# Chapter 23 Accelerated Destructive Degradation Test Data, Models, and Analysis ```Chapter 23
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 &amp; Sons Inc., 1998.
Work done jointly with Danny L. Kugler and Laura L. Kramer (Imaging
&amp; Printing Group, Hewlett-Packard)
January 13, 2014
3h 42min
23 - 1
Data, Models, and Data Analysis
Chapter 23 Objectives
reliability models.
• Show the connection between degradation reliability models
and failure-time reliability models.
• Present methods of data analysis and reliability inference
• Discuss the use of ADDT data to estimate acceleration
factors.
• Illustrate methods with examples.
23 - 2
• 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.
• 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
23 - 3
DegreesC
Number of Units Measured at Each
(Time, Temperature) Combination
70
6
60
50
25
6
4
9
6
6
6
6
8
8
8
7
6
12
16
8
0
2
4
Weeks
23 - 4
Scatter Plot
Linear–Linear Axes
Resp:Linear,Time:Linear
100
Newtons
80
60
40
50DegreesC
60DegreesC
70DegreesC
20
0
0
5
10
15
20
Weeks
23 - 5
Scatter Plot at Individual Conditions of Temperature
Linear–Linear Axes
Resp:Linear,Time:Linear, Dist:Normal
50DegreesC
60DegreesC
100
100
80
80
60
60
40
40
20
20
0
0
5
10
15
20
15
20
0
5
10
15
20
Newtons
0
70DegreesC
100
80
60
40
20
0
0
5
10
Weeks
23 - 6
Scatter Plot at Individual Conditions of Temperature
Square Root–Log Axes
Resp:Log,Time:Square root, Dist:Normal
50DegreesC
60DegreesC
80
40
20
10
80
40
20
10
5
10 15
0
5
10 15
Newtons
0
70DegreesC
80
40
20
10
0
5
10 15
Weeks
23 - 7
• 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.
• Though no commonly used in accelerated testing it is possible to have models with interactions among the accelerating
variables.
23 - 8
• Sample path model: For observation k at time τi and
accelerating variable(s) level xj the model is
yijk = hd(Dij ) + ǫijk
= &micro;ij + ǫijk , k = 1, . . . , nij
where &micro;ij = 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).
• The transformations for the observed degradation, Dij , ti,,
and the AccVarj might be suggested by physical/chemical
theory, past experience, or the data.
23 - 9
Degradation Path Model: Single Temperature Level
D(τ, x, β ) = exp[β0 + β1 exp(β2x)τ ]
√
τ = Weeks and x is Arrhenius-Transformed Temp
Linear–Linear Axes
100
70 DegreesC
Newtons
80
60
40
20
0
0
10
20
30
40
50
60
Weeks
23 - 10
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
70 DegreesC
60 DegreesC
50 DegreesC
Newtons
80
60
40
20
0
0
10
20
30
40
50
60
Weeks
23 - 11
Using Transformations
log[D(τ, x, β )] = β0 + β1 exp(β2x)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
23 - 12
A Class of Linear Degradation Models
• These models are of the form
yijk = &micro;ij + ǫijk
= β0 + β1 exp(β ′2xj )τi + ǫijk ,
k = 1, . . . , nij
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 single (scalar) AccVar xj , β ′2xj = β2xj .
◮ For multiple AccVar, xj , xj = (x2, . . . , xp)′, 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 + &middot; &middot; &middot; + βpxjp.
23 - 13
Interpretation of the Parameters
yijk = β0 + β1 exp(β ′2xj )τi + ǫijk
• β0 is path intercept parameter when τ = 0. For example,
◮ If τi =
√
ti then β0 is the intercept at time t = 0.
◮ If τi = log(ti) then β0 is intercept at time t = 1.
• The degradation rate at xj is υ(xj )(AccVarj ) = β1 exp(β ′2xj ).
• The sign (&plusmn;) of β1 determines if the degradation is increasing or decreasing in time.
• 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.
23 - 14
as a function of time and temperature is modeled by
yijk = β0 + β1 exp(β2xj )τi + ǫijk
where
yijk = log(Newtonsijk )
p
√
ti = Weeksi
τi =
xj = −11605/(◦Cj + 273.15)
(ǫijk /σ) ∼ Φnor (z).
The ǫijk term contains model and measurement errors.
23 - 15
Likelihood for Fixed AccVar level xj
• For the data at fixed xj of the AccVar with exact failure
times and right-censored observations, the likelihood is
nij Y Y
1
yijk − &micro;ij 1−δijk
yijk − &micro;ij δijk &times; 1−Φ
φ
Lj (θ |DATA) =
σ
σ
i k=1 σ
where &micro;ij ≡ &micro;(τi, xj , β ) = β0 + β1 exp(β ′2xj )τi, δijk indicates
whether observation yijk is a failure (δijk = 1) or a right
censored observation (δijk = 0), θ = (β0, β1, β ′2, σ), and nij
is the number of observations at (τi, xj ).
• For fixed xj , the model parameters are: the spread σ, the
intercept β0, and the slope of the line υj = β1 exp(β ′2xj ).
The parameter υj can be interpreted as the degradation
rate of &micro;(τ, xj , β ) with respect to the transformed time τ.
23 - 16
Normal Model ML Individual Line Fits
• For each temperature level three individual ML estimates
[j]
b [j].
are obtained: βb0 , υb [j], and σ
• A summary of the Normal model individual ML fits for the
ML Estimates
AccVarj
[j]
βb0
50◦C
4.490
60◦C
4.489
70◦C
4.400
υb [j]
−0.1088
−0.2089
−0.3626
sceυb[j]
0.01494
0.02214
0.01944
95% Approximate
Confidence Interval
for υ [j]
Lower
Upper
−0.1424
−0.08309
−0.2571
−0.4028
−0.16969
−0.32643
23 - 17
Individual Normal Model ML Fits
Square Root–Log Axes
[j]
b [j]τ
b [j] = βb0 + υ
&micro;
Resp:Log,Time:Square root, Dist:Normal
50DegreesC
60DegreesC
80
40
20
10
80
40
20
10
5
10 15
0
5
10 15
Newtons
0
70DegreesC
80
40
20
10
0
5
10 15
Weeks
23 - 18
Overlay of Individual Normal Model Fits
Square Root–Log Axes
[j]
b [j]τ
b [j] = βb0 + υ
&micro;
Resp:Log,Time:Square root, Dist:Normal
100
80
Newtons
60
50
40
30
20
50DegreesC
60DegreesC
70DegreesC
10
0
5
10
15
20
Weeks
23 - 19
• 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
• Because
log(| υj |) = log(| β1 |) + β ′2xj
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.
◮ 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.
23 - 20
Arrhenius Plot of |υb [j]|
Normal Model ML Estimates
Degradation rate versus DegreesC on Arrhenius Scale for
Resp:Log,Time:Square root,x:arrhenius, Dist:Normal
0.40
0.30
0.25
Slope
0.20
0.15
0.10
0.05
45
50
55
60
65
70
75
DegreesC on Arrhenius Scale
23 - 21
Likelihood for All Data With Right Censored Data
• For a sample of n units consisting of exact failure times and
right-censored observations, the likelihood can be expressed
as
L(θ |DATA) =
L(θ |DATA) =
Y
j
Lj (θ |DATA)
Y 1
ijk
σ
φ
yijk − &micro;ij δijk
σ
&times; 1−Φ
yijk − &micro;ij 1−δijk
σ
where θ = (β0, β1, β ′2, σ)′, and δijk indicates whether observation ijk is a failure (δijk = 1) or a right censored observation (δijk = 0).
• Note that &micro;ij is a nonlinear function of the accelerating
variable xj .
• For the AdhesiveBondB data where xj is a scalar,
&micro;ij = β0 + β1 exp(β2xj )τi
11605
xj = − ◦
.
Cj + 273.15
23 - 22
for the Acceleration Model Fit
• Parameter estimates
Parameter
β0
β1
β2
σ
95% Approximate
Confidence Interval
Lower
Upper
4.396
4.547
ML
Estimate
4.471
Standard
Error
0.03864
−8.641 &times; 108
1.595 &times; 109
−3.989 &times; 109
2.261 &times; 109
0.1580
0.01233
0.1356
0.1841
0.6364
0.05488
0.5375
0.7536
• 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,
b
υ(50)
= −0.1025
b
υ(70)
= −0.3883
23 - 23
Normal Model Arrhenius ML Fit
b ij = βb0 + βb1 exp(βb2xj )τi
&micro;
Resp:Log,Time:Square root,DegreesC:Arrhenius, Dist:Normal
100
80
Newtons
60
50
40
30
20
50DegreesC
60DegreesC
70DegreesC
25DegreesC
10
0
5
10
15
20
Weeks
23 - 24
Model Checking
Residual Plots
• Residuals versus fitted values.
• Residuals versus accelerating variables.
• Residuals versus time of exposure.
• Residuals versus observation order.
observations are taken sequentially.
This is useful when
• Residual probability plot.
23 - 25
Residuals Versus Fitted Values
Destructive Degradation Residuals versus Fitted Values
Resp:Log,Time:Square root,DegreesC:Arrhenius, Dist:Normal
4
Standardized Residuals
2
.95
0
.50
.05
-2
-4
-6
3.0
3.2
3.4
3.6
3.8
4.0
4.2
4.4
4.6
Fitted Values
23 - 26
Residuals Versus Temperature Conditions
Resp:Log,Time:Square root,DegreesC:Arrhenius, Dist:Normal
4
Standardized Residuals
2
.95
0
.50
.05
-2
-4
-6
50
55
60
65
70
DegreesC
23 - 27
Sev Residual Probability Plot
Destructive Degradation Residual Probability Plot with 95% Simultaneous Confidence Bands
Resp:Log,Time:Square root,DegreesC:Arrhenius, Dist:Normal
Normal Probability Plot
.9999
.999
Probability
.995
.98
.95
.9
.8
.6
.4
.2
.1
.05
.01
.002
.0005
.0001
-3
-2
-1
0
1
2
3
Standardized Residuals
23 - 28
• The standardized residuals look approximately like a random
sample from a NOR(0, 1).
• The vertical line at 0 is the median life of the standardized distribution. Then approximately 50% of the residuals
should be below that line.
• There appears to be some evidence of nonconstant variance, but it is not systematic with temperature or times.
23 - 29
Distribution of Degradation at (Possibly Transformed)
Time and AccVar (t, AccVar)
• The degradation distribution, for given time t and AccVar
vector x is
FY (y; τ, x) = P (Y ≤ y; τ, x) = Φ
&quot;
y − &micro;(τ, x, β )
σ
#
where y = hd(degradation), &micro;(τ, x, β ) = β0 + β1 exp(β ′2x)τ.
• The p quantiles of the distribution is yp = &micro;(t, x, β )+σΦ−1(p).
• The ML estimate of the degradation distribution for given
(t, x) is
b
y−&micro;
b
FY (y; τ, x) = Φ
b
σ
b x)τ , τ = h (t), x = h (AccVar),
b = βb0 + βb1 exp(β
where &micro;
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.
23 - 30
Given Time and Temperature (t, Temp)
• For the AdhesiveBondB data, the Normal model ML estimate of FY (y; τ, x) at time t and temperature ◦C is
b
y−&micro;
FbY (y; τ, x) = Φnor
b
σ
√
b
b
b
b = β0+β1 exp(β2x)τ , τ = Weeks, x = −11605/(◦C+
where &micro;
273.15), βb0 = 4.471, βb1 = −864064160, βb2 = 0.6364,
b = 0.1580
σ
• The ML estimate of the p quantile (log Newtons) is
b+σ
b Φ−1
ybp = &micro;
nor (p).
23 - 31
Induced Failure Time Distribution at
Fixed Values of (AccVar, Df )
• Observe that T ≤ t [i.e., ht(T ) ≤ τ ] is equivalent to observed
degradation less than Df (i.e., Y ≤ &micro;f ,) where &micro;f = hd(Df ).
Then
FT (t; x, β ) = FY (&micro;f ; x, β ) = Φ
τ −ν
= Φ
ς
&quot;
&micro;f − &micro;(τ, x, β )
σ
#
for t ≥ 0
where τ = ht(t),
(β0 − &micro;f ) exp(−β ′2x)
ν=
| β1 |
and
σ exp(−β ′2x)
ς=
.
| β1 |
• The failure time distribution is a mixed distribution with a
spike of Pr(T = 0) = Φ [(β0 − &micro;f )/σ ] at t = 0.
For t &gt; 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.
23 - 32
50
Newtons
100
ML Estimate Showing Proportion Failing as
a Function of Time at 25◦C
b Φ−1
ybp = βb0 + βb1 exp(βb2x)τ + σ
nor (p)
1%
0.1%
0
20
100
200
500
1000
Weeks
23 - 33
Quantiles for the Induced Failure Time Distribution at
AccVar vector x and Critical Condition Df
• For p ≥ Φ [(β0 − &micro;f )/σ ]
ht(tp) = τp = ν + ςΦ−1(p)
where
(β0 − &micro;f ) exp(−β ′2x)
ν=
| β1 |
Thus
σ exp(−β ′2x)
ς=
.
| β1 |
and
h
i
−1
−1
t p = ht
ν + ςΦ (p) .
• Substituting the expressions for ν and ς and after simplifications, shows that
log[ht(tp)] = log(τp) = −β ′2x + log
&quot;
(β0 − &micro;f ) + σΦ−1(p)
| β1 |
#
.
Thus, the log of the transformed failure-time distribution
quantiles are linear in the transformed AccVar vector x.
23 - 34
ML Estimates for Quantiles of the Induced Failure
Time Distribution at AccVar x and Critical
• For p ≥ Φ [(β0 − &micro;f )/σ ]
h
i
−1
−1
tbp = ht
νb + ςbΦ (p)
where h−1
t (&middot;) is the inverse of the ht (&middot;) function.
• An approximate 100(1−α)% confidence interval for tp based
on the large-sample approximate NOR(0, 1) distribution of
Zlog(bt ) = [log(tbp) − log(tp)]/scelog(bt ) is
p
p
[tp,
e
t̃p] = [tbp/w,
where w = exp[z(1−α/2)scebt /tbp].
tbp &times; w]
p
23 - 35
Failure Time Distribution Quantile Estimates
• For the AdhesiveBondB accelerated test data, ht(t) =
Then
log[
q
√
t.
#
−1
b
b Φ (p)
(β 0 − &micro; f ) + σ
.
tbp] = −βb2x + log
b
| β1 |
&quot;
This implies that
#
−1
b
b
(β0 − &micro;f ) + σ Φ (p)
.
log(tbp) = −2βb2x + 2 log
b
| β1 |
&quot;
Thus the logs of the quantiles of the failure-time distribution
are linear in the AccVar vector x.
23 - 36
Model Plot ML Estimate of Failure Time Distribution
as Functionn of Temperature
o
−1
b
b
b
b
b Φ (p)]/ | β1 |
log(tp) = −2β2x + 2 log [(β0 − &micro;f ) + σ
Resp:log,Time:Square root,DegreesC:arrhenius, Dist:normal
Failure−time distribution for degradation failure level of 40 Newtons
10000
Weeks
1000
100
10
90%
50%
10%
1
Probability spike at time zero = 3.6473e−007
20
30
40
50
60
70
80
DegreesC on Arrhenius Scale
23 - 37
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.
• Solving for τ (x) and τxU the equation
Df = D[τ (x), x, β ] = D[τ (xU ), xU , β ]
gives
τ (x U )
ht[t(xU )]
=
= exp[β ′2(x − xU )].
τ (x )
ht[t(x)]
Using τ (x) = ht[t(x)] = [t(x)]κ and solving for t(xU )/t(x)
yields
1
t(xU )
= exp β ′2(x − xU ) .
AF (x) =
t(x)
κ
23 - 38
ML Estimates of Acceleration Factors
√
• For the AdhesiveBondB, ht(t) = t, then κ = 1/2 and the
acceleration factor for temperature is
&quot;
AF (temp) = exp 2 β2
11605
11605
−
tempU K temp K
!#
where temp K = temp ◦C + 273.15 is temperature in the absolute Kelvin scale.
• The Normal model ML estimate are
&quot;
d (temp) = exp 2 βb
AF
2
11605
11605
−
tempU K temp K
!#
where βb2 = 0.6364.
• The ML estimate of the Arrhenius activation energy
is 2 &times; βb2 = 2 &times; 0.6364 = 1.2728
d (temp) is done easily using SPLIDA.
• The computation of AF
d (60) = 182.12
For example, AF
23 - 39
Normal Model Acceleration Factors
as a Function of Temperature
Acceleration Factor relative to 25 Degrees C
Arrhenius relationship with activation energy in units of eV,
200
1.2728
100
50
20
10
5
2
1
25
30
35
40
45
50
55
60
Degrees C
23 - 40
With Temperature and Relative Humidity as AccVar
yijk = β0 + β1 exp(β ′2xj )τi + ǫijk
= β0 + β1 exp(β2xj2 + β3xj3)τi + ǫijk
where
yijk = log(Newtonsijk )
p
√
τi =
ti = Weeksi
(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
RH
xj3 = log
.
1 − RH
23 - 41
• Objective: Assess the strength of an AdhesiveBond over
time.
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%.
• The test is destructive; each unit can be measured only
once.
• Experimental factors
◮ AccVar:
Temp (Temperature):
RH (Relative Humidity):
50◦C, 60◦C, 70◦C
20%, 80%
◮ Weeks : Measurements after 0, 1, 2, 4, 6, 8, 12, 16 Weeks
• Sample size: 6 units allocated at each (Weeks, RH, Temp)
combination. This gives a sample size equal
to: 6 &times; 8 &times; 2 &times; 3 = 288 units.
23 - 42
Structure of the Data
The test plan is a completely balanced 8 &times; 3 &times; 2 full factorial arrangement in Weeks, Temp, and RH, with 6 units
allocated at each combination of the experimental factors.
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
23 - 43
Acceleration Factors
√
• For the AdhesiveBondD, ht(t) = t, then acceleration factor for temperature and relative humidity is
t(xU )
1
AF (x) =
= exp β ′2(x − xU )
t(x)
κ
where κ = 1/2 because the square root transformation on
time. Then
(
!
11605
11605
AF (temp, RH) = exp 2 β2
−
+
tempU K temp K
!#)
&quot;
RH
RHU
− log
2 β3 log
1 − RH
1 − RHU
where tempU K = 25◦C + 273.15 and RHU = 50% are the
AccVar at use conditions.
• The estimation of AF (temp, RH) will be programmed in SPLIDA.
23 - 44
an Insulation Breakdown
Nelson 1990, Chapter 11, discusses the following linear
degradation model for time to an insulation Breakdown.
yijk = β0 + β1 exp(β2xj )τi + ǫijk
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
• 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.,
&micro;ij = β0 + β1 exp(β2xj2 + β3xj3 + β4xj2xj3) τi.
• Extrapolation is hazardous, especially with surfaces involving curvature.
• A physical model could suggest interactions.
• Current version of the SPLIDA software does not allow for
interactions.
23 - 46
Future Research in ADDT Data, Models, and Analysis
• Coarse data.
• Stochastic variability in the degradation response.
• Prediction in non-constant environments.
• Use of prior information.
• Random initiation times.
23 - 47
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
yij = β0 + β1 exp(β ′2xj )τi + 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
exp
′
xj − x̄ γ 2 τi − τ̄ + ǫij .
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.
23 - 49
Relationship Between Stable and Original Parameters
• Then
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 .
23 - 50
Induced Failure Time Distribution at
Use Conditions x and Critical Level Df
• In this case
τ −ν
FT (t, x, β ) = Φ
ς
for t ≥ 0
where
(β0 − &micro;f ) exp(−β ′2x)
ν=
| β1 |
and
σ exp(−β ′2x)
ς=
.
| β1 |
• The failure time distribution is a mixed distribution with a
spike of Φ [(&micro;f − β0)/σ ] = Φ (− ν/ς ) at t = 0.
For t &gt; 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
Induced Failure Time Distribution at
Use Conditions x and Critical Level Df
• In particular, the following follows:
and
τ −ν
ht(T ) ∼ Φ
,
ς
&quot;
y − &micro;(τ, x, β )
σ
#
t &gt; 0.
• This implies that ht(T ) is location-scale distributed with
parameters (ν, ς) and standardized distribution Φ(&middot;).
23 - 52
Density to Plot Induced Failure Time Distribution at
Use Conditions x and Critical Level Df
Let W = log(T ), one needs to plot fW (w) which is giving
by
exp(w)
τ −ν
fW (w, β ) =
&times;φ
ς
ς
where
dτ &times; dt t=exp(w)
for − ∞ &lt; w &lt; ∞
τ = 
ht[exp(w)]

1
if not transformation on time
dτ =
1

dt t=exp(w)
if square-root transformation on time

2τ
and
(β0 − &micro;f ) exp(−β ′2x)
ν=
| β1 |
and
σ exp(−β ′2x)
ς=
.
| β1 |
23 - 53
Induced Failure Time Distribution at
Use Conditions x and Critical Level Df
• In this case
−τ − ν
FT (t, x, β ) = 1 − Φ
ς
where as before
(β0 − &micro;f ) exp(−β ′2x)
ν=
| β1 |
and
for t ≥ 0
σ exp(−β ′2x)
ς=
.
| β1 |
• The failure time distribution is a mixed distribution with a
spike of 1 − Φ [(&micro;f − β0)/σ ] = 1 − Φ (− ν/ς ) at t = 0.
For t &gt; 0 the cdf is continuous and it agrees with the cdf
of continuous random variable.
23 - 54
Induced Failure Time Distribution at
Use Conditions x and Critical Level Df
• In particular the following follows:
&quot;
y − &micro;(τ, x, β )
σ
and
−τ − ν
−ht(T ) ∼ Φ
,
ς
#
t &gt; 0.
• This implies that that −ht(T ) is location-scale distributed
with parameters (ν, ς) and standardized distribution Φ(&middot;).
23 - 55
Density to Plot Induced Failure Time Distribution at
Use Conditions x and Critical Level Df
Let W = log(T ), you need to plot fW (w) which is giving by
exp(w)
−τ − ν
fW (w, β ) =
&times;φ
ς
ς
where
dτ &times; dt t=exp(w)
for − ∞ &lt; w &lt; ∞
τ = (
ht[exp(w)]
dτ 1
if not transformation on time
=
1
if square-root transformation on time
dt t=exp(w)
2τ
and
(β0 − &micro;f ) exp(−β ′2x)
ν=
| β1 |
and
σ exp(−β ′2x)
ς=
.
| β1 |
23 - 56
Quantiles of the Failure Time Distribution
• For decreasing degradation, the quantiles of the failure time
distribution are

ν

 0
if p ≤ Φ −
ς
tp =
h
i

 h−1 ν + ςΦ−1(p)
otherwise
t
• For increasing degradation, the quantiles of the failure time
distribution are

ν

 0
if p ≤ 1 − Φ −
ς
tp =
n h
io

 h−1 − ν + ςΦ−1(1 − p)
otherwise
t
23 - 57
```