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.
24 - 1
Work done jointly with Danny L. Kugler and Laura L. Kramer (Imaging
& Printing Group, Hewlett-Packard)
December 14, 2015
8h 10min
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.
24 - 3
• Monitor and audit a production process to identify changes
in design or process that might have a seriously negative
effect on product reliability.
A Class of Linear Degradation Models
• These models are of the form
yijk = µij + ǫijk
′ x )τ + ǫ
= β0 + β1 exp(β 2
j i
ijk
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.
i=2
◮ For a scalar x , β ′ x = β x .
2
j
j
j
2
′ = (β , β , . . . ),
◮ For multiple AccVar, xj = (xj1, . . . , xjp)′, β 2
2 3
′ x is a linear combination of the AccVar, i.e.,
and β 2
j
p
X
βixji = β2xj2 + β3xj3 + · · · + βpxjp.
β 2′ xj =
24 - 5
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.
24 - 2
• Present guidelines for developing practical ADDT plans with
good statistical properties.
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.
24 - 4
• 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.
Degradation Model for the AdhesiveBondB ADDT
yijk = β0 + β1 exp(β2xj )τi + ǫijk
• The degradation model for the planning and data analysis
is
where
yijk = log(Newtonsijk )
p
√
ti = Weeksi
τi =
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 .
24 - 7
• The range of AccVar conditions available for experimentation.
• The maximum length of the experiment.
Alternative Method for Specifying
the Planning Model Parameters
• Specify planning values for β0, β2, σ, say
β0✷ = 4.471
Weeks Aged
2 4
6 12
16
8
6
4
18
Totals
8
31
24
25
88
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.
24 - 8
Result: The planning information defines the degradation
curves at all levels of temperature. It also defines the failure
time distribution at a given temperature.
0
70 DegreesC
60 DegreesC
50 DegreesC
10
20
30
40
50
9
6
60
4
6
24 - 12
24 - 10
Degradation Paths from Planning Information
D(τ, x, β ✷) = exp[β0✷ + β1✷ exp(β2✷x)τ ]
√
Weeks, x is Arrhenius-Transformed Temp
Square Root–Log Axes
τ =
100
50
20
10
Weeks
6
6
Original AdhesiveBondB ADDT Plan
70
6
6
60
16
7
12
Weeks
6
8
4
8
8
2
8
25
0
50
Newtons
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
24 - 9
• The difference in the values for β1✷ obtained from the two
procedures is due to rounding in the specification of υ ✷ and
β2✷.
Original Test Plan
for the ADDT AdhesiveBondB Data
0
0
0
6
6
7
6
0
13
8
6
6
20
8
6
9
23
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
8
8
—
50
60
70
Totals
24 - 11
DegreesC
Comments/Questions on
the Original AdhesiveBondB ADDT Plan
24 - 13
• 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?
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.
24 - 15
• Choose initial sample size using large-sample approximations.
• Use simulation to compare alternative plans.
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
Want a Plan That
24 - 14
• Meets practical constraints and is intuitively appealing.
• Is robust to deviations from assumed inputs.
• Has reasonably good statistical properties.
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.
θ
Σb
"
∂g(θ )
.
∂θ
#
b)
• Large–sample approximate standard error of g(θ
∂θ
v
#
u"
u ∂g(θ ) ′
b =t
Ase[g(θ)]
where Σb is the inverse of the Fisher information matrix Iθ .
θ
Weeks
100
150
200
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
50
24 - 18
300
24 - 16
For the θ = (β1, β 2, β3, σ)′ parameterization, the computation of Σb can be numerically unstable. Transform to stable
θ
parameters.
72
68
64
60
56
54
52
50
48
46
44
0
Simulation of Original Test Plan
Degradation 0.05 Quantile Versus Time
• Monte Carlo simulation is a powerful tool.
Newtons
80
Weeks
100
120
140
160 180
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
60
220
3000
4000
3000
5000
0.60
beta2
0.70
Focus quantity 1: 1588
beta2: 0.6509
24 - 19
0.80
b
b
• In repeated samples V
is random and if V
>
log(b
tr
log(b
tp )
p)
V✷ b then the ratio R = t˜p/tp will be greater than RT .
log(tp )
e
r
e
24 - 23
t˜p/tp will be greater than RT with a prob-
ability approximately equal to 0.5.
• The ratio R =
• In repeated samples approximately 100(1 − α)% of the intervals will contain tp.
z2
V✷
(1−α/2) log(b
tp )
n=
[log(RT )]2
provides confidence intervals for tp having the following characteristics:
24 - 21
Mean of the simulation estimates
0.50
260
Simulation of Original Test Plan
FT (t) Estimate at 25◦C with Df = 40 Newtons
.01
.005
.002
.001
.0005
.0001
.00005
.00001
.000003
.000001
.0000003
.0000001
.00000003
.00000001
40
2000
Simulation of Original Test Plan
Joint Distribution of tb.05 and βb2 Estimates
1000
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
0.80
0.75
0.70
0.65
0.60
0.55
0.50
0
1000
t_0.05 at 25 DegreesC for a failure def of 40 Newtons
0
Statistical Properties of a Test Plan
With a Given Sample Size n
t_0.05 at 25 DegreesC for a failure def of 40 Newtons
15
5
0
20
30
DegreesC
40
50
60
70
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
Simulation of Original Test Plan
Failure-Time 0.05 Quantile Estimates Versus
Temperature
10000
1000
100
10
1
#
!
r
24 - 20
b
˜ p) = log(tbp)± √1 z
log(tp), log(t
V
= log(tbp)±log(R
log(b
tp )
n (1−α/2)
e
[tbp/R, tbpR]
p
.
e
80%
24 - 24
3.0
24 - 22
, R with RT , and
log(b
tp )
p
r
s
√
b
= t˜p/tp.
R = exp (1/ n)z(1−α/2) V
log(b
t )
p
log(b
tp )
2
z(1−α/2)
V✷
[log(RT )]2
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
Sample Size to Estimate
the AdhesiveBondB t0.05 Quantile
at 25◦C when Df = 40 Newtons
solve for n to get
5000
10000
2000
500
1000
200
2.5
99%
2.0
Confidence Interval Precision Factor R
1.5
50
1.0
95%
90%
10
20
100
n=
b
d /tb2 with V✷
• Replace V
= nVar
p
b
log(b
t )
t
where
Exponentiation yields a confidence interval for tp
"
• Approximate 100(1 − α)% confidence interval for log(tp):
ADDT Sample Size Determination for a Quantile tp
Weeks
Sample Size
8 12
4
0
Fraction Failing
Test plans with a sample size of
beta2
A Two-Variables Linear Degradation Model
With Temperature and Relative Humidity as AccVar
= β0 + β1 exp(β2xj2 + β3xj3)τi + ǫijk
′ x )τ + ǫ
yijk = β0 + β1 exp(β 2
j i
ijk
• AdhesiveBondD model
where
yijk = log(Newtonsijk )
p
√
ti = Weeksi
τi =
(ǫijk /σ) ∼ Φnor (z).
24 - 25
• 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
Original Test Plan
for the AdhesiveBondD ADDT Data
Temp
◦C
6
6
6
6
6
6
36
0
6
6
6
6
6
6
36
1
Weeks Aged
2
4
6
8
12
16
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
6
6
6
6
6
6
36
6
6
6
6
6
6
36
48
48
48
48
48
48
288
Totals
• A 8 × 3 × 2 full factorial arrangement in Weeks, Temp, and
RH, with 6 units at each factor combination.
RH
%
20
50
60
70
80
50
60
70
Totals
24 - 26
• 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.
Choosing Experimental Variable Definition
to Minimize Interaction Effects
• Care should be used in defining experimental variables.
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.
• Guidance on variable definition and possible transformation
of the response and the experimental models should, as
much as possible, be taken from mechanistic models.
100
•
•
•
•
•
Volts
300
•
•
500
•
24 - 30
24 - 28
• Knowledge from mechanistic models is also useful for planning experiments.
• Models without statistical interactions simplify modeling,
interpretation, explanation, and experimental design.
• Proper choice can reduce the occurrence or importance of
statistical interactions.
• 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.
24 - 27
Size (mm)
• Choose initial sample size using large-sample approximations.
• Use simulation to compare alternative plans.
•
•
•
200
•
Comparison of Experimental Layout with
Volts/mm Versus Size and Volts Versus Size
•
•
•
150
Size (mm)
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).
•
100
•
•
3.0
Voltage Stress (Volts/mm)
50
2.5
1.5
1.0
• 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
3.0
2.0
2.5
2.0
1.5
1.0
Size (mm)
•
50 100
Volts
•
200
•
•
•
•
•
•
•
•
300
•
•
200
•
•
Voltage Stress
(Volts/mm)
100
Technical Details
300
•
•
3.0
•
2.5
1.0
1.5
2.0
3.0
• The asymptotic standard error for gb and log(gb) are
1 q
Vgb
Ase(gb) = √
n
q
1 Vgb
1 q
Vlog(gb) = √
Ase[log(gb)] = √
.
n
n g
• Easy to choose n to control Ase.
24 - 31
24 - 35
To compute these variance factors one uses planning values
for θ (denoted by θ ✷) as discussed later.
• 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.
Given a specified model and parameter values (but without need to specify sample size), one can compute scaled
asymptotic variances.
Asymptotic Approximate Standard Errors for a
Function of the Parameters g(θ )
24 - 33
The following slides give technical details used in SPLIDA
to implement the methodology.
400
Comparison of Experimental Layout with
Volts versus Size and Volts/mm versus Size
•
Size (mm)
2.5
2.0
1.5
1.0
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).
24 - 32
• Plans allowing for censoring and coarse data (accounting
for measurement limitations).
ADDT Model Asymptotic Variances
#
θ
"
X
∂ 2Lij (θ )
∂ 2L(θ )
πij E −
=n
∂ θ∂ θ′
∂ θ∂ θ′
ij
#
Under certain regularity conditions the following results hold
asymptotically (large sample)
θ
"
b ∼
• θ
˙ MVN(θ , Σb ), where Σb = Iθ−1, and
Iθ = E −
πij Iij
#′
θ
Σb
"
#
∂g(θ )
.
∂θ
Avar(gb).
24 - 34
where πij is the proportion of observations allocated to
(ti, xj ).
"
∂g(θ )
∂θ
b) ∼
• For a scalar gb = g(θ
˙ NOR[g(θ ), Avar(gb)], where
Avar(gb) =
!2
• 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
ADDT Model Fisher Information Matrix
• The Fisher information matrix, I , is
θ
"
"
#
#
X
∂ 2Lij
∂ 2L
πij E −
=n
∂ θ∂ θ′
∂ θ∂ θ′
ij
ij
X
Iθ = E −
= n
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
′
f11(ξ ij )uij uij
f12(ξ ij )uij
1

Iij = 2 
σ
symmetric
f22(ξ ij )






= 







1





exp(β ′ xj )τi

=
2


′ x )τ
β1 xj exp(β 2

j i



.

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
A Reparameterization of the
Linear ADDT Model for Numerical Stability
• The model is as before
′ x )τ + σǫ
yij = β0 + β1 exp(β 2
j i
ij
exp
xj − x̄ γ 2 τi − τ̄ + 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
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
′ x ), for the values of x stress variables in the
β1 exp(β 2
j
j
data set.
24 - 38
• The vector ϕ = (γ0, γ1, γ 2, σ)′ denotes the stable parameters.
∂µij
∂β0
∂µij
∂β1
∂µij
∂ β2
24 - 37
∂µij
∂γ0
∂µij
∂γ1
∂µij
∂γ2
2

24 - 40
• For decreasing degradation ht(tp) = ν+ςΦ−1(p), for p > Φ −

tp

2


b
 Avar[ht(tp)].
 ∂ht (tbp) 
 Avar(tbp).
∂ tbp ✷ 

Avar[ht(tbp)] = 

ht (tp✷)
 ∂h−1(z)  t
Avar(tbp) = 
∂z
24 - 42
• This allows to write Avar[ht(tbp)] as a function of Σb or as
θ
a function of Σϕ
b.
Thus
Then ignoring the probability spike, we get
Asymptotic Variance of ht(tbp)
• For test planning, the Fisher matrix is evaluated at the
planning values θ ✷.





= 



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





1




h
i




.

exp γ ′ (x − x̄) τ − τ̄ 
v ij
j
i
2
=

h
i



′ (x − x̄) τ
γ1 (xj − x̄) exp γ 2

j
i
where and f11(ξ ij ), f12(ξ ij ), f22(ξ ij ) are the LSINF elements for the distribution Φ at the experimental conditions
f11(ξ ij )v ij v ′ f12(ξ ij )v ij
1
ij

Iij = 2 
σ
symmetric
f22(ξ ij )

• The contribution Iij to the Fisher information matrix is
ADDT Fisher Matrix (Transformed Time Scale)
• For test planning the Fisher matrix is evaluated at planning
values θ ✷.
24 - 39
Relationship Between the Stable Parameters ϕ and
the Original Parameters θ
• It can be shown that
′ x̄)τ̄
γ0 = β0 + β1 exp(β 2
′ x̄)
γ1 = β1 exp(β 2
γ 2 = β 2.
• Solving for the βs
β2 = γ 2
β1 = γ1 exp(−x̄′γ 2)
β0 = γ0 − γ1τ̄ .
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


∂β
∂β
∂β
0
0
0
0



 ∂γ
∂γ
∂γ
0
1
2



 ∂β1 ∂β1 ∂β1


0

 ∂γ
A = 

0 ∂γ1 ∂ γ 2



 ∂ β2 ∂ β2 ∂ β2
0 


 ∂γ
∂γ
∂γ
0
1
2
0
0
0′
1


1 −τ̄
0′
0


′
′
′
 0 exp[−γ 2x̄] −γ1x̄ exp[−γ 2x̄] 0 
= 

 0 0
I
0 
0 0
0′
1
24 - 41
0 is a vector of zeros and I is an identity matrix with the
same row dimension as x.
ν
.
ς
and
"
cθ =
#
∂ht(tp)
.
∂θ
Asymptotic Variance of ht(tbp) in Funtion of
the θ and ϕ Parameters.
!′
!2
cθ′ Σθb cθ
∂ht(tp)
∂ht(tp)
= A′
= A ′ cθ .
∂θ
∂θ
• Define
∂ht(tp)
cϕ =
∂ϕ
It can be shown that
cϕ =
∂θ
∂ϕ
∂z
∂ht−1(z)
∂z
!2
24 - 43
24 - 45
t, transformation,
= 2ht(tp✷).
√
′ Σ c .
cϕ
b ϕ
ϕ
!2
∂ht−1(z)
′
cθ′ AΣϕ
b A cθ
∂ht−1(z)
∂z
• Then direct computations yield
Avar(tbp) =
=
=
Derivatives
ht (tp✷)
• For the square root of time, ht(t) =
∂z 2
∂ht−1(z)
=
= 2z ∂z
∂z
• The elements of cθ are:
∂ht(tp)
1
= −
′ x)
∂β0
β1 exp(β 2
ht(tp)
∂ht(tp)
= −
∂β1
β1
∂ht(tp)
= − x ht(tp)
∂β2
∂ht(tp)
Φ−1(p)
= −
′ x)
∂σ
β1 exp(β 2
Linearization of the Degradation Model
h
i
τi − τ̄c
• A first order approximation about ϕ✷ gives
n
i
o
∂µij
∂µij
+ γ2 − γ2✷
+ σǫijk
zijk = γ0 + γ1 − γ1✷
∂γ
∂γ
✷ 2
✷1
+ σǫijk
+ γ2 − γ2✷ v2ij
= γ0 + γ1 − γ1✷ v1ij
where
h
zijk = yijk − exp γ2✷ xj − x̄c
✷
v1ij
= exp γ2✷(xj − x̄c) τi − τ̄c
h
i
✷
= γ1✷ (xj − x̄c) exp γ2✷(xj − x̄c) τi
v2ij
f22(ξ ij )
24 - 47
• The asymptotic variance for this model is


−1
′
f11(ξ ij )v ij v ij
f
(σ ✷)2 X
12 (ξ ij )v ij 

πij 

n 
symmetric
Σϕ
b =
ij
i−1
(σ ✷)2 h
,
F ϕ ✷ (ξ )
=
n
✷ , v ✷ )′ and ϕ✷ = (1, γ ✷ , γ ✷ , γ ✷ , σ ✷ )′ .
where v ij = (1, v1ij
0 1 2
2ij
∂ht(z)
∂z
!2

n
′ Σ c
cϕ
Avar(tbp) =
b ϕ
ϕ
σ2
′ F (ξ )−1 c
= cϕ
ϕ
ϕ

f11(ξ )v v ′ f12(ξ ij )v ij
X
ij
ij
ij
.
πij 
symmetric
f22(ξ ij )
ij
Standardized Asymptotic Variance Formula
n
σ2
• It can be shown that
where
F ϕ (ξ ) =
1
′ (x − x̄ )]
γ1 exp[γ 2
c
24 - 44
• The standardized information matrix Fϕ(ξ ) depends only
on the test plan (ξ ij , πij ), ϕ, and the derivatives v ij .
Derivatives-Continued
= −
= − (x − x̄c)ht(tp)
=
1
= −
′ (x − x̄ )]
γ1 exp[γ 2
c
"
#
1
τ̄c
′ (x − x̄ )] − ht (tp )
γ1 exp[γ 2
c
• The elements of cϕ are:
∂ht(tp)
∂γ0
∂ht(tp)
∂γ1
∂ht(tp)
∂γ2
∂ht(tp)
∂σ
where (x̄c, τ̄c) is a centroid of the data.
24 - 46
• The elements of cθ and cϕ were obtained directly. A check
for the expressions is
cϕ = A ′ cθ
Optimal Test Plans
• Statistically optimum plans are a current research topic.
They are not practical plans but provide insight into the
problem.
i−1
c ϕ✷ .
• Statistical optimum plans can be obtained numerically under the following criteria:
h
′
argξ,π min Avar(tbp; ξ , π ) = argξ,π min cϕ
✷ F ϕ ✷ (ξ )
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
where c and Fϕ are evaluated at ϕ✷.
24 - 49
dc(ξ , v ) = cϕ ′[Fϕ(ξ )]−1Fϕ(ξ v )[Fϕ(ξ )]−1 cϕ−cϕ ′ [Fϕ(ξ )]−1 cϕ
• The test plan (ξ copt, πcopt) minimizes Avar(tbp; ξ , π ) iff
v
sup dc(ξ copt, x) = 0.
Standardization of Explanatory Variables
• Simple scaling of the explanatory variables and time gives
s )′ w ]v + σǫ
yijk = β0s + β1s exp[(β 2
j i
ijk
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 =
x − xL
U
τi
vi =
, 0 ≤ vi ≤ 1.
τH
β0s = β0
β1s = τL exp[β2xL] β1
β2s = (xH − xL)β2
24 - 51
xL, xH are the smallest and largest possible value of the
explanatory variable x and τH is the largest possible time in
the experiment.
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