size and Volts Versus Size
•
100
•
•
300
500
Volts
24 - 30
•
•
•
•
•
•
•
200
300
3.0
•
•
•
2.0
•
•
•
50 100
Volts
1.0
1.5
Size (mm)
2.0
1.5
1.0
Size (mm)
•
2.5
•
2.5
3.0
Comparison of Experimental Layout with
Volts versus Size and Volts/mm versus Size
•
•
•
100
200
300
400
Voltage Stress
(Volts/mm)
24 - 31
Areas for Future Research in ADDT Planning
• Optimum plans (to provide insight and general rules).
• Optimized compromise plans (using practical constraints).
• Plans that allow the use of prior information on activation
energy (i.e., plans for Bayesian analysis).
• Plans allowing for censoring and coarse data (accounting
for measurement limitations).
24 - 32
Technical Details
The following slides give technical details used in SPLIDA
to implement the methodology.
24 - 33
ADDT Model Asymptotic Variances
Under certain regularity conditions the following results hold
asymptotically (large sample)
b ∼
• θ
˙ MVN(θ , Σb ), where Σb = Iθ−1, and
θ
θ
"
Iθ = E −
∂ 2L(θ )
∂ θ∂ θ′
#
"
X
∂ 2Lij (θ )
=n
πij E −
′
∂
θ
∂
θ
ij
#
where πij is the proportion of observations allocated to
(ti, xj ).
b) ∼
• For a scalar gb = g(θ
˙ NOR[g(θ ), Avar(gb)], where
Avar(gb) =
"
∂g(θ )
∂θ
#′
"
#
∂g(θ )
.
θ
∂θ
Σb
• When g(θ ) is positive for all θ , then
b )] ∼
log[g(θ
˙ NOR{log[g(θ )], Avar[log(gb)]}, where
Avar[log(gb)] =
1
g
!2
Avar(gb).
24 - 34
Asymptotic Approximate Standard Errors for a
Function of the Parameters g(θ )
Given a specified model and parameter values (but without need to specify sample size), one can compute scaled
asymptotic variances.
• The variance factors Vgb = nAvar(gb) and Vlog(gb) = nAvar[log(gb)]
may depend on the actual value of θ but they do not depend
on n.
To compute these variance factors one uses planning values
for θ (denoted by θ ✷) as discussed later.
• The asymptotic standard error for gb and log(gb) are
1 q
Ase(gb) = √
Vgb
n
q
1 Vgb
1 q
Vlog(gb) = √
.
Ase[log(gb)] = √
n
n g
• Easy to choose n to control Ase.
24 - 35
ADDT Model Fisher Information Matrix
• The Fisher information matrix, Iθ , is
"
∂ 2L
#
X
"
∂ 2L
ij
=n
πij E −
∂ θ∂ θ′
∂ θ∂ θ′
ij
X
= n
πij Iij
Iθ = E −
#
ij
where L = log[L(θ )], Lij is the contribution of a single
observation at ξ ij = (ti, xj ) to the log-likelihood, and πij is
the proportion of observations made at ξ ij .
E is the expectation operator, the index on the summation operation runs over the distinct time, and stress level
combinations ξ ij .
Iij is the contribution of one observation at ξ ij to Iθ .
24 - 36
Contributions to the ADDT Model
Fisher Information Matrix
• The contribution Iij to the Fisher information matrix is

′
1  f11(ξ ij )uij uij
Iij = 2
σ
symmetric
f12(ξ ij )uij
f22(ξ ij )


where and f11(ξ ij ), f12(ξ ij ), f22(ξ ij ) are the LSINF elements for the distribution Φ at the experimental conditions
ξ ij and uij is the vector of partial derivatives of the degradation path with respect to the β parameters

uij




= 




∂µij
∂β0
∂µij
∂β1
∂µij
∂ β2




1



′ x )τ
=
exp(
β


2 j i



β1 xj exp(β ′2xj )τi




.

• For test planning the Fisher matrix is evaluated at planning
values θ ✷.
24 - 37
A Reparameterization of the
Linear ADDT Model for Numerical Stability
• The model is as before
yij = β0 + β1 exp(β ′2xj )τi + σǫij
• Suppose that x̄ is the centroid of the stress variables
[i.e., x̄ = (x̄1, . . . , x̄k )′] and τ̄ is an average transformed
time. Then the model can be reparameterized as
yij = γ0 + γ1
exp
′
xj − x̄ γ 2 τi − τ̄ + σǫij
where γ0 is the intercept for the average stress line (i.e.,
degradation line for x̄) at τ̄ ; γ1 is the slope of the average
stress line; and γ 2 = β 2 are the regression coefficients
corresponding to the x variables.
• It can be shown that γ1 is the geometric mean of the slopes
β1 exp(β ′2xj ), for the values of xj stress variables in the
data set.
• The vector ϕ = (γ0, γ1, γ 2, σ)′ denotes the stable parameters.
24 - 38
Relationship Between the Stable Parameters ϕ and
the Original Parameters θ
• It can be shown that
γ0 = β0 + β1 exp(β ′2x̄)τ̄
γ1 = β1 exp(β ′2x̄)
γ 2 = β 2.
• Solving for the βs
β2 = γ 2
β1 = γ1 exp(−x̄′γ 2)
β0 = γ0 − γ1τ̄ .
24 - 39
ADDT Fisher Matrix (Transformed Time Scale)
• The contribution Iij to the Fisher information matrix is

′
1  f11(ξ ij )v ij v ij
Iij = 2
σ
symmetric
f12(ξ ij )v ij
f22(ξ ij )


where and f11(ξ ij ), f12(ξ ij ), f22(ξ ij ) are the LSINF elements for the distribution Φ at the experimental conditions
ξ ij and v ij is the vector of partial derivatives of the degradation path with respect to the γ = (γ0, γ1, γ 2) parameters

v ij




= 




∂µij
∂γ0
∂µij
∂γ1
∂µij
∂γ2



1



i
h


′
=
exp γ 2(xj − x̄) τi − τ̄



i
h


γ1 (xj − x̄) exp γ ′2(xj − x̄) τi




.


• For test planning, the Fisher matrix is evaluated at the
planning values θ ✷.
24 - 40
Alternative Expression for
the Variance-Covariance Matrix
• Let Σγb be the variance-covariance matrix in the stable parameterization and Σb the corresponding matrix in the origθ
inal parameterization.
• Then Σb = AΣγb A′, where
θ






A = 








= 


∂β0 ∂β0 ∂β0
0 

∂γ0 ∂γ1 ∂ γ 2

∂β1 ∂β1 ∂β1

0 

∂γ0 ∂γ1 ∂ γ 2


∂ β2 ∂ β2 ∂ β2
0 


∂γ0 ∂γ1 ∂ γ 2
0
0
0′
1
1 −τ̄
0′
0 exp[−γ ′2x̄] −γ1x̄′ exp[−γ ′2x̄]
0 0
I
0 0
0′
0
0
0
1





0 is a vector of zeros and I is an identity matrix with the
same row dimension as x.
24 - 41
Asymptotic Variance of ht(tbp)
ν
• For decreasing degradation ht(tp) = ν+ςΦ−1(p), for p > Φ −
.
ς
Then ignoring the probability spike, we get
2

Thus
 ∂ht (tbp) Avar[ht(tbp)] = 

∂ tbp t✷
p

 ∂h−1
(z)
t
Avar(tbp) = 

∂z

 Avar(tbp).

2
ht (t✷
p)

 Avar[ht(tbp)].

• This allows to write Avar[ht(tbp)] as a function of Σb or as
θ
a function of Σϕ
b.
24 - 42
Asymptotic Variance of ht(tbp) in Funtion of
the θ and ϕ Parameters.
• Define
∂ht(tp)
cϕ =
∂ϕ
It can be shown that
cϕ =
∂θ
∂ϕ
!′
and
∂ht(tp)
cθ =
.
∂θ
"
#
∂ht(tp)
∂ht(tp)
= A′
= A ′ cθ .
∂θ
∂θ
• Then direct computations yield
Avar(tbp) =
=
=
!2
−1
∂ht (z)
c′θ Σθb cθ
∂z
!2
−1
∂ht (z)
′
c′θ AΣϕ
b A cθ
∂z
!2
−1
∂ht (z)
c′ϕ Σϕ
b cϕ .
∂z
24 - 43
Standardized Asymptotic Variance Formula
• It can be shown that
n
σ2
∂ht(z)
∂z
!2
Avar(tbp) =
n
σ2
c′ϕ Σϕ
b cϕ
−1
′
= c ϕ F ϕ (ξ )
cϕ
where

f11(ξ ij )v ij v ′ij
X
F ϕ (ξ ) =
πij 
symmetric
ij
f12(ξ ij )v ij
f22(ξ ij )

.
• The standardized information matrix Fϕ(ξ ) depends only
on the test plan (ξ ij , πij ), ϕ, and the derivatives v ij .
24 - 44
Derivatives
• For the square root of time, ht(t) =
−1
∂ht (z)
∂z 2
=
= 2z ∂z
∂z
√
t, transformation,
= 2ht(t✷
p ).
ht (t✷
p)
• The elements of cθ are:
∂ht(tp)
1
= −
∂β0
β1 exp(β ′2x)
ht(tp)
∂ht(tp)
= −
∂β1
β1
∂ht(tp)
= − x ht(tp)
∂β2
∂ht(tp)
Φ−1(p)
= −
∂σ
β1 exp(β ′2x)
24 - 45
Derivatives-Continued
• The elements of cϕ are:
∂ht(tp)
∂γ0
∂ht(tp)
∂γ1
∂ht(tp)
∂γ2
∂ht(tp)
∂σ
1
= −
γ1 exp[γ ′2(x − x̄c)]
#
"
1
τ̄c
− ht(tp)
=
′
γ1 exp[γ 2(x − x̄c)]
= − (x − x̄c)ht(tp)
= −
1
γ1 exp[γ ′2(x − x̄c)]
where (x̄c, τ̄c) is a centroid of the data.
• The elements of cθ and cϕ were obtained directly. A check
for the expressions is
cϕ = A ′ cθ
24 - 46
Linearization of the Degradation Model
• A first order approximation about ϕ✷ gives
zijk = γ0 + γ1
= γ0 + γ1
where
zijk
✷
v1ij
✷
v2ij
∂µij
∂µij
✷
✷
− γ1
+ γ2 − γ2
+ σǫijk
∂γ1
∂γ2
✷ + γ − γ ✷ v ✷ + σǫ
− γ1✷ v1ij
2
ijk
2
2ij
n
h
i
o
✷
= yijk − exp γ2 xj − x̄c τi − τ̄c
i
h
✷
= exp γ2 (xj − x̄c) τi − τ̄c
i
h
✷
✷
= γ1 (xj − x̄c) exp γ2 (xj − x̄c) τi
• The asymptotic variance for this model is


f11(ξ ij )v ij v ′ij
(σ ✷)2 X
πij 
Σϕ
b =
n  ij
symmetric
i−1
(σ ✷)2 h
,
F ϕ ✷ (ξ )
=
−1
f12(ξ ij )v ij 


f22(ξ ij )
n
✷ , v ✷ )′ and ϕ✷ = (1, γ ✷ , γ ✷ , γ ✷ , σ ✷ )′ .
where v ij = (1, v1ij
0 1 2
2ij
24 - 47
Optimal Test Plans
• Statistically optimum plans are a current research topic.
They are not practical plans but provide insight into the
problem.
• Statistical optimum plans can be obtained numerically under the following criteria:
i−1
h
′
argξ,π min Avar(tbp; ξ , π ) = argξ,π min cϕ✷ Fϕ✷ (ξ )
c ϕ✷ .
Notice that the derivatives, c and the information matrix,
F , are evaluated at ϕ✷.
• The General Equivalence Theorem can be used to check
the optimality of the test plan obtained from the numerical
algorithm.
24 - 48
The General Equivalence Theorem to Check Optimal
Test Plans
• Defining ξ v as a plan with all units at v and
dc(ξ , v ) = cϕ ′[Fϕ(ξ )]−1Fϕ(ξ v )[Fϕ(ξ )]−1 cϕ−cϕ ′ [Fϕ(ξ )]−1 cϕ
where c and Fϕ are evaluated at ϕ✷.
• The test plan (ξ copt, πcopt) minimizes Avar(tbp; ξ , π ) iff
sup dc(ξ copt, x) = 0.
v
24 - 49
General Guidelines
• Reduce lowest level of the accelerating variable (to minimize
extrapolation)-subject to seeing some action.
• Measure test units at three or four different times for each
level of the AccVar.
• Choose initial sample size using large-sample approximations.
• Use simulation to compare alternative plans.
24 - 50
Standardization of Explanatory Variables
• Simple scaling of the explanatory variables and time gives
yijk = β0s + β1s exp[(β s2)′wj ]vi + σǫijk
where wj are the scaled explanatory variables and vi is the
scaled time variable (all explanatory variables scaled in [0, 1]).
• When xj = xj is a scalar, then
xj − xL
, 0 ≤ wj ≤ 1.
wj =
xU − xL
τi
vi =
, 0 ≤ vi ≤ 1.
τH
β0s = β0
β1s = τL exp[β2xL] β1
β2s = (xH − xL)β2
xL, xH are the smallest and largest possible value of the
explanatory variable x and τH is the largest possible time in
the experiment.
24 - 51