Chapter 20
Planning Accelerated Life Tests
William Q. Meeker and Luis A. Escobar
Iowa State University and Louisiana State University
20 - 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
Possible Reasons for Conducting an Accelerated Test
Accelerated tests (ATs) are used for different purposes.
These include:
• ATs designed to identify failure modes and other weaknesses in product design.
• ATs for improving reliability
• ATs to assess the durability of materials and components.
#
20 - 3
• ATs to monitor and audit a production process to identify
changes in design or process that might have a seriously
negative effect on product reliability.
Model and Assumptions
"
log(t) − µ
σ
• Failure-time distribution is loglocation-scale
Pr(T ≤ t) = F (t; µ, σ) = Φ
11605
.
temp ◦C + 273.15
• µ = µ(x) = β0 + β1x, where
x=
• σ does not depend on the experimental variables.
• Units tested simultaneously until censoring time tc.
• Observations statistically independent.
20 - 5
Planning Accelerated Life Tests
Chapter 20 Objectives
• Outline reasons and practical issues in planning ALTs.
• Describe criteria for ALT planning.
• Illustrate how to evaluate the properties of ALTs.
• Describe methods of constructing and choosing among ALT
plans
◮ One-variable plans.
◮ Two-variable plans.
20 - 2
• Present guidelines for developing practical ALT plans with
good statistical properties.
Motivation/Example
Reliability Assessment of an Adhesive Bond
20 - 4
• Need: Estimate of the B10 of failure-time distribution at
50◦C (expect ≥ 10 years).
• Constraints
◮ 300 test units.
◮ 6 months for testing.
• 50◦C test expected to yield little relevant data.
Assumed Planning Information for the
Adhesive Bond Experiment
The objective is finding a test plan to estimate B10 with
good precision.
• Weibull failure-time distribution with same shape parameter
at each level of temperature σ and location scale parameter
µ(x) = β0 + β1x, where x is ◦C in the Arrhenius scale.
• .1% failing in 6 months at 50◦C.
• 90% failing in 6 months at 120◦C.
Result: Defines failure probability in 6 months at all levels of
temperature. If σ is given also, defines all model parameters.
20 - 6
Temp
◦C
50
110
130
150
Failure
Probability
pi
0.001
0.60
1.00
1.00
60
100
100
Expected
Number Failing
E(ri)
Engineers’ Originally Proposed Test Plan
for the Adhesive Bond
100
100
100
Allocation
Proportion Number
πi
ni
1/3
1/3
1/3
20 - 7
Critique of Engineers’ Original Proposed Plan
• Arrhenius model in doubt at high temperatures (above 120◦C).
• Question ability to extrapolate to 50◦C.
• Data much above the B10 are of limited value.
Suggestion for improvement:
Simulation of Engineers’
Modified Traditional ALT Plan
60
80
90
100
Degrees C on Arrhenius scale
70
120
Precision factors R for quantile estimates at 50 Degrees C
R( 0.1 quantile)= 2.288
R( 0.5 quantile)= 2.484
R(Ea)= 1.165
Levels = 80,100,120 Degrees C, n=100,100,100
Censor time=183,183,183, parameters= -16.74,0.7265,0.5999
50
Results based on 300 simulations
Lines shown for 50 simulations
40
10%
20 - 11
20 - 9
• Test at lower more realistic temperatures (even if only small
fraction will fail).
20
10
5
200
100
50
2000
1000
500
20000
10000
5000
50000
• Larger allocation to lower temperatures.
Days
10
10
10
10
10
10
5
4
3
2
1
0
40
60
100
Degrees C
80
120
140
20 - 8
160
10%
5%
50%
Adhesive Bond
Engineers’ Originally Proposed Test Plan
n = 300, πi = 1/3 at each 110◦C, 130◦C, 150◦C
Temp
◦C
50
80
100
120
"
.04
.29
.90
Failure
Probability
pi
4
29
90
Expected
Number Failing
E(ri)
∂g(θ )
∂θ
b)
• Asymptotic variance of g(θ
b =
Avar[g(θ)]
#′
θ
"
#
∂g(θ )
.
∂θ
20 - 10
20 - 12
• Use Monte Carlo simulation. Specific results, provides picture of data, requires much computer time.
Simple to compute (with software) and general results.
Σb
• Properties depend on test plan, model and (unknown) parameter values. Need planning values.
Assume inferences needed on a function g(θ ) (one-to-one
and all the first derivatives with respect to the elements of θ
exist, and are continuous).
Methods of Evaluating Test Plan Properties
For this plan and the Weibull-Arrhenius model, Ase[log(tb.1(50))] =
.4167. The asymptotic precision factor for a 95% confidence
interval of t.1(50) is R = exp(1.96 × Ase) = 2.26.
Allocation
Proportion Number
πi
ni
0
1/3
100
1/3
100
1/3
100
Engineers’ Modified Traditional ALT Plan with a
Maximum Test Temperature of 120◦C
Days
Statistically Optimum Plan for the Adhesive Bond
• Objective: Estimate B10 at 50◦C with minimum variance.
• Constraint: Maximum testing temperature of 120◦C.
• Inputs: Failure probabilities pU = .001 and pH = .90.
20 - 13
Adhesive Bond
Weibull Distribution Statistically Optimum Plan
5
4
3
2
60
80
100
120
140
20 - 15
160
10%
5%
50%
Allocations: πLow = .71 at 95◦C, πHigh = .29 at 120◦C
10
10
10
10
1
0
40
Degrees C
Expected
Number Failing
E(ri)
38
78
1.0
0.8
0.6
0.4
0.2
0.4
0.6
ξL
.2
.1
.5
1
0.8
2
1.0
20 - 14
Contour Plot Showing
log10{Avar[log(tb.1)]/ min Avar[log(tb.1)]}
as Function of ξL, πL to Find the Optimum ALT Plan
πL
0.2
0.0
0.0
60
80
90
100
Degrees C on Arrhenius scale
70
120
Precision factors R for quantile estimates at 50 Degrees C
R( 0.1 quantile)= 2.103
R( 0.5 quantile)= 2.309
R(Ea)= 1.155
Levels = 95,120 Degrees C, n=212,88
Censor time=183,183, parameters= -16.74,0.7265,0.5999
50
Results based on 300 simulations
Lines shown for 50 simulations
40
10%
20 - 16
Simulation of the
Weibull Distribution Statistically Optimum Plan
50000
20000
10000
5000
2000
1000
500
200
100
50
20
10
5
10
10
10
10
10
10
5
4
3
2
1
0
40
60
100
Degrees C
80
120
140
20 - 18
160
10%
5%
50%
Allocations: πLow = .74 at 78◦C, πHigh = .26 at 120◦C
Adhesive Bond
Lognormal Distribution Statistically Optimum Plan
Days
Days
10
10
ni
Failure
Probability
pi
.001
.18
.90
Weibull Distribution
Statistically Optimum Plan
πi
213
87
Allocation
Proportion Number
.71
.29
20 - 17
For this plan and the Weibull-Arrhenius model, Ase[log(tb.1(50))] =
.3794
50
95
120
Temp
◦C
Days
Temp
◦C
50
78
120
Failure
Probability
pi
.001
.13
.90
30
69
Expected
Number Failing
E(ri)
Lognormal Distribution
Statistically Optimum Plan
233
77
Allocation
Proportion Number
πi
ni
.74
.26
20 - 19
For this plan and the Lognormal-Arrhenius model, Ase[log(tb.1(50))] =
.2002
Want a Plan That
20 - 21
• Meets practical constraints and is intuitively appealing.
• Is robust to deviations from assumed inputs.
• Has reasonably good statistical properties.
Types of Accelerated Life Test Plans
• Optimum plans—Maximize statistical precision.
• Traditional plans—Equal spacing and allocation; may be
inefficient.
• Optimized (best) compromise plans—require at least 3
levels of the accelerating variable (e.g., 20% constrained at
middle) and optimize lower level and allocation.
20 - 23
Critique of the Statistically Optimum Plan
• Still too much temperature extrapolation (to 50◦C).
• Only two levels of temperature.
• Optimum Weibull and lognormal plans quite different
◮ 95◦C and 120◦C for Weibull versus.
◮ 78◦C and 120◦C for lognormal.
20 - 20
In general, optimum plans not robust to model departures.
Criteria for Test Planning
Subject to constraints in time, sample size and ranges of
experimental variables,
• Minimize Var[log(tbp)] under the assumed model.
• Maximize the determinant of the Fisher information matrix.
• Minimize Var[log(tbp)] under more general or higher-order
model(s) (for robustness).
20 - 22
• Control the expected number of failures at each experimental condition (since a small expected number of failures
at critical experimental conditions suggests potential for a
failed experiment).
General Guidelines for Planning ALTs
(Suggested from Optimum Plan Theory)
• Choose the highest level of the accelerating variable to be
as high as possible.
• Lowest level of the accelerating variable can be optimized.
• Allocate more units to lower levels of the accelerating variable.
• Test-plan properties and optimum plans depend on unknown inputs.
20 - 24
Practical Guidelines for Compromise ALT Plans
• Use three or four levels of the accelerating variable.
• Limit high level of the accelerating variable to maximum
reasonable condition.
• Reduce lowest level of the accelerating variable (to minimize
extrapolation)—subject to seeing some action.
80
100
120
140
50%
10%
5%
160
20 - 25
• Allocate more units to lower levels of the accelerating variable.
• Use statistically optimum plan as a starting point.
• Evaluate plans in various meaningful ways.
5
4
3
2
1
0
60
Adhesive Bond
Adjusted Compromise Weibull ALT Plan
πLow = .52, πMid = .20, πHigh = .28
10
10
10
10
10
10
40
Degrees C
20 - 27
156
60
84
Allocation
Proportion Number
πi
ni
.52
.20
.28
Failure
Probability
pi
.001
.03
.24
.90
5
14
76
Expected
Number Failing
E(ri)
Adjusted Compromise Weibull ALT Plan for the
Adhesive Bond
(20% Constrained Allocation at Middle)
Temp
◦C
50
78
98
120
Simulation of the Adhesive Bond
Compromise Weibull ALT Plan
Levels = 78,98,120 Degrees C, n=155,60,84
Censor time=183,183,183, parameters= -16.74,0.7265,0.5999
80
90
100
120
Precision factors R for quantile estimates at 50 Degrees C
R( 0.1 quantile)= 2.381
R( 0.5 quantile)= 2.645
R(Ea)= 1.177
70
Degrees C on Arrhenius scale
60
Results based on 300 simulations
Lines shown for 50 simulations
50
10%
20 - 28
20 - 26
For this plan with the Weibull-Arrhenius model, Ase[log(tb.1(50))] =
.4375.
50000
20000
10000
5000
2000
1000
500
200
100
50
20
10
5
40
Basic Issue 2: Allocation of Test Units
◮ Tends to equalize the number of failures at experimental
conditions.
20 - 30
• Need to constrain a certain percentage of units to the middle level of the accelerating variable.
◮ Suggested by statistically optimum plan.
• Extrapolation in time (assumed failure-time distribution).
20 - 29
• Low level of the acceleration variable—expect to extrapolate in time.
• Middle and high levels of the acceleration variable—expect
to interpolate in time.
◮ Testing more units near the use conditions is intuitively
appealing.
• Allocate more test units to low rather than high levels of
the accelerating variable.
Days
Suggested Plan:
• Extrapolation in the acceleration variable (assumed temperaturetime relationship).
Need to Balance:
Basic Issue 1: Choose Levels of Accelerating Variables
Days
• Constraints on test positions (instead of test units): Consider replacement after 100p% failures at each level of accelerating variable.
Generalizations and Comments
• Increases asymptotic variance of estimator of B10 at 50◦C
by 33% (if assumptions are correct).
• Continue tests at each level of accelerating variable until at
least 100p% units have failed.
Properties of Compromise ALT Plans
Relative to Statistically Optimum Plans
However it also,
• Include some tests at the use conditions.
• Need to quantify robustness.
20 - 32
• Desire to estimate reliability (instead of a quantile) at use
conditions.
• Fine tune with computer evaluation and/or simulation of
user-suggested plans.
• Reduces low test temperature to 78◦C (from 95◦C).
• Uses three levels of accelerating variable, instead of two
levels.
• Is more robust to departures from assumptions and uncertain inputs.
20 - 31
Choosing Experimental Variable Definition
to Minimize Interaction Effects
• Guidance on variable definition and possible transformation
of the response and the experimental models should, as
much as possible, be taken from mechanistic models.
• Care should be used in defining experimental variables.
• Moderate increases in two accelerating variables may be
safer than using a large amount of a single accelerating
variable.
• Proper choice can reduce the occurrence or importance of
statistical interactions.
ALT with Two or More Variables
• There may be interest in assessing the effect of nonaccelerating variables.
Size (mm)
•
•
•
200
100
•
•
•
•
•
•
Volts
300
•
•
500
•
Comparison of Experimental Layout with
Volts/mm Versus Size and Volts Versus Size
20 - 36
20 - 34
• Knowledge from mechanistic models is also useful for planning experiments.
• Models without statistical interactions simplify modeling,
interpretation, explanation, and experimental design.
20 - 33
• There may be interest in assessing joint effects of two more
accelerating variables.
Examples of Choosing Experimental Variable
Definition to Minimize Interaction Effects
•
•
•
150
Size (mm)
• For humidity testing of corrosion mechanism use RH and
temperature (not vapor pressure and temperature)
•
100
•
•
3.0
Voltage Stress (Volts/mm)
50
2.5
1.5
1.0
• For testing dielectrics, use size and volts stress (e.g., mm
and volts/mm instead of mm and volts)
• For light exposure, use aperture and total light energy
(not aperture and exposure time)
• To evaluate the adequacy of large-sample approximations
with censored data, use % failing and expected number
failing (not % failing and sample size).
20 - 35
3.0
2.0
2.5
2.0
1.5
1.0
•
50 100
•
•
•
•
300
•
Pr(T ≤ t) = F (t; µ, σ) = Φ
•
•
"
•
•
300
•
•
•
200
•
100
Voltage Stress
(Volts/mm)
#
log(t) − µ
.
σ
Multiple Variable ALT
Model and Assumptions
Volts
200
•
•
3.0
•
2.5
1.0
1.5
2.0
3.0
• σ does not depend on the experimental variables.
• Units tested simultaneously until censoring time tc.
0.0
-6
0.2
p=.1
0.6
Standardized Factor 1
0.4
p=.01
140
0.8
180
p=.5 p=.99
160
1.0
200
p=1
1.2
Insulation ALT
3 × 3 VPM × THICK Factorial Test Plan
-0.2
-6
p=3.2x10
pUU=1.8x10
120
Electrical Stress (volts per mm)
100
20 - 37
20 - 39
20 - 41
Standardized Factor 2
• Observations statistically independent.
0.4
0.35
0.3
0.25
0.2
0.15
80
-2
-1
0
1
• µ = µ(x) is a function of the accelerating (or other experimental) variables.
400
Comparison of Experimental Layout with
Volts versus Size and Volts/mm versus Size
•
Size (mm)
2.5
2.0
1.5
1.0
• Failure-time distribution
Size (mm)
Thickness (cm)
Insulation ALT
From Chapter 6 of Nelson (1990) and
Escobar and Meeker (1995)
#
20 - 40
20 - 38
• Engineers needed rapid assessment of insulation life at use
conditions.
• 1000/10000 hours available for testing.
• 170 test units available for testing.
• Possible experimental variables:
◮ VPM (Volts/mm) [accelerating].
◮ THICK (cm) [nonaccelerating].
◮ TEMP (◦C) [accelerating].
Models Used in Examples
µ = β0 + β1 log(VPM)
"
11605
temp ◦C + 273.15
µ = β0 + β1 log(VPM) + β2 log(THICK)
µ = β0 + β1 log(VPM) + β2
σ constant.
The ALT Design Problem
• Design test plan to estimate life at the
use conditions of VPMU = 80 volts/mm,
THICKU = 0.266 cm, TEMPU = 120 ◦C.
• Interest centers
on a quantile
h
i in lower tail of life distribution,
tp = exp µ(xU ) + Φ−1(p)σ .
• Need to choose levels of the accelerating variable(s) x1, . . . , xk
and allocations π1, · · · , πk to those conditions. Equal allocation can be a poor choice.
20 - 42
Multi-Variable Experimental Region
• Maximum levels for all variables:
VPMH = 200 volts/mm
THICKH = 0.355 cm
TEMPH = 260 ◦C.
• Explicit minimum levels for all experimental variables:
VPMA = 80volts/mm
THICKA = 0.163cm
TEMPA = 120◦C
(also stricter implicit limits for VPM and TEMP).
20 - 43
• May need to restrict highest combinations of accelerating
variables; e.g., constrain by equal failure-probability line
(by using a maximum failure probability constraint p∗ or
equivalently a standardized censored failure time ζ ∗ constraint).
Degenerate and Nondegenerate Test Plans to
Estimate tp
Degenerate plans:
• Test all units at xU .
• Test two (or more) combinations of the experimental
variables on a line with slope s passing through xU .
Nondegenerate practical plans:
20 - 45
• Test at three (or more) noncollinear combinations of the
experimental variables in the plane.
Splitting Degenerate Plans
• It is possible to split a degenerate plan into a nondegenerate
optimum test plan (maintaining optimum Var[log(tbp)]).
• Use secondary criteria to chose best split plan.
• Split xi = (x1i, x2i)′ with allocation πi into
xi1 = (x1i1, x2i1)′ and xi2 = (x1i2, x2i2)′
with allocations πi1 and πi2 (where πi1 + πi2 = πi)
0.4
0.35
0.3
0.25
0.2
0.15
0.15
0.2
0.25
0.3
0.35
0.4
-0.2
0.0
-6
0.2
0.4
0.6
140
0.8
160
p=.33
Standardized Factor 1
120
Electrical Stress (volts per mm)
100
180
1.0
200
p=1
1.2
Insulation ALT
VPM × THICK Optimum Test Plan
pUU=1.8x10
-0.2
pUU=1.8x10
0.0
-6
80
0.2
0.4
0.6
140
0.8
160
p=.28
Standardized Factor 1
120
Electrical Stress (volts per mm)
100
180
p=1
200
1.0
1.2
Insulation ALT VPM × THICK
Optimum Test Plan with p∗/ζ ∗ constraint
20 - 44
20 - 46
• Optimum degenerate plan corresponds to a single-variable
optimum.
• Some units tested at highest levels of accelerating variables.
• Otherwise there is at least one optimum degenerate test
plan in the x1 × x2 plane.
• When acceleration does not help sufficiently, it is optimum
to test all units at the use conditions.
Optimum Degenerate Plan: Technical Results
80
1
0
-1
-2
Standardized Factor 2
20 - 48
Standardized Factor 2
1
0
-1
-2
πi1xi1 + πi2xi2 = πixi.
• Can introduce a p∗ constraint
[or a ζ ∗ constraint where p∗ = Φ(ζ ∗)].
20 - 47
• Can also split compromise plans and maintain Var[log(tbp)].
Thickness (cm)
Thickness (cm)
0.0
-6
pUU=1.8x10
80
140
160
p=.33
Standardized Factor 1
0.5
120
180
p=.99
Electrical Stress (volts per mm)
100
0.2
0.4
p=.5 p=.99
0.6
Standardized Factor 1
p=.1
0.8
160
1.0
200 220
180
1.0
200
p=1
p=1
Insulation ALT VPM × TEMP
3 × 3 Factorial Test Plan
0.0
p=.01
140
1.2
Insulation ALT VPM × THICK 20%
Compromise Test Plan with p∗/ζ ∗ constraint
0.4
0.3
0.35
0.25
-0.2
p=.00016
120
Electrical Stress (volts per mm)
100
0.2
0.4
0.6
Standardized Factor 1
0.8
1.0
200
p=1
1.2
20 - 49
20 - 51
Plan
|F |
Interaction
Model
0.0
∞
∞
0.0
0.0
0.0
1.2 × 10−5
80.1
7.3 × 10−4
∞
1.7 × 10−5
145
V[log(b
tp )]
Comparison of Test Plans and Properties
for the VPM×THICK ALT
No Interaction
Model
|F |
80.1
0.0
138
2.4 × 10−3
Optimum degenerate
No ζ ∗
131
1.6 × 10−3
0.0
144
Optimum split
No ζ ∗
131
9710
V[log(b
tp )]
Optimum degenerate
ζ ∗ = 2.5454
0.0
1.2 × 10−4
3×3
Factorial
from Nelson (1990)
Optimum split
ζ ∗ = 2.5454
96.1
102
0.2
0.4
0.6
Standardized Factor 1
0.8
160
1.0
200
p=1
20 - 50
7.0 × 10−3
0.0
p=.39
140
180
Insulation ALT VPM × TEMP
Optimum Test Plan
96.1
1.2
20% Compromise degenerate
ζ ∗ = 4.04
-0.2
20% Compromise split
ζ ∗ = 4.04
260
240
220
200
180
160
120
Electrical Stress (volts per mm)
140
-06
pUU=1.8x10
80
|F |
Interaction
Model
V[log(b
tp )]
Comparison of Test Plan Properties
for the VPM×TEMP ALT
|F |
No Interaction
Model
20 - 52
50.5
1.3 × 10−3
0.0
1613
∞
∞
3.0 × 10−6
V[log(b
tp )]
50.5
0.0
430
0.0
2.7 × 10−6
Optimum degenerate
No ζ ∗
54.7
2.0 × 10−3
5768
349
Optimum split
No ζ ∗
54.7
0.0
324
1.7 × 10−3
20% Compromise degenerate
No ζ ∗
77.7
1.2 × 10−3
77.3
20% Compromise split
No ζ ∗
77.7
0.0
20% Compromise degenerate
ζ ∗ = 5.0
20 - 54
1.7 × 10−6
0.0
0.0
20% Compromise split
ζ ∗ = 5.0
3×3
Factorial Adapted
from Nelson (1990)
Plan
120
100
1.2
0.2
0.15
260
240
220
200
180
-6
pUU=1.8x10
80
0.0
p=1
160
Standardized Factor 2
1.0
0.8
0.6
0.2
0.4
0.0
160
140
120
-0.2
p=.36
140
180
Insulation ALT VPM × TEMP
20% Compromise Test Plan with p∗/ζ ∗ constraint
260
240
220
200
180
160
120
Electrical Stress (volts per mm)
140
-06
pUU=1.8x10
100
120
80
20 - 53
Standardized Factor 2
-0.2
1
0
1.2
1.0
0.8
0.6
0.2
0.4
0.0
-0.2
Temperature (degrees C)
Standardized Factor 2
Standardized Factor 2
-1
-2
1.2
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
Thickness (cm)
Temperature (degrees C)
Temperature (degrees C)
Extensions of Results to Other Problems
• With one accelerating and several other regular experimental variables, replicate single-variable ALT at each combination of the regular experimental variables.
• Can use a fractional factorial for the regular experimental
variables.
• If the approximate effect of a regular experimental variable
is known, can tilt factorial to improve precision.
• With two or more accelerating variables, our results show
how to tilt the traditional factorial plans to restrict extrapolation and maintain statistical efficiency.
20 - 55