Chapter 23
Accelerated Destructive Degradation Test Data,
Models, and Analysis
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.
& Printing Group, Hewlett-Packard)
Weeks Aged
2 4
6 12
16
8
6
4
18
7
6
0
13
0
0
6
6
8
6
9
23
8
6
6
20
Weeks
8
31
24
25
88
Totals
15
20
23 - 1
23 - 3
Accelerated Destructive Degradation Tests
Data, Models, and Data Analysis
Chapter 23 Objectives
• Describe useful accelerated destructive degradation test (ADDT)
reliability models.
• Show the connection between degradation reliability models
and failure-time reliability models.
• Present methods of data analysis and reliability inference
for ADDT.
4
6
9
6
23 - 4
23 - 2
• Discuss the use of ADDT data to estimate acceleration
factors.
• Illustrate methods with examples.
6
6
AdhesiveBondB ADDT Plan
Number of Units Measured at Each
(Time, Temperature) Combination
70
6
6
60
7
16
8
12
Weeks
6
8
4
8
8
2
50
25
0
0
AdhesiveBondB Data
Resp:Linear,Time:Linear, Dist:Normal
50DegreesC
80
60
100
40
60
20
40
15
20
0
70DegreesC
10
15
20
5
0
60
80
100
40
20
0
0
20
10
100
5
Weeks
80
0
5
60DegreesC
10
15
23 - 6
20
AdhesiveBondB ADDT Data
Scatter Plot at Individual Conditions of Temperature
Linear–Linear Axes
DegreesC
Work done jointly with Danny L. Kugler and Laura L. Kramer (Imaging
December 14, 2015
8h 10min
AdhesiveBondB ADDT Data
• Objective: Assess the strength of an AdhesiveBond over
time; estimate the proportion of devices with a strength
below 40 Newtons after 5 years of operation (approximately
260 weeks) at 25◦C.
• The test is destructive; each unit can be measured only
once. There were 6 right censored observations.
0
Temp
◦C
8
50DegreesC
60DegreesC
70DegreesC
5
10
AdhesiveBondB Data
Destructive Degradation Scatter Plot
Resp:Linear,Time:Linear
AdhesiveBondB ADDT Data
Scatter Plot
Linear–Linear Axes
8
—
50
60
70
Totals
100
80
60
40
20
0
0
23 - 5
Newtons
• 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.
Newtons
0
0
80
40
20
10
0
AdhesiveBondB Data
Resp:Log,Time:Square root, Dist:Normal
10 15
50DegreesC
5
10 15
70DegreesC
5
10 15
60DegreesC
5
23 - 7
AdhesiveBondB ADDT Data
Scatter Plot at Individual Conditions of Temperature
Square Root–Log Axes
80
40
20
10
80
40
20
10
Weeks
Statistical Model for ADDT Data
= µij + ǫijk ,
k = 1, . . . , nij
= hd(Dij ) + ǫijk
• Sample path model: For observation k at time τi and
accelerating variable(s) level xj the model is
yijk
where µij = hd(Dij ) and yijk are, respectively, monotone
increasing transformations of Dij and the observed degradation.
nij is the number of observations at time ti and AccVarj
vector level xj .
ǫijk is a residual deviation which describes unit-to-unit variability with (ǫijk /σ) ∼ Φ(z).
23 - 9
• The transformations for the observed degradation, Dij , ti,,
and the AccVarj might be suggested by physical/chemical
theory, past experience, or the data.
10
20
30
70 DegreesC
60 DegreesC
50 DegreesC
40
50
60
23 - 11
Degradation Path Model: Multiple Temperature Levels
D(τ, x, β ) = exp[β0 + β1 exp(β2x)τ ]
√
τ = Weeks, x is Arrhenius-Transformed Temp
Linear–Linear Axes
100
80
60
40
20
0
0
Weeks
Model for Accelerated Degradation Path
• Actual degradation path model: Actual path of a unit at
ti and accelerating variable(s) level (combinations) AccVarj
(e.g., temperature or temperature and relative humidity) is
Dij = D(τi, xj , β )
where τi = ht(ti) is a known monotone increasing transformations of ti and xj = ha(AccVarj ) are known transformations of the accelerating variables at the jth level (or
combination of levels) AccVarj .
When there is no possibility of confusion, τi and xj are called
the time and the AccVar, respectively.
• Rates in the model are with respect to τ = ht(t).
• Path parameters: the elements of β are fixed but unknown.
23 - 8
• Though no commonly used in accelerated testing it is possible to have models with interactions among the accelerating
variables.
100
80
60
40
20
0
0
0
10
20
70 DegreesC
60 DegreesC
50 DegreesC
10
30
40
70 DegreesC
Weeks
20
30
50
40
50
60
60
Using Transformations
to Linearize a Degradation Model
log[D(τ, x, β )] = β0 + β1 exp(β2x)τ
√
Weeks, x is Arrhenius-Transformed Temp
Square Root–Log Axes
100
50
20
10
Weeks
23 - 12
23 - 10
Degradation Path Model: Single Temperature Level
D(τ, x, β ) = exp[β0 + β1 exp(β2x)τ ]
√
Weeks and x is Arrhenius-Transformed Temp
Linear–Linear Axes
τ =
τ =
Newtons
Newtons
Newtons
Newtons
k = 1, . . . , nij
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
23 - 13
◮ For a single (scalar) AccVar x , β ′ x = β x .
2
j
j
j
2
′ = (β , β , . . . ),
◮ For multiple AccVar, xj , xj = (x2, . . . , xp)′, 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 =
Linear Degradation Model for
the AdhesiveBondB Data
yijk = β0 + β1 exp(β2xj )τi + ǫijk
For the AdhesiveBondB data, the strength of the adhesive
as a function of time and temperature is modeled by
where
yijk = log(Newtonsijk )
p
√
ti = Weeksi
τi =
xj = −11605/(◦Cj + 273.15)
(ǫijk /σ) ∼ Φnor (z).
23 - 15
The ǫijk term contains model and measurement errors.
AdhesiveBondB ADDT Data
Normal Model ML Individual Line Fits
• For each temperature level three individual ML estimates
[j]
b [j].
are obtained: βb0 , υb [j], and σ
4.490
−0.1088
−0.3626
−0.2089
υb [j]
0.01944
0.02214
0.01494
sceυb[j]
ML Estimates
Upper
−0.08309
−0.16969
−0.1424
−0.32643
−0.2571
−0.4028
Lower
95% Approximate
Confidence Interval
for υ [j]
• A summary of the Normal model individual ML fits for the
AdhesiveBondB data is
4.489
[j]
50◦C
βb0
60◦C
4.400
AccVarj
70◦C
23 - 17
Linear Degradation Model
Interpretation of the Parameters
′ x )τ + ǫ
yijk = β0 + β1 exp(β 2
j i
ijk
For the linear degradation model
√
ti then β0 is the intercept at time t = 0.
• β0 is path intercept parameter when τ = 0. For example,
◮ If τi =
◮ If τi = log(ti) then β0 is intercept at time t = 1.
′ x ).
• The degradation rate at xj is υ(xj )(AccVarj ) = β1 exp(β 2
j
• The sign (±) of β1 determines if the degradation is increasing or decreasing in time.
23 - 14
• For a power transformation of time τ = tκ, the components
of the vector parameter β 2 are related to the amount of acceleration obtained by increasing the accelerating variables
AccVar.
ADDT Individual Analysis
Likelihood for Fixed AccVar level xj
i k=1 σ
nij Y Y
1
φ
1−δijk
δ
ijk
y
y
ijk − µij
ijk − µij
× 1−Φ
σ
σ
• For the data at fixed xj of the AccVar with exact failure
times and right-censored observations, the likelihood is
Lj (θ |DATA) =
′ x )τ , δ
where µij ≡ µ(τi, xj , β ) = β0 + β1 exp(β 2
j i
ijk indicates
whether observation yijk is a failure (δijk = 1) or a right
′ , σ), and n
censored observation (δijk = 0), θ = (β0, β1, β 2
ij
is the number of observations at (τi, xj ).
80
40
20
10
80
40
20
10
0
0
10 15
50DegreesC
5
10 15
70DegreesC
5
Weeks
80
40
20
10
0
AdhesiveBondB Data
Resp:Log,Time:Square root, Dist:Normal
5
10 15
60DegreesC
AdhesiveBondB ADDT Data
Individual Normal Model ML Fits
Square Root–Log Axes
[j]
b [j]τ
b [j] = βb0 + υ
µ
23 - 18
23 - 16
• For fixed xj , the model parameters are: the spread σ, the
′ x ).
intercept β0, and the slope of the line υj = β1 exp(β 2
j
The parameter υj can be interpreted as the degradation
rate of µ(τ, xj , β ) with respect to the transformed time τ.
Newtons
50DegreesC
60DegreesC
70DegreesC
Weeks
5
10
15
AdhesiveBondB Data
Destructive Degradation Individual Regression Analyses
Resp:Log,Time:Square root, Dist:Normal
50
55
60
65
70
Degradation rate versus DegreesC on Arrhenius Scale for
AdhesiveBondB Data
Resp:Log,Time:Square root,x:arrhenius, Dist:Normal
Arrhenius Plot of |υb [j]|
Individual Degradation Rates
Normal Model ML Estimates
0
AdhesiveBondB ADDT Data
Overlay of Individual Normal Model Fits
Square Root–Log Axes
[j]
b [j] = βb0 + υ
b [j]τ
µ
100
80
60
50
40
30
20
10
0.40
0.30
0.25
0.20
0.15
0.10
0.05
45
DegreesC on Arrhenius Scale
Standard
Error
0.03864
20
75
23 - 19
23 - 21
95% Approximate
Confidence Interval
Lower
Upper
4.396
4.547
AdhesiveBondB Data ML Estimates
for the Acceleration Model Fit
ML
Estimate
4.471
• Parameter estimates
Parameter
β0
0.7536
2.261 × 109
0.5375
0.1841
−3.989 × 109
0.05488
0.1356
1.595 × 109
0.6364
0.01233
−8.641 × 108
β2
0.1580
β1
σ
b
υ(50)
= −0.1025
b
υ(70)
= −0.3883
23 - 23
• Normal model ML for the slopes (degradation rates) at
b ◦Cj ) = βb1 exp(βb2xj ),
each temperature are obtained from υ(
where xj = −11605/(◦Cj + 273.15). In this case for the four
temperatures of interest the ML estimates are
b
υ(25)
= −0.0151,
b
υ(60)
= −0.2035,
Individual Degradation Rate Estimates
• The ML estimates υb [j] (slopes of the individual lines) can
be used to identify the relationship between the degradation
rate and the AccVar.
• When the degradation is decreasing, use absolute values of
the degradation rate.
• Because
′x
log(| υj |) = log(| β1 |) + β 2
j
the surface log(| υb [j] |) versus the AccVar xj should be approximately linear in the xj if the model relating degradation
rate and the AccVar is adequate. Then
◮ For a single AccVar xj , the plot of log(| υb [j] |) versus xj
should be approximately linear.
23 - 20
◮ For an AccVar vector xj , the plot of
log(| υb [j] |) versus any of the accelerating variables, conditional on fixed values of the remaining accelerating
variables, should be approximately linear.
Likelihood for All Data With Right Censored Data
σ
φ
σ
× 1−Φ
σ
yijk − µij 1−δijk
• For a sample of n units consisting of exact failure times and
right-censored observations, the likelihood can be expressed
as
ijk
Y
Lj (θ |DATA)
L(θ |DATA) =
j
Y 1 yijk − µij δijk
L(θ |DATA) =
′ , σ)′ , and δ
where θ = (β0, β1, β 2
ijk indicates whether observation ijk is a failure (δijk = 1) or a right censored observation (δijk = 0).
µij = β0 + β1 exp(β2xj )τi
11605
.
xj = −
◦ C + 273.15
j
50DegreesC
60DegreesC
70DegreesC
25DegreesC
5
10
15
AdhesiveBondB data
Destructive Degradation Regression Analyses
Resp:Log,Time:Square root,DegreesC:Arrhenius, Dist:Normal
AdhesiveBondB ADDT Data
Normal Model Arrhenius ML Fit
b ij = βb0 + βb1 exp(βb2xj )τi
µ
0
Weeks
20
23 - 24
23 - 22
• Note that µij is a nonlinear function of the accelerating
variable xj .
10
20
30
40
60
50
80
100
• For the AdhesiveBondB data where xj is a scalar,
Newtons
Newtons
Slope
Model Checking
Residual Plots
• Residuals versus fitted values.
60
DegreesC
65
70
.05
.50
.95
23 - 25
This is useful when
• Residuals versus accelerating variables.
• Residuals versus time of exposure.
• Residuals versus observation order.
observations are taken sequentially.
• Residual probability plot.
55
AdhesiveBondB Data
Destructive Degradation Residuals versus DegreesC
Resp:Log,Time:Square root,DegreesC:Arrhenius, Dist:Normal
AdhesiveBondB ADDT Data
Residuals Versus Temperature Conditions
4
2
0
-2
-4
-6
50
23 - 27
23 - 29
• There appears to be some evidence of nonconstant variance, but it is not systematic with temperature or times.
• The vertical line at 0 is the median life of the standardized distribution. Then approximately 50% of the residuals
should be below that line.
• The standardized residuals look approximately like a random
sample from a NOR(0, 1).
Some Comments on the AdhesiveBondB Residuals
Standardized Residuals
.002
.0005
.0001
.01
.2
.1
.05
.4
.6
.98
.95
.9
.8
.995
.999
.9999
Standardized Residuals
4
2
0
-2
-4
-6
3.2
3.4
3.6
Fitted Values
3.8
4.0
4.2
4.4
AdhesiveBondB Data
Destructive Degradation Residuals versus Fitted Values
Resp:Log,Time:Square root,DegreesC:Arrhenius, Dist:Normal
AdhesiveBondB ADDT Data
Residuals Versus Fitted Values
3.0
AdhesiveBondB ADDT Data
Sev Residual Probability Plot
-2
-1
Standardized Residuals
0
1
2
3
.95
.50
.05
23 - 28
23 - 26
4.6
AdhesiveBondB Data
Destructive Degradation Residual Probability Plot with 95% Simultaneous Confidence Bands
Resp:Log,Time:Square root,DegreesC:Arrhenius, Dist:Normal
Normal Probability Plot
-3
b
y−µ
b
σ
#
23 - 30
b x)τ , τ = h (t), x = h (AccVar),
b = βb0 + βb1 exp(β
where µ
a
t
2
b ,σ
and βb0, βb1, β
2 b are ML estimates of the corresponding parameters. When AccVar is a vector, ha is generally a different function for each of its elements.
FbY (y; τ, x) = Φ
• The ML estimate of the degradation distribution for given
(t, x) is
• The p quantiles of the distribution is yp = µ(t, x, β )+σΦ−1(p).
y − µ(τ, x, β )
FY (y; τ, x) = P (Y ≤ y; τ, x) = Φ
σ
′ x)τ.
where y = hd(degradation), µ(τ, x, β ) = β0 + β1 exp(β 2
"
• The degradation distribution, for given time t and AccVar
vector x is
Distribution of Degradation at (Possibly Transformed)
Time and AccVar (t, AccVar)
Probability
AdhesiveBondB ADDT Data
Distribution of Degradation at
Given Time and Temperature (t, Temp)
• For the AdhesiveBondB data, the Normal model ML esti
mate of FY (y; τ, x) at time t and temperature ◦C is
−1 (p).
b+σ
b Φnor
ybp = µ
100
200
Weeks
500
1%
0.1%
1000
23 - 33
23 - 31
b
y−µ
FbY (y; τ, x) = Φnor
b
σ
√
b = βb0+βb1 exp(βb2x)τ , τ = Weeks, x = −11605/(◦C+
where µ
273.15), βb0 = 4.471, βb1 = −864064160, βb2 = 0.6364,
b = 0.1580
σ
20
ML Estimate Showing Proportion Failing as
a Function of Time at 25◦C
−1
b Φnor
ybp = βb0 + βb1 exp(βb2x)τ + σ
(p)
0
• For p ≥ Φ [(β0 − µf )/σ]
i
h
tbp = ht−1 νb + ςbΦ−1(p)
t̃p] = [tbp/w,
tbp × w]
Zlog(bt ) = [log(tbp) − log(tp)]/scelog(bt ) is
p
p
e
[tp,
p
where w = exp[z(1−α/2)scebt /tbp].
23 - 35
• An approximate 100(1−α)% confidence interval for tp based
on the large-sample approximate NOR(0, 1) distribution of
where ht−1(·) is the inverse of the ht(·) function.
ML Estimates for Quantiles of the Induced Failure
Time Distribution at AccVar x and Critical
Degradation Df
Decreasing Linear Degradation
Newtons
• The ML estimate of the p quantile (log Newtons) is
100
50
Induced Failure Time Distribution at
Fixed Values of (AccVar, Df )
for Decreasing Linear Degradation
"
ς=
for t ≥ 0
and
#
′ x)
σ exp(−β 2
.
| β1 |
µf − µ(τ, x, β )
σ
• Observe that T ≤ t [i.e., ht(T ) ≤ τ ] is equivalent to observed
degradation less than Df (i.e., Y ≤ µf ,) where µf = hd(Df ).
Then
τ −ν
ς
FT (t; x, β ) = FY (µf ; x, β ) = Φ
= Φ
′ x)
(β0 − µf ) exp(−β 2
| β1 |
where τ = ht(t),
ν=
• The failure time distribution is a mixed distribution with a
spike of Pr(T = 0) = Φ [(β0 − µf )/σ] at t = 0.
23 - 32
For t > 0 the cdf is continuous and it agrees with the cdf
of a log-location-scale variable with standardized cdf Φ(z),
location ν and scale ς.
Quantiles for the Induced Failure Time Distribution at
AccVar vector x and Critical Condition Df
Decreasing Linear Degradation
h
′ x)
(β0 − µf ) exp(−β 2
| β1 |
and
i
ς=
′ x)
σ exp(−β 2
.
| β1 |
ht(tp) = τp = ν + ςΦ−1(p)
• For p ≥ Φ [(β0 − µf )/σ]
where
ν=
Thus
tp = ht−1 ν + ςΦ−1(p) .
"
(β0 − µf ) + σΦ−1(p)
.
| β1 |
#
• Substituting the expressions for ν and ς and after simplifications, shows that
′ x + log
log[ht(tp)] = log(τp) = −β 2
| βb1 |
#
.
#
√
t.
23 - 34
Thus, the log of the transformed failure-time distribution
quantiles are linear in the transformed AccVar vector x.
AdhesiveBondB ADDT Data
Failure Time Distribution Quantile Estimates
"
"
b Φ−1(p)
(βb0 − µf ) + σ
• For the AdhesiveBondB accelerated test data, ht(t) =
Then
q
log[ tbp] = −βb2x + log
This implies that
b Φ−1(p)
(βb0 − µf ) + σ
log(tbp) = −2βb2x + 2 log
.
| βb1 |
Thus the logs of the quantiles of the failure-time distribution
are linear in the AccVar vector x.
23 - 36
50
60
70
10%
90%
50%
Model plot for AdhesiveBondB data
Resp:log,Time:Square root,DegreesC:arrhenius, Dist:normal
Failure−time distribution for degradation failure level of 40 Newtons
40
80
23 - 37
AdhesiveBondB Data
Model Plot ML Estimate of Failure Time Distribution
as Functionn of Temperature
o
b Φ−1(p)]/ | βb1 |
log(tbp) = −2βb2x + 2 log [(βb0 − µf ) + σ
10000
1000
100
30
Probability spike at time zero = 3.6473e−007
20
′ x )τ + ǫ
= β0 + β1 exp(β 2
j i
ijk
!#
!#
= β0 + β1 exp(β2xj2 + β3xj3)τi + ǫijk
23 - 41
Acceleration Factors
• Here we consider time power transformations, i.e., τ =
ht(t) = tκ, where κ 6= 0.
• To obtain the accelerating effect of the AccVar x, let τ (x)
and τ (xU ) be the (transformed) times to reach the critical
degradation Df when the (transformed) accelerating variable take values x and xU , respectively.
Df = D[τ (x), x, β ] = D[τ (xU ), xU , β ]
• Solving for τ (x) and τxU the equation
gives
ht[t(xU )]
τ (x U )
′ (x − x )].
=
= exp[β 2
U
τ (x )
ht[t(x)]
1.2728
30
35
40
45
50
55
60
23 - 40
23 - 38
Using τ (x) = ht[t(x)] = [t(x)]κ and solving for t(xU )/t(x)
yields
t(xU )
1 ′
AF (x) =
= exp β 2
(x − x U ) .
t(x)
κ
25
Arrhenius relationship with activation energy in units of eV,
AdhesiveBondB Data
Normal Model Acceleration Factors
as a Function of Temperature
200
100
50
20
10
5
2
1
Degrees C
AdhesiveBondD ADDT Data
• Experimental factors
◮ AccVar:
Temp (Temperature):
RH (Relative Humidity):
23 - 42
• Sample size: 6 units allocated at each (Weeks, RH, Temp)
combination. This gives a sample size equal
to: 6 × 8 × 2 × 3 = 288 units.
◮ Weeks : Measurements after 0, 1, 2, 4, 6, 8, 12, 16 Weeks
50◦C, 60◦C, 70◦C
20%, 80%
• The test is destructive; each unit can be measured only
once.
There is interest in estimating the proportion of devices with
a strength below 25 Newtons after 5 years of operation at
room temperature of 25◦C and relative humidity of 50%.
• Objective: Assess the strength of an AdhesiveBond over
time.
Acceleration Factor relative to 25 Degrees C
10
1
DegreesC on Arrhenius Scale
AdhesiveBondB ADDT Data
ML Estimates of Acceleration Factors
"
AF (temp) = exp 2 β2
"
11605
11605
−
tempU K temp K
• The Normal model ML estimate are
d (temp) = exp 2 βb
AF
2
where βb2 = 0.6364.
• The ML estimate of the Arrhenius activation energy
is 2 × βb2 = 2 × 0.6364 = 1.2728
yijk
yijk = log(Newtonsijk )
p
√
ti = Weeksi
τi =
(ǫijk /σ) ∼ Φnor (z).
11605
xj2 = − ◦
Cj + 273.15
RH
xj3 = log
.
1 − RH
• The AccVar transformations are Arrhenius for Temp (Temperature) and logit for RH (Relative Humidity), i.e.,
where
• AdhesiveBondD Model
A Two-Variables Linear Degradation Model
With Temperature and Relative Humidity as AccVar
d (temp) is done easily using SPLIDA.
• The computation of AF
d (60) = 182.12
For example, AF
23 - 39
where temp K = temp ◦C + 273.15 is temperature in the absolute Kelvin scale.
11605
11605
−
tempU K temp K
√
• For the AdhesiveBondB, ht(t) = t, then κ = 1/2 and the
acceleration factor for temperature is
Weeks
AdhesiveBondD ADDT Data
Structure of the Data
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
23 - 43
48
48
48
48
48
48
288
Totals
The test plan is a completely balanced 8 × 3 × 2 full factorial arrangement in Weeks, Temp, and RH, with 6 units
allocated at each combination of the experimental factors.
0
6
6
6
6
6
6
36
Temp
◦C
6
6
6
6
6
6
36
RH
%
20
50
60
70
80
50
60
70
Totals
Another Linear Degradation Model for
an Insulation Breakdown
yijk = β0 + β1 exp(β2xj )τi + ǫijk
Nelson 1990, Chapter 11, discusses the following linear
degradation model for time to an insulation Breakdown.
where
yijk = log[(Breakdown Voltage)ijk ]
τi = ti = Weeksi
xj = −11605/(◦Cj + 273.15)
(ǫijk /σ) ∼ Φnor (z).
Notice that in this instance the variable time is used in the
linear scale.
23 - 45
Future Research in ADDT Data, Models, and Analysis
• Nonlinear degradation models.
• Coarse data.
• Stochastic variability in the degradation response.
• Prediction in non-constant environments.
• Use of prior information.
• Random initiation times.
23 - 47
AdhesiveBondD ADDT Data
Acceleration Factors
t(xU )
1 ′
= exp β 2
(x − x U )
t(x)
κ
√
• For the AdhesiveBondD, ht(t) = t, then acceleration factor for temperature and relative humidity is
AF (x) =
(
!
where κ = 1/2 because the square root transformation on
time. Then
11605
11605
−
+
AF (temp, RH) = exp 2 β2
temp K temp K
U
"
!#)
RH
RHU
− log
1 − RH
1 − RHU
2 β3 log
where tempU K = 25◦C + 273.15 and RHU = 50% are the
AccVar at use conditions.
23 - 44
• The estimation of AF (temp, RH) will be programmed in SPLIDA.
ADT Models with Interactions
• Most accelerated test models used in practice do not contain interaction terms.
• Interaction terms imply curvature in the degradation rate
versus the AccVar surface, i.e.,
µij = β0 + β1 exp(β2xj2 + β3xj3 + β4xj2xj3) τi.
• Extrapolation is hazardous, especially with surfaces involving curvature.
• A physical model could suggest interactions.
23 - 46
• Current version of the SPLIDA software does not allow for
interactions.
Technical Details
The following slides give technical details used in SPLIDA
to implement the methodology.
23 - 48
A Reparameterization of the
Linear ADDT Model for Numerical Stability
• The model is as before
′ x )τ + ǫ .
yij = β0 + β1 exp(β 2
j i
ij
exp
xj − x̄ γ 2 τi − τ̄ + ǫij .
′
• 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.
′ x)
σ exp(−β 2
for t ≥ 0
ς=
| β1 |
.
23 - 49
• It can be shown that γ1 is the geometric mean of the slopes
′ x ), for the values of x stress variables in the data
β1 exp(β 2
j
j
set.
and
τ −ν
ς
Induced Failure Time Distribution at
Use Conditions x and Critical Level Df
Decreasing Linear Degradation
FT (t, x, β ) = Φ
′ x)
(β0 − µf ) exp(−β 2
• In this case
where
ν=
| β1 |
• The failure time distribution is a mixed distribution with a
spike of Φ [(µf − β0)/σ] = Φ (− ν/ς) at t = 0.
For t > 0 the cdf is continuous and it agrees with the cdf of a
location-scale variable with standardized cdf Φ(z), location
ν and scale ς.
23 - 51
Density to Plot Induced Failure Time Distribution at
Use Conditions x and Critical Level Df
Decreasing Linear Degradation
dt t=exp(w)
dτ × ς=
for − ∞ < w < ∞
′ x)
σ exp(−β 2
.
| β1 |
if square-root transformation on time
τ −ν
exp(w)
×φ
ς
ς
Let W = log(T ), one needs to plot fW (w) which is giving
by
fW (w, β ) =
where
1

1
2τ


and
τ = 
ht[exp(w)]
if not transformation on time
=
′ x)
(β0 − µf ) exp(−β 2
| β1 |
dt t=exp(w)
dτ and
ν=
23 - 53
"
t > 0.
#
23 - 50
Relationship Between Stable and Original Parameters
• Then
′ 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 .
τ −ν
,
ς
y − µ(τ, x, β )
σ
Induced Failure Time Distribution at
Use Conditions x and Critical Level Df
Decreasing Linear Degradation (Continued)
Y = hd(degradation) ∼ Φ
• In particular, the following follows:
and
ht(T ) ∼ Φ
ς=
for t ≥ 0
′ x)
σ exp(−β 2
.
| β1 |
23 - 52
• This implies that ht(T ) is location-scale distributed with
parameters (ν, ς) and standardized distribution Φ(·).
and
−τ − ν
ς
Induced Failure Time Distribution at
Use Conditions x and Critical Level Df
Increasing Linear Degradation
• In this case
FT (t, x, β ) = 1 − Φ
′ x)
(β0 − µf ) exp(−β 2
| β1 |
where as before
ν=
• The failure time distribution is a mixed distribution with a
spike of 1 − Φ [(µf − β0)/σ] = 1 − Φ (− ν/ς) at t = 0.
For t > 0 the cdf is continuous and it agrees with the cdf
of continuous random variable.
23 - 54
"
t > 0.
y − µ(τ, x, β )
σ
#
Induced Failure Time Distribution at
Use Conditions x and Critical Level Df
Increasing Linear Degradation
Y = hd(degradation) ∼ Φ
• In particular the following follows:
and
−τ − ν
,
−ht(T ) ∼ Φ
ς
23 - 55
• This implies that that −ht(T ) is location-scale distributed
with parameters (ν, ς) and standardized distribution Φ(·).
Quantiles of the Failure Time Distribution



0
h
i

 h−1 ν + ςΦ−1(p)
t
ν
if p ≤ Φ −
ς
otherwise
• For decreasing degradation, the quantiles of the failure time
distribution are
tp =



0
n h
io
 −1

ht
− ν + ςΦ−1(1 − p)
otherwise
if p ≤ 1 − Φ −
ν
ς
• For increasing degradation, the quantiles of the failure time
distribution are
tp =
23 - 57
Density to Plot Induced Failure Time Distribution at
Use Conditions x and Critical Level Df
Increasing Linear Degradation
−τ − ν
exp(w)
×φ
ς
ς
× Let W = log(T ), you need to plot fW (w) which is giving by
dτ for − ∞ < w < ∞
dt t=exp(w)
fW (w, β ) =
where
1
2τ
and
ς=
′ x)
σ exp(−β 2
.
| β1 |
τ = (
ht[exp(w)]
1
if not transformation on time
if square-root transformation on time
=
| β1 |
′ x)
(β0 − µf ) exp(−β 2
dt t=exp(w)
dτ and
ν=
23 - 56