Chapter 21
Analysis of Accelerated Degradation Data
William Q. Meeker and Luis A. Escobar
Iowa State University and Louisiana State University
21 - 1
Copyright 1998-2008 W. Q. Meeker and L. A. Escobar.
Based on the authors’ text Statistical Methods for Reliability
Data, John Wiley & Sons Inc. 1998.
December 14, 2015
8h 10min
Background
Today’s manufacturers face strong pressure to:
• Develop newer, higher technology products in record time.
• Improve productivity, product field reliability, and overall
quality.
• Increased need for up-front testing of materials, components and systems.
21 - 3
• Accelerated degradation tests can be useful for such upfront testing.
Advantages of Using Degradation Data
Over Failure-Time Data
• Degradation is natural response for some tests.
• Useful reliability inferences even with 0 failures.
• More justification and credibility for extrapolative acceleration models.
(Modeling closer to physics-of-failure)
• Can be more informative than failure-time data.
(Reduction to failure-time data loses information)
21 - 5
Chapter 21
Analysis of Accelerated Degradation Data
Objectives
• Show how accelerated degradation tests can be used to
assess and improve product reliability.
• Present models, methods of analysis, and methods of inference for accelerated degradation tests.
• Show how to analyze data from accelerated degradation
tests.
4000
6000
8000
173 Degrees C
133 Degrees C
83 Degrees C
21 - 4
10000
21 - 2
• Compare accelerated degradation test methods with traditional accelerated life test methods using failure-time data.
2000
Percent Increase in Resistance Over Time
for Carbon-Film Resistors
(Shiomi and Yanagisawa 1979)
10.0
5.0
1.0
0.5
0
Hours
Limitations of Degradation Data
• Degradation level may not correlate well with failure.
21 - 6
• Analyses more complicated; requires statistical methods not
yet widely available.
(Modern computing capabilities should help here)
• Substantial measurement error can diminish the information
in degradation data.
• Obtaining degradation data may have an effect on future
product degradation (e.g., taking apart a motor to measure
wear).
• Some degradation measurements are destructive (destructive degradation tests).
• Degradation data may be difficult or impossible to obtain.
Percent Increase
1000
Hours
2000
3000
Percent Increase in Operating Current
for GaAs Lasers Tested at 80◦C
0
Device-B Power Drop
Simple One-Step Chemical Reaction
Leading to Failure
4000
21 - 7
• A1(t) is the amount of harmful material available for reaction at time t
k
dA2
= k1 A 1
dt
8000
21 - 9
• A2(t) is proportional to the amount of failure-causing compounds at time t.
• Chemical reaction:
1
A1 −→
A2
• Power drop proportional to A2(t)
and
Device-B Power Drop
D(t) = D∞[1 − exp(−R t)]
Fixed D∞ and Rate R
dA1
= −k1A1
dt
• The rate equations for this reaction are
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237 Degrees C
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195 Degrees C
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Device-B Power Drop
Accelerated Degradation Test Results
at 150◦C, 195◦C, and 237◦C
(Use Conditions 80◦C)
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
-1.2
-1.4
Hours
Device-B Power Drop
Simple One-Step Chemical Reaction
Leading to Failure (continued)
• Solution to differential equations:
A1(t) = A1(0) exp(−k1t)
A2(t) = A2(0) + A1(0)[1 − exp(−k1t)]
where A1(0) and A2(0) are initial conditions.
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21 - 8
• If A2(0) = 0, then D∞ = limt→∞ A2(t) = A1(0) and the
solution for A2(t) (the function of primary interest) can be
reexpressed as
A2(t) = A1(t)[1 − exp(−k1t)]
D(t) = D∞ [1 − exp (−R t)]
where D(t) = A2(t) is the degradation at time t and R = k1
is the reaction rate.
-0.8
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-1.0
Hours
0
2000
Hours
4000
6000
Device-B Power Drop
D(t) = D∞[1 − exp(−R t)]
Variability in Asymptote D∞
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8000
21 - 10
• A simple 1-step diffusion process has the same solution.
0.0
2000
-0.2
-0.4
0.0
-0.6
Percent Increase in Operating Current
Power drop in dB
Power drop in dB
Power drop in dB
15
10
5
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-0.2
-0.4
-0.8
-0.6
-1.0
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-0.4
-0.6
-0.8
-1.0
2000
4000
6000
!
Device-B Power Drop
D(t) = D∞[1 − exp(−R t)]
Variability in Asymptote D∞ and Rate R
0
Hours
Acceleration of Degradation
−Ea
kB(temp + 273.15)
R(temp)
R(tempU )
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8000
• The Acceleration Factor between temp and tempU is
AF (temp) = AF (temp, tempU , Ea) =
When temp > tempU , AF (temp, tempU , Ea) > 1.
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-1.0
-1.5
0
4000
195 Degrees C
237 Degrees C
2000
Hours
150 Degrees C
80 Degrees C
6000
21 - 17
8000
Illustration of the Effect of Arrhenius Temperature
Dependence on the Degradation Caused by a
Single-Step Chemical Reaction
D(t; temp) = D∞ × {1 − exp [−{RU × AF (temp)} × t]}
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• The pre-exponential factor γ0 and the reaction activation
energy Ea are characteristics of the product or material.
◦ C.
where temp is temperature in ◦C and kB = 8.6 × 10−5 is
Boltzmann’s constant in units of electron volts (eV) per
R(temp) = γ0 exp
• The Arrhenius model describing the effect that temperature has on the rate of a simple one-step chemical reaction
is
Power drop in dB
Power drop in dB
Model for Degradation Data
Dij = D(tij , β1i, . . . , βki)
• Actual degradation path model: Actual path of unit ith
at time tij is
• Path parameters: β1i, . . . , βki may be random from unitto-unit or fixed in the population/process.
ǫij ∼ NID(0, σǫ2),
i = 1, . . . , n,
j = 1, . . . , mi.
• Sample path model: Sample degradation path of unit ith
at tij (the jth inspection time for unit i) is
yij = Dij +ǫij ,
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• Can use transformations on the response, time, or random
parameters, as suggested by physical/chemical theory, past
experience, or the data.
Arrhenius Model Temperature Effect
on Time to an Event
• Re-expressing the single-step chemical reaction degradation
path model to allow for acceleration:
D(t; temp) = D∞ × {1 − exp [−{RU × AF (temp)} × t]}
where RU is the rate reaction at tempU .
• Failure defined by D(t) < Df .
h
AF (temp)
f
− R1 log 1 − DD∞
U
i
=
T (tempU )
AF (temp)
• Equating D(T ; temp) to Df and solving for T gives the failure
time at temperature temp as
T (temp) =
21 - 16
• Thus the simple degradation process induces a Scale Accelerated Failure Time (SAFT) model.
Device-B Power Drop
Degradation Model and Parameters
• Basic parameters: RU = R(80), D∞, Ea.
• Estimation parameters:
β1 = log[R(195)], β2 = log(−D∞), and β3 = Ea.
• Assume that (β1, β2) follow a bivariate normal distribution.
• Assume that activation energy β3 = Ea is a fixed (but unknown) characteristic of Device-B.
• Variability in path model parameters: (β1, β2, β3) ∼ MVN(µβ , Σβ
[but Var(β3)=0].
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


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.15021 −.02918 0


Σ̂β = 
−.02918
.01809 0  ,
0
0 0
Device-B Power Drop Data
Approximate ML Estimates
(Computed with Program of Pinheiro and Bates 1995)

−7.572


µ̂β = 
.3510  ,
.6670
b ǫ = .0233,
σ
Loglikelihood = 1201.8.
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-6.5
Plot of β1 = log[R(195)] Versus β2 = log(−D∞)
for the i = 1, . . . , 34 Sample Paths from Device-B
ρbβ1β2 = −.56
0.6
0.4
-8.5
Beta1
150
Hours
5x10^3
195 Degrees C
5x10^2
100
5x10^4
80
5x10^5
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8000
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1000
Bootstrap Sample Estimates of F (t) at 80◦C
0
Device-B Power Drop
D(t) = D∞[1 − exp(−R t)]
Variability in Asymptote D∞ and rate R
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Device-B Power Drop Observations and Fitted
Degradation Model for the i = 1, . . . , 34 Sample Paths
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Power drop in dB
Power drop in dB
Proportion Failing
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21 - 23
Estimates of the Device-B Life Distributions at 80,
100, 150, and 190◦C, Based on the Degradation Data
Beta2
Proportion Failing
50
200
Thousands of Hours
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500
21 - 25
1000
80% and 90% Bias-Corrected Percentile Bootstrap
Confidence Intervals for F (t) at 80◦C
1.0
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140
160
Degrees C
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Scatterplot of Device-B Failure-Time Data with
Failure Defined as Power Drop Below −.5 dB
10000
5000
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100
20
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120
237 Degrees C
10^2
195
10^3
Hours
10^4
150 Degrees C
10^5
80 Degrees C
Lognormal-Arrhenius Model Fit to the Device-B
Failure-Time Data with Degradation Model Estimates
.99
.98
.9
.95
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.1
.05
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.01
.005
.001
10^1
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Device-B Power Drop
Accelerated Degradation Test Results
at 150◦C, 195◦C, and 237◦C
(Use Conditions 80◦C)
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237 Degrees C
10^1
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237 Degrees C
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195
10^3
500
1000
195 Degrees C
Hours
10^4
150 Degrees C
Hours
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150 Degrees C
2000
21 - 26
21 - 28
10000
80 Degrees C
10^5
21 - 30
Weibull-Arrhenius Model Fit to the Device-B
Failure-Time Data with Degradation Model Estimates
10
Lognormal-Arrhenius Model Fit to the Device-B
Failure-Time Data
Power drop in dB
Proportion Failing
Proportion Failing
Proportion Failing
Hours
Proportion Failing
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Plasma Concentrations of Indomethicin Following
Intravenous Injection
Fitted Biexponential Model
2.5
2.0
1.5
1.0
0.5
0.0
Hours
Approximate Accelerated Degradation Analysis
The simple method for degradation data analysis extends
directly to accelerated degradation analysis.
• For each sample path one uses the algorithm described to
predict the failure times.
• These data can be analyzed using the methods to analyze
ALT data.
21 - 33
• It is important to remember, however, that such an analysis
has the same limitations described in for the simple analysis
of degradation data.
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10 g
600
100 g
Scar Width Resulting from a Metal-to-Metal Sliding
Test for Different Applied Weights
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Theophylline Serum Concentrations
Fitted Curves for a First-Order Compartment Model
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8
6
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2
0
0
Sliding Metal Wear Data Analysis
21 - 34
• The predicted pseudo failure times were obtained by using
ordinary least squares to fit a line through each sample path
on the log-log scale and extrapolating to the time at which
the scar width would be 50 microns.
• The sliding test was conducted over a range of different
applied weights in order to study the effect of weight and
to gain a better understanding of the wear mechanism.
• An experiment was conducted to test the wear resistance
of a particular metal alloy.
Concentration (mg/L)
60
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Cycles
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21 - 36
6000
Metal-to-Metal Sliding Test for Different Applied
Weights
Extrapolation to Failure Definition
(Using linear regression on linear axes)
Microns
Concentration (mg/L)
Microns
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Scar Width Resulting from a Metal-to-Metal Sliding
Test for Different Applied Weights
(Using log-log Axes)
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21 - 38
8
Metal-to-Metal Sliding Test for Different Applied
Weights
Extrapolation to Failure Definition
(Using linear regression on log-log axes)
100
50
20
10
5
2
0
10
0
•
••
20
40
•
•
•
Grams
60
80
100
••
21 - 40
120
10%
50%
90%
Pseudo Failure Time to 50 Microns Scar Width Versus
Applied Weight for the Metal-to-Metal Sliding Test
10000
5000
2000
1000
500
200
.995
.99
.98
.9
.95
.8
.7
.6
.5
.4
.3
.2
.1
.05
.02
.01
.005
.002
.0005
1000
100 g
50 g
Time
2000
3000
10 g
4000
5000
21 - 42
6000
Lognormal Probability Plot Showing the Lognormal
Regression Model ML Estimates of Time to 50
Microns Width for Each Weight
Cycles
5
2
2
Cycles
Pseudo Failure Times
724
718
659
677
3216 1729 2234 1689
3981 4600 5718 4487
Metal-Wear Failure Times in Hours
Grams
100
50
10
100 g
50 g
Time
2000
Lognormal Probability Plot Showing the ML Estimates
of Time to 50 Microns Width for Each Weight
.995
.99
.98
.95
.9
.8
.7
.6
.5
.4
.3
.2
.1
.05
.02
.01
.005
.002
.0005
1000
21 - 41
Microns
Proportion Failing
Microns
Proportion Failing
Other Topics in Chapter 21
• Choice of parameter transformation in the estimation/bootstrap
procedure.
• Stochastic process degradation models.
Test planning case study in Chapter 22.
21 - 43