Chapter 19
Analyzing Accelerated Life Test Data
William Q. Meeker and Luis A. Escobar
Iowa State University and Louisiana State University
19 - 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
Example: Temperature-Accelerated Life Test on
Device-A (from Hooper and Amster 1990)
Data: Singly right censored observations from a temperatureaccelerated life test.
Purpose: To determine if the device would meet its hazard
function objective at 10,000 and 30,000 hours at operating
temperature of 10◦C.
200
500
1000
2000
5000
10000
20000
50000
0
0/30
20
9/20
60
Degrees C
40
10/100
80
14/15
Device-A Hours Versus Temperature
(Hooper and Amster 1990)
19 - 5
100
19 - 3
We will show how to fit an accelerated life regression model
to these data to answer this and other questions.
Hours
Chapter 19
Analyzing Accelerated Life Test Data
Objectives
• Describe and illustrate nonparametric and graphical methods of analyzing and presenting accelerated life test data.
• Describe and illustrate maximum likelihood methods of analyzing and making inferences from accelerated life test data.
• Illustrate different kinds of data and ALT models.
• Discuss some specialized applications of accelerated testing.
• Describe pitfalls in accelerated testing.
19 - 2
Hours Versus Temperature Data from a
Temperature-Accelerated Life Test on Device-A
Hours
Censored
Status
1
1
...
90
30
60
60
60
...
60
40
40
...
40
10
15
20
100
30
14/15
9/20
10/100
0/30
In Subexperiment
Units
Failures
5000
Failed
Failed
...
Censored
...
11
80
80
80
80
...
80
Temperature
◦C
1298
1390
...
5000
Failed
Failed
Failed
...
Censored
1
1
1
1
...
1
Number
of Devices
581
925
1432
...
5000
Failed
Failed
Failed
Failed
...
Censored
19 - 4
283
361
515
638
...
5000
ALT Data Plot
• Examine a scatter plot of lifetime versus stress data.
• Use different symbols for censored observations.
Note: Heavy censoring makes it difficult to identify the form
of the life/stress relationship from this plot.
19 - 6
40 Deg C
1000
2000
5000
10000
20000
19 - 7
50000
19 - 9
80 Deg C
60 Deg C
200
40 Deg C
500
1000
2000
i
5000
10000
20000
19 - 8
50000
Weibull Multiple Probability Plot Giving Individual
Weibull Fits to Each Level of Temperature for
Device-A ALT Data
h
i
log(t)−µ
bi
c [T (temp ) ≤ t] = Φ
Pr
,
i = 40, 60, 80
sev
i
σ
b
.98
.9
.7
.5
.3
.2
.1
.05
.02
.01
.005
.003
.001
100
Hours
.59
1.72
95% Approximate
Confidence Interval
Lower
Upper
8.9
10.6
2.0
9.3
.27
8.0
7.5
Standard
Error
.42
.35
.70
1.17
1.0
8.64
.32
.55
6.7
ML
Estimate
9.81
µ
1.19
.16
.21
Parameter
µ
σ
.80
7.08
σ
σ
µ
19 - 12
The individual loglikelihoods were L40 = −115.46, L60 = −89.72,
and L80 = −115.58. The confidence intervals are based on the
normal approximation method.
80◦C
60◦C
40◦C
Device-A ALT Lognormal ML Estimation Results at
Individual Temperatures
19 - 10
Note: Either the lognormal or the Weibull distribution
model provides a reasonable description for the device-A
data. But the lognormal distribution provides a better fit
to the individual subexperiments.
• Try to identify a distributional model that fits the data well
at all of the stress-levels.
• Compute nonparametric estimates Fb for each level of accelerating variable; plot on a single probability plot.
ALT Multiple Probability Plot of Nonparametric
Estimates at Individual Levels of Accelerating Variable
.0001
.0002
.0005
Proportion Failing
Strategy for Analyzing ALT Data
For ALT data consisting of a number of subexperiments,
each having been run at a particular set of conditions:
• Examine the data graphically: Scatter and probability plots.
• Use a multiple probability plot to study the data from the
individual subexperiments.
• Fit an overall model involving a life/stress relationship.
• Perform residual analysis and other diagnostic checks.
• Perform a sensitivity analysis.
i
• Assess the reasonableness of using the ALT data to make
the desired inferences.
80 Deg C
60 Deg C
200
500
Lognormal Multiple Probability Plot Giving Individual
Lognormal Fits to Each Level of Temperature for
Device-A ALT Data
h
i
log(t)−µ
bi
c [T (temp ) ≤ t] = Φ
Pr
,
i = 40, 60, 80
nor
i
σ
b
.98
.9
.95
.8
.7
.6
.5
.4
.3
.2
.1
.05
.02
.01
.005
.002
100
Hours
b i, σ
b i).
◮ Compute ML estimates (µ
19 - 11
Note: There are some small differences among the slopes
that could be due to sampling error.
• Assess the commonly used assumption that σi does not
depend on Tempi and that Tempi only affects µi.
b i, σ
b i) cdfs on same plot.
◮ Plot the LOGNOR(µ
Let Ti be the failure time at temperature Tempi. For the
lognormal, Ti ∼ LOGNOR(µi, σi), assumed model:
• For each individual level of accelerating variable compute
the ML estimates.
ALT Multiple Probability Plot of ML Estimates at
Individual Levels of Accelerating Variable
.0001
.0005
Proportion Failing
#
The Arrhenius-Lognormal Regression Model
"
log(t) − µ(x)
Pr[T (temp) ≤ t] = Φnor
σ
••
••
••
•
•
60
•
••
••
••
••
80
10%
5%
1%
19 - 13
100
19 - 15
80 Deg C
200
60 Deg C
500
40 Deg C
1000
2000
5000
10000
10 Deg C
20000
19 - 14
50000
Lognormal Multiple Probability Plot of the
Arrhenius-Lognormal Log-Linear Regression Model Fit
to the Device-A ALT Data
h
i
log(t)−µ
b(x)
c [T (temp) ≤ t] = Φ
b
Pr
,
µ(x)
= βb0 + βb1x
nor
σ
b
.98
.9
.95
.8
.7
.6
.5
.4
.3
.2
.1
.05
.02
.01
.005
.002
100
Hours
.98
.63
ML
Estimate
−13.5
.13
.08
Standard
Error
2.9
.75
.47
1.28
.79
95% Approximate
Confidence Intervals
Lower
Upper
−19.1
−7.8
σ
h
b ).
◮ Compute ML estimates (βb0, βb1, σ
h
b i) = βb0 + βb1xi, σ
b
◮ Plot the LOGNOR µ(x
plot.
i
cdfs on same
19 - 18
−1
b for various values
(p)σ
◮ Plot tbp(x) = exp βb0 + βb1x + Φnor
of p and a range of values of x.
i
Let T (xi) be the failure time at xi = 11605/(Tempi +
273.15). For the, T (xi) ∼ LOGNOR(µ(xi) = β0 + β1xi, σ),
lognormal SAFT assumed model:
• To make inferences at levels of accelerating variable not
used in the ALT, use a life/stress relationship to fit all the
data.
ALT Multiple Probability Plot of ML Estimates with
an Assumed Life/Stress Relationship
19 - 16
The loglikelihood is L = −321.7. The confidence intervals are
based on the normal approximation method.
β1
Parameter
β0
ML Estimation Results for the Device-A ALT Data
and the Arrhenius-Lognormal Regression Model
.0001
.0005
Proportion Failing
The Arrhenius-lognormal regression model is
where
• µ(x) = β0 + β1x,
• x = 11605/(temp K) = 11605/(temp ◦C + 273.15),
• β1 = Ea is the activation energy, and
• σ assumed to be constant.
•
•
•
40
b
µ(x)
= βb0 + βb1x
Scatter plot showing the Arrhenius-Lognormal
Log-Linear Regression Model
Fit to the Device-A ALT Data
20
−1
b
b,
log[tbp(x)] = µ(x)
+ Φnor
(p)σ
50000
20000
10000
5000
2000
500
1000
200
50
100
0
Degrees C on Arrhenius scale
19 - 17
• −2(Lconst − Lunconst) = −2(−321.7 + 320.76) = 1.88 <
2
χ(.75,3)
= 4.1, indicating that there is no evidence of inadequacy of the constrained model, relative to the unconstrained model.
Lconst = −321.7
Lunconst = L40 + L60 + L80 = −320.76
• Use likelihood ratio test to check for lack of fit with respect
to the constraints.
• The Arrhenius-lognormal regression model can be viewed
as a constrained model (µ linear in x and σ constant).
• Distributions fit to individual levels of temperature can be
viewed as an unconstrained model.
Analytical Comparison of Individual and
Arrhenius-Lognormal Model ML Estimates of
Device-A Data
Hours
ML Estimation for the Device-A Lognormal
Distribution F (30,000) at 10◦C
= −13.469 + .6279 × 11605/(10 + 273.15) = 12.2641
b
µ(x)
= βb0 + βb1x
#
=
"
.287 .048
.048 .0176
1
#
.
b σ
b = [log(30,000) − 12.2641]/.9778
ζce = [log(te) − µ]/
= −2.000
"
d µ)
d µ,
b
b σ
b)
Var(
Cov(
d µ,
d σ
b σ
b ) Var(
b)
Cov(
Fb (30,000) = Φnor (ζce) = Φnor (−2.000) = .02281
Σ̂µb,σb =
)
19 - 19
φ(ζce) d
2
c2Var(
cCov(
d σ
d µ,
b)
b σ
b) + ζ
b + 2ζ
Var(µ)
sceFb =
e
e
b
σ
φ(−2.000) h
=
.286 + 2 × (−2.000) × .047 + (−2.000)2 × .0176
.9778
= .0225.
Checking Model Assumptions
•
•
0.10
•
••
••
••
0.20
•
••
••
••
••
0.50
•
••
• •
••
•
•
•
•
• •
1.00
Standardized Residuals
•
2.00
5.00
Confidence Interval for the Device-A Lognormal
Distribution F (30,000) at 10◦C
"
Fb
,
Fb + (1 − Fb ) × w
.14]
#
.02281
.02281 + (1 − .02281)/w
Fb
Fb + (1 − Fb )/w
.02281
,
.02281 + (1 − .02281) × w
= [.0032,
=
Fe (te )] =
A 95% normal-approximation confidence interval based on
the assumption that Zlogit(Fb) ∼ NOR(0, 1) is
[F (te ),
e
where
= exp{(1.96 × .0225)/[.02281(1 − .02281)]} = 7.232.
w = exp{(z(1−α/2)sceFb )/[Fb (1 − Fb )]}
19 - 20
This wide interval reflects sampling uncertainty when activation energy is unknown. The interval does not reflect model
uncertainty. With given activation energy, the confidence
intervals would be much narrower.
•
••
••
•
•
•
•
3
2x10
3
••
••
••
•
•
5x10
3
10
4
••
•
•
2x10
Fitted Values
4
5x10
4
10
5
2x10
5
5
19 - 22
5x10
Plot of Standardized Residuals Versus Fitted Values
for the Arrhenius-Lognormal Log-Linear Regression
Model Fit to the Device-A ALT Data
5.00
2.00
1.00
0.50
0.20
0.10
0.05
0.02
10
Some Practical Suggestions
• See Nelson (1990).
19 - 24
• Design tests to limit the amount extrapolation needed for
desired inferences.
• Seek physical justification for life/stress relationships.
• Use results from physical failure mode analysis.
• Seek physical understanding of cause of failure.
• Use pilot experiments; evaluate the effect of stress on degradation and life.
• Build on previous experience with similar products and materials.
Standardized Residuals
It is important to check model assumptions by using residual
analysis and other model diagnostics
(
log[t(xi)] − βb0 − βb1xi
b
σ
• Define standardized residuals as
exp
where t(xi) is a failure time at xi.
• Residuals corresponding to censored observations are called
censored standardized residuals.
• Plot residuals versus the fitted values given by exp βb0 + βb1xi .
• Do a probability plot of the residuals.
19 - 21
Note: For the Device-A data, these plots do not conflict
with the model assumptions.
.9
.8
.7
.6
.5
.4
.3
.2
.1
.05
.02
.01
.005
.002
0.05
19 - 23
Probability Plot of the Residuals from the
Arrhenius-Lognormal Log-Linear Regression Model fit
to the Device-A ALT Data
Probability
4
3
2
1
0
•
••
•
•
•
100
•
••
•
••
•
•
••
•
•
•
•
150
i
•
••
••
•
kV/mm
200
250
300
19 - 26
350 400
•
••
•
•
•
•
Breakdown Times in Minutes of a Mylar-Polyurethane
Insulating Structure (from Kalkanis and Rosso 1989)
10
10
10
10
10
-1
10
.99
.98
.9
.95
.8
.7
.6
.5
.4
.3
.2
.1
.05
.02
.01
.99
.98
.9
.95
.8
.7
.6
.5
.4
.3
.2
.1
.05
.02
.01
10
10
-1
-2
361.4 kV/mm
10
361.4
-1
0
10
0
10
10
219.0
1
1
219.0
10
i
157.1
Minutes
10
2
3
10
100.3
Minutes
10
2
122.4
10
4
3
100.3
10
10
5
50 kV/mm
6
10
10
10
4
7
19 - 30
Lognormal Probability Plot of the Inverse Power
Relationship-Lognormal Model Fitted to the
Mylar-Polyurethane Data Including 361.4 kV/mm
h
i
log(t)−µ
bi
c [T (temp ) ≤ t] = Φ
Pr
,
i = 100.3, . . . , 361.4
nor
i
σ
b
19 - 28
Lognormal Probability Plot of the Individual Tests in
the Mylar-Polyurethane ALT
i
h
log(t)−µ
bi
c [T (temp ) ≤ t] = Φ
,
i = 100.3, . . . , 361.4
Pr
nor
i
σ
b
Minutes
Inferences from AT Experiments
• Inferences or predictions from ATs require important assumptions about:
◮ Focused correctly on relevant failure modes.
◮ Adequacy of AT model for extrapolation.
◮ AT manufacturing testing processes can be related to
actual manufacturing/use of product.
• Important sources of variability usually overlooked.
• Deming would call ATs analytic studies
(see Hahn and Meeker 1993, American Statistician).
19 - 25
Accelerated Life Test of a Mylar-Polyurethane
Laminated Direct Current High Voltage Insulating
Structure
• Data from Kalkanis and Rosso (1989)
• Time to dielectric breakdown of units tested at 100.3, 122.4,
157.1, 219.0, and 361.4 kV/mm.
• Needed to evaluate the reliability of the insulating structure and to estimate the life distribution at system design
voltages (e.g. 50 kV/mm).
• Except for the highest level of voltage, the relation between
log life and log voltage appears to be approximately linear.
19 - 27
• Failure mechanism probably different at 361.4 kV/mm.
#
Inverse Power Relationship-Lognormal Model
• The inverse power relationship-lognormal model is
"
log(t) − µ(x)
F (t) = Pr[T (volt) ≤ t] = Φnor
σ
where µ(x) = β0 + β1x, and x = log(Voltage Stress).
• σ assumed to be constant.
19 - 29
Proportion Failing
Proportion Failing
50
•
•• ••
•• ••••
•
100
•
•
••
•
200
•
••
••
•
500
10%
50%
90%
b
µ(x)
= βb0 + βb1x
•
•
••
••
kV/mm on log scale
19 - 31
Plot of Inverse Power Relationship-Lognormal Model
Fitted to the Mylar-Polyurethane Data
Including 361.4 kV/mm
8
7
6
5
4
3
2
1
0
20
−1
b
b,
log[tbp(x)] = µ(x)
+ Φnor
(p)σ
10
10
10
10
10
10
10
10
-1
10
10
0
219.0
1
10
157.1
2
122.4
10
100.3
10
3
4
50 kV/mm
10
10
5
19 - 33
Lognormal Probability Plot of the Inverse Power
Relationship-Lognormal Model Fitted to the
Mylar-Polyurethane Data W/O the
361.4 kV/mm Data
i
h
log(t)−µ
b(x)
c [T (temp) ≤ t] = Φ
b
,
µ(x)
= βb0 + βb1x
Pr
nor
σ
b
10
ML
Estimate
27.5
.12
.60
Standard
Error
3.0
.83
−5.46
1.32
−3.11
1
•
•
•
••
•
•
•
5
20
••
•
•
••
•
•
100
Fitted Values
•
•
••
•
••
•
•
500
•
•
••
•
•
•
•
2000
•
••
•
•
•
19 - 32
Lognormal Plot of the Standardized Residuals versus
b
exp(µ(x))
for the Inverse Power
Relationship-Lognormal Model Fitted to the
Mylar-Polyurethane Data with the 361.4 kV/mm Data
20.0
10.0
5.0
2.0
1.0
0.5
0.2
0.1
8
7
6
5
4
3
2
1
0
50
•
•• ••
•• ••••
•
100
•
•
••
•
200
•
••
••
•
500
10%
50%
90%
b
µ(x)
= βb0 + βb1x
•
•
••
••
kV/mm on log scale
19 - 34
Plot of Inverse Power Relationship-Lognormal Model
Fitted to the Mylar-Polyurethane Data (also Showing
361.4 kV/mm Data Omitted from the ML Estimation)
10
10
10
10
10
10
10
10
10
-1
10
20
−1
b
b,
log[tbp(x)] = µ(x)
+ Φnor
(p)σ
10.0
5.0
2.0
1.0
0.5
0.2
0.1
50
••
•
•
••
•
•
100
200
•
•
••
•
••
•
•
Fitted Values
500
•
•
••
•
•
•
•
1000
•
•
•
•
•
•
2000
19 - 36
5000
Lognormal Plot of the Standardized Residuals versus
b for the Inverse Power Relationship-Lognormal
exp(µ)
Model Fitted to the Mylar-Polyurethane Data W/O
the 361.4 kV/mm Data
Minutes
.99
.98
.9
.95
.8
.7
.6
.5
.4
.3
.2
.1
.05
.02
.01
Minutes
1.05
−4.29
95% Approximate
Confidence Intervals
Lower
Upper
21.6
33.4
Inverse Power Relationship-Lognormal Model ML
Estimation Results for the Mylar-Polyurethane
ALT Data
Parameter
β0
σ
β1
The loglikelihood is L = −271.4. The confidence intervals are
based on the normal approximation method.
19 - 35
Standardized Residuals
Standardized Residuals
Minutes
Proportion Failing
Analysis of Interval ALT Data on a New-Technology
IC Device
• Tests were run at 150, 175, 200, 250, and 300◦C.
• Developers interested in estimating activation energy of the
suspected failure mode and the long-life reliability.
• Failures had been found only at the two higher temperatures.
• After early failures at 250 and 300◦C, there was some concern that no failures would be observed at 175◦C before
decision time.
i
19 - 37
• Thus the 200◦C test was started later than the others.
100
200
500
2000
300 Degrees C
1000
Hours
250 Degrees C
5000
19 - 39
10000
Lognormal Probability Plot of the Failures at 250 and
300◦C for the New-Technology Integrated Circuit
Device ALT Experiment
h
i
log(t)−µ
bi
c [T (temp ) ≤ t] = Φ
Pr
,
i = 250, 300
nor
i
σ
b
.9
.95
.8
.7
.6
.5
.4
.3
.2
.1
.05
.02
.01
.005
.002
.9
.95
.8
.7
.6
.5
.4
.3
.2
.1
.05
.02
.01
.005
.002
.0005
.0001
10
300 Deg C
2
250
3
10
200
10
4
175
Hours
150
10
5
6
100
10
10
7
19 - 41
Lognormal Probability Plot Showing the
Arrhenius-Lognormal Model ML Estimation Results for
the New-Technology IC Device
h
i
log(t)−µ
b(x)
c [T (temp) ≤ t] = Φ
b
Pr
,
µ(x)
= βb0 + βb1x
nor
σ
b
.0001
.0005
Proportion Failing
Proportion Failing
Status
Right Censored
Right Censored
Right Censored
Failed
Failed
Failed
Right Censored
Failed
Failed
Failed
Right Censored
Number of
Devices
50
50
50
1
3
5
41
4
27
16
3
Temperature
◦C
150
175
200
250
250
250
250
300
300
300
300
New-Technology IC Device ALT Data
Hours
Lower Upper
1536
1536
96
384
788
788
1536
1536
2304
2304
384
788
1536
1536
192
384
788
19 - 38
Individual Lognormal ML Estimation Results for the
New-Technology IC Device
250◦C
σ
σ
µ
Parameter
µ
.46
.87
6.56
ML
Estimate
8.54
.05
.26
.07
Standard
Error
.33
.36
.48
6.4
.58
1.57
6.7
95% Approximate
Confidence Intervals
Lower
Upper
7.9
9.2
300◦C
The loglikelihood were L250 = −32.16 and L300 = −53.85.
The confidence intervals are based on the normal approximation
method.
19 - 40
Arrhenius-Lognormal Model ML Estimation Results for
the New-Technology IC Device
.83
ML
Estimate
−10.2
.06
.07
Standard
Error
1.5
.42
.68
.64
.97
95% Approximate
Confidence Intervals
Lower
Upper
−13.2
−7.2
.52
Parameter
β0
σ
β1
The loglikelihood is L = −88.36.
Comparing the constrained and unconstrained models Luconst =
L250+L300 = −86.01 and for the constrained model, Lconst =
−88.36. The comparison has just one degree of freedom and
2
−2(−88.36 + 86.01) = 4.7 > χ(.95,1)
= 3.84, again indicating
that there is some lack of fit in the constant-σ Arrheniuslognormal model.
19 - 42
x
x
x
300
50%
10%
1%
350
b
µ(x)
= βb0 + βb1x
x
x
x
250
19 - 43
300 Deg C
2
3
10
250
200
10
4
175
150
10
5
6
100
10
10
7
19 - 44
Lognormal Probability Plot Showing the
Arrhenius-Lognormal Model ML Estimation Results for
the New-Technology IC Device with Given Ea = .8
h
i
log(t)−µ
b(x)
c [T (temp) ≤ t] = Φ
b
Pr
,
µ(x)
= βb0 + Ea x
nor
σ
b
.9
.95
.8
.7
.6
.5
.4
.3
.2
.1
.05
.02
.01
.005
.002
10
Hours
19 - 46
19 - 48
• Plan ATs that will allow effective prediction of failure time
distributions at desired ratio (or ratios) of time scales.
• With multiple time scales, understand ratio or ratios of time
scales that arise in actual use.
• Need to use the appropriate time scale(s) for evaluation of
each failure mechanism.
Suggestions:
Dealing with Multiple Time Scales
◮ Use sensitivity analysis to assess the effect of departures
from model assumptions and uncertainty in other inputs.
◮ Use confidence intervals to quantify statistical uncertainty.
• Suggestions:
• Confidence intervals ignore model uncertainty (which can
be tremendously amplified by extrapolation in Accelerated
Testing).
• Standard statistical confidence intervals quantify uncertainty
arising from limited data.
• There is uncertainty in statistical estimates.
Pitfall 2: Failure to Properly Quantify Uncertainty
.0001
.0005
Proportion Failing
Arrhenius Plot Showing ALT Data and the
Arrhenius-Lognormal Model ML Estimation Results for
the New-Technology IC Device.
7
6
5
4
3
2
200
Degrees C on Arrhenius scale
150
−1
b
b,
log[tbp(x)] = µ(x)
+ Φnor
(p)σ
10
10
10
10
10
10
100
• Suggestions:
◮ Determine failure mode of failing units.
◮ Analyze failure modes separately.
Pitfall 3: Multiple Time Scales
• Composite material
19 - 47
◮ On/off cycles of a print operation (thermal cycling of
substrate and printhead lamination).
◮ Characters or pages printed (ink supply, resistor degradation).
◮ Real time (corrosion)
• Inkjet pen
◮ On-off cycles induce thermal and mechanical shocks that
can lead to fatigue cracks.
◮ Filament evaporates during burn time.
• Incandescent light bulbs
◮ Stress cycles during use lead to initiation and growth of
cracks.
◮ Chemical degradation over time changes material ductility.
19 - 45
• Other failure modes, if not recognized in data analysis, can
lead to incorrect conclusions.
• High levels of accelerating factors can induce failure modes
that would not be observed at normal operating conditions
(or otherwise change the life-acceleration factor relationship).
Pitfall 1: Multiple (Unrecognized) Failure Modes
Hours
10
10
10
10
10
10
6
5
4
3
2
1
60
Degrees C
80
100
120
Temperature-Accelerated Life Test
for an IC Device
40
Pitfall 4: Masked Failure Mode
◮ Identify failure modes of all failures.
Comparison of Two Products II
Questionable Comparison
120
10%
10%
Vendor 2
10%
140
19 - 49
19 - 51
6
5
4
3
2
1
6
5
4
3
2
1
60
60
80
Degrees C
Degrees C
80
100
100
10%
Mode 2
10%
Mode 1
120
10%
140
19 - 50
19 - 52
140
Vendor 1
10%
Vendor 2
120
Comparison of Two Products I
Simple Comparison
40
Unmasked Failure Mode with Lower Activation Energy
for an IC Device
10
10
10
10
10
10
10
10
10
10
10
10
40
Pitfall 5: Faulty Comparison
• It is sometimes claimed that Accelerated Testing is not useful for predicting reliability, but is useful for comparing alternatives.
19 - 54
◮ Understand the physical reason for any differences.
◮ Analyze failure modes separately.
◮ Identify failure modes of all failures.
◮ Know (anticipate) different failure modes.
• Comparisons can, however, also be misleading.
1
2
5
140
• Beware of comparing products that have different kinds of
failures.
100
4
80
Degrees C
19 - 53
• Suggestions:
60
Hours
Hours
3
6
◮ Analyze failure modes separately.
10
10
10
10
10
10
40
Vendor 1
◮ Limit acceleration and test at levels of accelerating variables such that each failure mode will be observed at
two or more levels of the accelerating variable.
◮ Know (anticipate) different failure modes.
• Suggestions:
• Masked failure modes could dominate in the field.
• Masked failure modes may be the first one to show up in
the field.
• Accelerated test may focus on one known failure mode,
masking another!
Hours
Hours
Pitfall 6: Acceleration Factors
Can Cause Deceleration!
• Increased temperature in an accelerated circuit-pack reliability audit resulted in fewer failures than in the field because
of lower humidity in the accelerated test.
Pitfall 7: Untested Design/Production Changes
• Lead-acid battery cell designed for 40 years of service.
• New epoxy seal to inhibit creep of electrolyte up the positive
post.
• Accelerated life test described in published article demonstrated 40 year life under normal operating conditions.
• Automobile air conditioners failed due to a material drying
out degradation, lack of use in winter (not seen in continuous accelerated testing).
• Improper epoxy cure combined with charge/discharge cycles
hastened failure.
• First failure after 2 years of service; third and fourth failures
followed shortly thereafter.
• 200,000 units in service after 2 years of manufacturing.
• Inkjet pens fail from infrequent use.
• Entire population had to be replaced with a re-designed cell.
• Higher than usual use rate of a mechanical device in an accelerated test inhibited a corrosion mechanism that eventually caused serious field problems.
• Suggestion: Understand failure mechanisms and how they
are affected by experimental variables.
19 - 56
5
4
3
2
1
0
5 Degrees C
45 Degrees C
85 Degrees C
30
996 cen 196 cen
40
174 cen
496 cen
48 cen
49 cen
50
49 cen
156 cen
60
70
19 - 59
0
20
40
60
80
100
.2
.1
.05
.03
.02
.01
.005
.003
.001
.0005
.0003
4/1000
35
10
4/200
40
85 Deg C, 35 V
85 Deg C, 40.6 V
85 Deg C, 46.5 V
85 Deg C, 51.5 V
45 Deg C, 46.5 V
5 Deg C, 62.5 V
50
2/50
6/502
1/175
45
200
4/53
50
Volts
Hours
500
2000
55
1/50
i
5000
60
18/174
20000
65
19 - 60
50000
Weibull Probability Plot for the Individual Voltage and
Temperature Level Combinations for the Tantalum
Capacitors ALT, with ML Estimates of Weibull cdfs
h
i
log(t)−µ
bi
c [T (temp ) ≤ t] = Φ
Pr
sev
i
σ
b
19 - 58
Tantalum Capacitors ALT Design Showing Fraction
Failing at Each Point
19 - 55
Temperature/Voltage ALT Data on
Tantalum Electrolytic Capacitors
• Two-factor ALT
• Non-rectangular unbalanced design
• Much censoring
• The Weibull distribution seems to provide a reasonable model
for the failures at those conditions with enough failures to
make a judgment.
• Temperature effect is not as strong.
19 - 57
• Data analyzed in Singpurwalla, Castellino, and Goldschen
(1975)
10
10
10
10
10
10
20
Volts
Degrees C
Proportion Failing
Scatter Plot of Failures in the Tantalum Capacitors
ALT Showing Hours to Failure Versus Voltage with
Temperature Indicated by Different Symbols
Hours
Model 1:
µ(x) = β0 + β1x1 + β2x2
Tantalum Capacitors ALT
Weibull/Arrhenius/Inverse Power Relationship Models
Model 2:
µ(x) = β0 + β1x1 + β2x2 + β3x1x2
where x1 = log(volt), x2 = 11605/(temp K), and β2 = Ea.
• Coefficients of the regression model are highly sensitive to
whether the interaction term is included in the model or
not (because of the nonrectangular design with highly unbalanced allocation).
• Data provide no evidence of interaction.
2
85 Deg C, 35 V
85 Deg C, 40.6 V
85 Deg C, 46.5 V
85 Deg C, 51.5 V
45 Deg C, 46.5 V
5 Deg C, 62.5 V
10
10
3
4
30 Deg C, 24 V
10
10
5
19 - 61
−20.1
ML
Estimate
84.4
4.4
Standard
Error
13.6
1.72
−.04
−28.8
.69
−11.4
95% Approximate
Confidence Interval
Lower
Upper
57.8
111.
β1
.36
.19
3.16
.33
72.35
135.1
2.33
−32.5
−292.3
σ
26.7
.40
11.6
109.0
−1.35
19.9
3.3
1.72
−2.8
-78.6
.80
β0
β2
Parameter
β0
Tantalum Capacitor ALT Weibull-Inverse Power
Relationship Regression ML Estimation Results
Model 1
Model 2
5.13
.36
β1
β2
2.33
−1.17
3.16
σ
19 - 62
β3
Loglikelihoods L1 = −539.63 and L2 = −538.40
1
2
3
4
5
6
7
8
9
10
10
10
10
10
10
10
10
10
10
10
20
85 Deg C
45 Deg C
5 Deg C
30
40
Volts
85
50
70
5 Degrees C
45
60
19 - 64
ML Estimates of t0.01 for the Tantalum Capacitor Life
Using the Arrhenius-Inverse Power Relationship
Weibull Model
Hours
• Strong evidence for an important voltage effect on life.
1
19 - 65
19 - 63
Weibull Multiple Probability Plot of the Tantalum
Capacitor ALT Data Arrhenius-Inverse Power
Relationship Weibull Model (with no Interaction)
i
h
log(t)−µ
b(x)
c [T (temp) ≤ t] = Φ
b x) = βb0 + βb1x1 + βb2x2
, µ(
Pr
sev
σ
b
.5
.2
.1
.05
.02
.01
.005
.003
.001
10
Hours
Other Topics in Chapter 19
• Environmental stress screening.
• Burn-in.
• Environmental stress and STRIFE testing.
• Highly accelerated life tests (HALT).
Discussion of
.000005
.000003
.00001
.00002
.00005
.0001
.0002
.0005
Proportion Failing