Planning 2-Stage Accelerated Life Tests
LC Tang (董润楨), Ph.D
Department of Industrial and Systems Engineering
National University of Singapore
Overview
• Planning a sequential Accelerated Life Test (ALT)
• Motivation of using an Auxiliary Stress (AS)
• An integrated planning framework for sequential ALT
with an AS
• Numerical illustrations
A Constant-Stress ALT
Time
Probability
distributions
Maximum
Test
Duration
Life-stress
relationship
Use
Stress
Low
Stress
Mid
Stress
High
Stress
Stress Level
A Scale-Accelerated Weibull Lifetime Model
• Standardization of stress x  x  s    s  sk    s0  sk 1
s0 : use stress sk : the highest stress
• Weibull lifetime distribution at any stress
 log T  -  
 SEV 




• A scale-accelerated failure time model
  0  1 x , 1  0

 is a constant independent of stress
Motivations of Sequential ALT Planning
• ALT planning based on the Maximum Likelihood theory
Step 1:
Step 2:
Step 3:
Specify ALT model
parameter values
Minimize the asymptotic
variance of ML estimator
Evaluate the plan
using simulations
• Locally optimal for specified model parameters  0 , 1, 
• Problems:
– There often exists a high margin of specification error
– Developed plans are usually sensitive to the specified value
A Framework of Sequential ALT Planning
Information
Planning information
e.g. test duration, specified
parameter values, etc.
Information on the slope
parameter 1
Planning Procedure
Plan & Perform the test at the
highest stress to quickly obtain
failures
Preliminary information on (  0 ,  )

Planning information
e.g. test duration, number of
stress levels, sample sizes,
etc.
•
•
Plan the tests at lower stress levels
Tang, L.C. and Liu, X. (2010) “Planning for Sequential Accelerated Life Tests”, Journal of
Quality Technology, 42, 103-118.
Liu, X. and Tang, L.C. (2009) “A Sequential Constant-Stress Accelerated Life Testing Scheme
and Its Bayesian Inference”, Quality and Reliability Engineering International, 25, 91-109.
Part I
Planning Sequential Constant-Stress Accelerated
Life Tests
Sample Size at the Highest Stress Level
Specify
 the values of  0 (or  H ) and 
 the censoring time cH
 the expected number of failures RH
Sample Size:
RH
p
nH
Page 8

p  1  exp   cH / exp(  H ) 
1/ 

Inference at the Highest Stress Level
Time in log-scale
 θH   l θH ; DH  θH   H , 
Stress
0
High
Low
1
Use
Inference at the Highest Stress Level
ˆ )
θˆ H | y H ~ N (θˆ H , Σ
H
where
θˆ H  arg max  (θ H )
ˆ  Iˆ 1
Σ
H
Generalized MLE
Covariance matrix
H
Iˆ H  [ 2l (θ H ; D H ) / θ H 2 ]θ
H
θˆ H
Observed information
Page 10
Construction of Prior Distributions
  k , 
  i ,  for xi  0,1
  H , 
Information on
the value of 1
 i   k  1 xi

 is a constant
Construction of Priors at Low Stresses
for any i  1,..., H  1, there exists a one-one transformation θi  (θH )
with non-vanishing Jacobian θH / θi ,such that
1
 (θi )    ( i  1 xi ,  ) 
1
1 ~U ( 1 , 1 )

 ( i  1 xi )
dF ( 1 )
i
 ( i  ˆ H ) 2 
erf ( i )  erf ( i )
 exp  

ˆ
23/ 2 ( var(ˆ H ))1/ 2 (i  i )
2
var(

)

H 
where
i  ˆ H  1 xi , i  ˆ H  1 xi , i  ˆ H  1 xi ,   cov(ˆ H , ˆ H ) / var1/ 2 ( ˆ H ) var1/ 2 (ˆ H )
i  var1/ 2 (ˆ H )  i  var1/ 2 (ˆ H )   ( i  ˆ H )  var1/ 2 ( ˆ H )
 
(2 var( ˆ H ) var(ˆ H )(1   2 ))1/ 2

i
i  var1/ 2 (ˆ H )  i  var1/ 2 (ˆ H )   ( i  ˆ H )  var1/ 2 ( ˆ H )
 
(2 var( ˆ H ) var(ˆ H )(1   2 ))1/ 2

i
Page 12
Illustration of the Sequential ALT
Plan & Run the
test at the highest
stress
Time in log-scale
Deduction of Prior
Distributions
Stress
0
High
Page 13
Low
1
Use
Pre-Posterior
Analysis &
Optimization
The Bayesian Optimization Criterion
Given the information obtained under the highest stress, the
optimum sample allocation and stress combinations for tests
under lower stresses are chosen to minimize the pre-posterior
expectation of the posterior variance of certain life percentile
under use stress over the specified range of β1
Min C (ξ ) = E1 {var( y p (1))}
= E1 {c var(θ0 )cT } c  [1, log( log(1  p))]
Page 14
Problem Formulation
Design Matrix

X 1
x1
1
xH 1
1
xH

T
 E1 (var( y p ( x1 )))





Λ
E1 (var( y p ( xH 1 ))) 



var( y p (0)) 

Min E1 (var( y p (1)))  1( XT Λ 1X) 11T
s.t.
x {( x1 , x2 ,..., xH 1 ) 
π {(1 ,  2 ,...,  H ) 
H 1
H
: 0  xi  1} and xH  0
:  i  i  1 and 0   i  1}
Page 15
Pre-Posterior Analysis
1
E1 (var( y p ( xi )))  
1  1

Σ i  I θi  I
1


1
cΣi cd 1

i 1
where
  2 log l  θi  

 2 log l  θi 
I θi  E  
; y   
f  y dy
2
2

θi
θi


2

log   θ i
i
I 
θi 2
Page 16
Information
contained in
the prior
density
Information
expected to
obtain at
stress level i
Adhesive Bond Test (Meeker and Escobar 1998)
• Total Sample Size: 300
• Total Testing Duration: 6 months =183days
• Standardized Testing Region: 0  xH  x  xU  1
• Assumptions:
log(T ) ~  SEV   ,  
  log A 
Activation energy, Ea
1

Boltzmann constant, k B  8.6171105 T
(Arrhenius)

    0  1 x,

 is a constant
Page 17
0  log A  Ea k B 1  sH , 1  Ea k B 1  ( s0  sH )
Planning at the Highest Temp
Planning
information:
 H  4.72
  0.6
RH  15
cH  60
50 samples are needed
Page 18
RH  nH p
Posterior Density
Simulated Failure times:
33.3, 48.4, 39.3, 58.8, 47.4, 60.0, 33.6, 19.4, 38.0, 28.6, 60.0, 53.2, 17.7, 25.4, 44.5,
34.6, 16.9, 60.0, 31.7, 60.0 ,49.2, 60.0, 10.953, 60.0, 18.8, 3.3, 1.4, 17.3, 46.8, 40.9,
60.0, 28.4, 60.0, 4.2, 21.9, 49.6, 20.6, 60.0, 46.6, 6.4, 25.2, 60.0, 13.6, 29.5, 60.0,
60.0, 31.3, 29.4, 54.3, 34.0
Page 19
Normal Approximation
ˆ )
θˆ H | data ~ N (θˆ H , Σ
H
ˆ  Iˆ 1 and Iˆ  [ 2l (θ ) / θ 2 ]
where Σ
H
H
H
H
H θ
H
θˆ H
Page 20
Planning of an ALT with 2 Stress Levels

 Sample size
nL  300  nH  250
100
 Test duration
cL  183  cH  123
 Posterior density at xH

ˆ
θ H | y H ~ N θˆ H , Σ
H
 Specified range of 1
1  3.84,5.12
(i.e.Ea   0.6, 0.8)

E1 var  y p 1  in log-scale
Planning Information:
10

1
0.1
0.01
0.1
0.2
0.3
0.4
High
0.5
0.6
xL
0.7
0.8
0.9
1
Low
Page 21
Effects of the pre-specified slope parameter
Suppose we raise the expectation of the product reliability
Ea 0.6,0.8  Ea 0.6,0.9 i.e. 1 3.84,5.12  1 3.84,5.76


E1 var  y p 1  in log-scale
Effect:
100
10
Beta1 ranges from
3.84 to 5.76
1
0.1
Run the test under a
higher stress to
produce more
failures
Beta1 ranges from
3.84 to 5.12
0.01
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
High
xL
Low
Page 22
Plan an ALT with 3 stress levels
Planning Information:
Min E1 (var( y0.1 (1)); xL ,  )
nL  250, cL  123
   H ,   , 1  3.84,5.12 
s.t.
nL  M  (1   )  p ( xL )  RL
Additional constraints:
nL  M    p ( xM )  RM
nL  250  (1   )
0  x H  x L  2 xM  1
nM  250  
xM 
for 0    1
0   1
xL
2
Minimum number of failure
RL and RM
where
p( xL )  


 (1  exp( exp( ))) ( ,  )d d 
)    (1  exp( exp( ))) (  ,  )d d 
L
 0
p( xM

L
L
L
L

 0
M
M
M
M
Page 23
M
The feasible region
Page 24
Interior Penalty Function Method
Page 25
Page 26
Inference from Test Results
Simulated failure times (assume 0  4, 1  4,   4)
Stress
Sample
Test
Expected
Simulated
Observed
Level
Size
Duration
Failures
Failure Times
Failures
15
33.3, 48.4, 39.3, 58.8, 47.4, 60.0, 33.6, 19.4, 38.0,
28.6, 60.0, 53.2, 17.7, 25.4, 44.5, 34.6, 16.9, 60.0,
31.7, 60.0 ,49.2, 60.0, 10.953, 60.0, 18.8, 3.3, 1.4,
17.3, 46.8, 40.9, 60.0, 28.4, 60.0, 4.2, 21.9, 49.6,
20.6, 60.0, 46.6, 6.4, 25.2, 60.0, 13.6, 29.5, 60.0,
60.0, 31.3, 29.4, 54.3, 34.0
38
High
xH  0
Mid
xM  0.39
Low
xL  0.78
50
20
230
60
123
123
5
46.1 62.5 86.2 98.9 101.7 123
(×224)
5
5
22.8 44.8 59.1 84.4 87.7
105.2 123 (×224)
6
Page 27
Inference
• Results obtained under the high stress
   H , 

0.0112 0.0003 
~ N  3.87, 0.65 , 

0.0003
0.0086



• Results obtained under the mid and low stress
   M , 
Increasing
Decreasing

 0.0156 0.0016  
~ N  5.28, 0.594  , 

0.0016
0.0080



   L , 

 0.0377 0.0060  
~ N   7.24, 0.664  , 

0.0060
0.0042



Simulation Study
Planning information:
Total Sample Size: 300
Total Test Duration: 183
Pre-specified ALT model parameters: 9 scenarios are considered
*For sequential plans:
We set the expected number of failures at the high stress level
at 15 within 60 days
*For each simulation scenario:
a. both sequential and non-sequential plans are generated;
b. failure data are generate according to the optimum plans;
c. 10th percentile are use stress are estimated;
d. repeat b and c for 100 times, and move to another scenario
Page 29
Simulation Design Table
- k %: the specified value is k% lower than the true value
+k %: the specified value is k% higher than the true value
(0):
the specified value is the true value
Scenarios
Pre-specified
Pre-specified
Pre-specified 1
Pre-specified 1
(non-sequential)
(sequential)
0

1
(0)
(0)
(0)
- 20 % ~ + 20 %
2
- 25 %
- 25 %
- 20 %
- 20 % ~ + 20 %
3
- 25 %
- 25 %
+ 20 %
- 20 % ~ + 20 %
4
- 25 %
+ 25 %
- 20 %
- 20 % ~ + 20 %
5
- 25 %
+ 25 %
+ 20 %
- 20 % ~ + 20 %
6
+ 25 %
- 25 %
- 20 %
- 20 % ~ + 20 %
7
+ 25 %
- 25 %
+ 20 %
- 20 % ~ + 20 %
8
+ 25 %
+ 25 %
- 20 %
- 20 % ~ + 20 %
9
+ 25 %
+ 25 %
+ 20 %
- 20 % ~ + 20 %
Page 30
Simulation Results
Page 31
Precision
1. Sequential plans yields more precise estimation
Variance
2. Sequential plans gives a conservative sense of statistical
precision: Sample variance > Asymptotic variance
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Sample variance
(non-sequential plan)
Asymptotic variance
(sequential plan)
Sample variance
(sequential plan)
0
Page 32
Asymptotic variance
(non-sequential plan)
1
2
3
4
5
6
7
Simulation scenarios
8
9
10
Effect of Parameter Mis-specification on Precision
Sequential Plans
Non-sequential Plans
Effect on
the
expected
variance
Effect on
the
observed
variance
0.270
0.1945
0.016
-0.2075
0.044
0.043
-0.007
-0.1180
For sequential plan:
0.083
0.0655
Since
0.035
0.0185
1.
-0.031
-0.009
Model parameters 0 and  are
estimated at stage one;
For non-sequential plan:
2.
An interval value of 1 is used
Results are sensitive to the specified
model parameters 0 and 1 .
Hence, the plan robustness to the misspecification of model parameters
Page 33
has been enhanced
0
1

0 * 1
 0 *
1 *
0 * 1 *
Effect on the
expected
variance
Effect on the
observed
variance
0

-0.053
-0.038
(0, 0.0001)
(- 0.0001,0)
 0 *
(- 0.0001,0)
(- 0.0001,0)
Robustness
Define the Relative Error (RE) as:
sample variance - asymptotic variance asymptotic variance
3. Sequential plans is more robust to mis-specification of model
parameters
2.5
RE
(non-sequential plan)
2
RE
1.5
RE
1
(sequential plan)
0.5
0
0
Page 34
1
2
3
4
5
6
7
Simulation scenarios
8
9
10
Effect of Parameter Mis-specification on the Relative Error (RE)
Non-sequential Plans
Sequential Plans
Effect
0
1

0 * 1
 0 *
1 *
0 * 1 *
Effect
0
-0.7684

 0 *
-0.7187
-0.2367
0.0011
(0, 0.0001)
(0, 0.0001)
0.4905
0.1334
For sequential plan:
0.1201
Since
-0.0532
1.
Model parameters 0 and  are
estimated at stage one;
For non-sequential plan:
2.
An interval value of 1 is used
RE is sensitive to the pre-specified
model parameters 0 and 1 .
Hence, the plan robustness to the misspecification of model parameters
has been enhanced
Page 35
Simulation Results
4. Sequential plans reduce the degree of extropolation;
5. Sequential plans are especially robust to mis-specification of the
intercept parameters (scenarios 6-9) due to the preliminary test
under the high stress
Optimum low stress
(non-sequential plan)
120
Temperature
100
80
60
40
20
Use stress
Optimum low stress
(sequential plan)
0
0
Page 36
1
2
3
4
5
6
7
Simulation scenarios
8
9
10
Effect of Parameter Mis-specification on the Optimum Low Stress level
Non-sequential Plans
Sequential Plans
Effect
0
1

0 * 1
 0 *
1 *
0 * 1 *
Effect
0
12.25

 0 *
-13.25
3.25
-5
(- 0.0001,0)
(- 0.0001,0)
-1.25
3.25
For sequential plan:
1.75
Since
0.75
1.
Model parameters 0 and  are
estimated at stage one;
For non-sequential plan:
2.
An interval value of 1 is used
RE is sensitive to the pre-specified
model parameters 0 and 1 .
Hence, the plan robustness to the misspecification of model parameters
has been enhanced
Page 37
Comparison with 4:2:1 Plan
ASR 
c:
|Ase  Ase when model parameters are correctly specified|
Ase when model parameters are correctly specified
Test duration at the highest stress level
Extension from 2-Stage Planning to a Full
Sequential Planning
•
•
•
•
•
2-Stage Planning
Prior distributions for all low stresses are constructed
simultaneously (all-at-one)
Tests at all low stresses are planned simultaneously
Full Sequential Planning
Only the prior distribution for one low stress is constructed
Only the test at one low stresses are planned
More tests at low stresses are planned iteratively
The basic framework still works !
Part II
Planning Sequential Constant-Stress Accelerated Life Tests
with
Stepwise Loaded Auxiliary Stress
Liu X and Tang LC (2010), “Planning sequential constant-stress accelerated life tests with
stepwise loaded auxiliary acceleration factor”, Journal of Statistical Planning and Inference,
140, 1968-1985.
Motivations of an Auxiliary Stress
• Testing more units near the use condition is intuitively
appealing, because more testing is being done closer to the use
condition 
• Failures are elusive at low stress levels for highly reliable
testing items 
– the lowest stress level is forced to be elevated, resulting in high,
sometimes intolerable, degree of extrapolation in estimating product
reliability at use stress
Illustration
Low degree of extrapolation
with zero failure
Time
high degree of extrapolation
with more failures
Maximum
Test
Duration
Use
Stress
Candidate low
stress 1
Candidate low
stress 2
High
Stress
Stress Level
Auxiliary Stress
• An Auxiliary Stress (AS), with roughly known effect on product life, is
introduced to further amplify the failure probability at low stress levels
• Examples of possible AS:
– In the reliability test of micro relays operating at difference levels of
silicone vapor (ppm), the usage rate (Hz) might be used as an auxiliary
factor (Yang 2005).
– In the temperature-accelerated life test, the humidity level controlled in
the testing chamber might be used as an AS (Livingston 2000).
– Dimension of testing samples (Bai and Yun 1996)
• Joseph and Wu (2004) and Jeng et al. (2008) proposed a method known as
the Failure Amplification Method (FAMe) for the Design of Experiments.
– FAMe was developed for system optimization while ALT is used for
reliability estimation at user condition through extrapolation.
Model Extension
• Standardization of the level of AS
h  (v  vuse )  (vmax  vuse )1
vuse : use stress vmax : the highest stress
• The extended testing region: [0,1]2
• A scale-accelerated failure time model
  0  1 x   2 h

 is a constant independent of stress
Examples:
Hallberg-Peck model
Higher usage rate model (Yang 2005)
An Integrated Framework of Sequential ALT Planning
with an Auxiliary Stress
Planning Information
e.g. Sample size; Test
duration; Specified model
parameters
Step 1: Plan and perform the life
test at the highest stress level
Step 2: Compute the number of
failures at low stresses
Is an AS needed?
No
yes
Is an AS available?
yes
Step 3b: Plan the tests at low
stresses with an AS
i.e. optimize sample allocation,
stress combinations, and the
loading profile of AS
No
Step 3a: Plan the tests at low
stresses without an AS
i.e. optimize sample allocation,
and stress combinations
Step 1
Planning & Inference
at the Highest Temperature Level
ALT for Electronic Controller
Experiment Target:
To demontrate the 10% life quantile at use condition exceeds 2 years
Stress Factor:
Temperature (other factors, such as humidity, voltage, etc are set to use level)
Planning information and Assumptions:
1). 120 sample units and 75 days are available.
2). The use temperature is 450 C  318K
The highest temperature allowed in the test is 850 C  358K
3). Failure time T follows Weibull distribution

F  t     log  t     

4).  is a constant, independent of temperature;  follows Arrhenius stress-life
relationship
i  log A 
Activation energy, Ea 1
   0  1  si
Boltzmann constant, k B Ti
where  0  log A 1  Ea si  11605 / Ti
Test Planning at the Highest Stress
Planning Inputs:
target number of failures: rk
  1/ 

 k   
 Rk  exp   ck exp    
parameter values: k (  0 ), 
   
 
confidence level: 
censoring time: ck
Planning Output: nk
rk 1
C
i 0
i
nk
nk i
1

R
R
 k  k  1
i
Risk of see less
failures than expected
(Binomial Bogey test, Yang 2007)
Testing Output: ˆ H ( ˆ0 ) or ˆ
Results
Planning
information:
 k  7.5
  0.677
rk  6
ck  720hr
  0.9
44 samples are needed
Data Obtained at the Highest Stress
Weibull Probability Plot for
Observed Failure Data
Time-to-failure (hours)
79.559 210.47
400.56 491.41
673.98
109.4
204.7
425.32
117.15 328.99
720×29
Note: This is just a
particular run
590.03
138.94
149.95
643.31
351.87
0
-0.5
-1
-1.5
-2
-2.5
R² = 0.8995
1.7
2.2
2.7
3.2
Statistical Inference at the Highest Stress
Posterior distribution derived from a constant prior :
  θk ; y   l  θk ; y 

 y j  k  
   1  y j  k

 ck  k 
  exp  j  log   
 exp 

1


exp






j








j 1




 

where θk  ( k , );   0 if the data is censored, otherwise   1
nk
Normal Approximation to the Posterior distribution (Berger 1985)
ˆ )  N (θˆ ,[ Iˆ ]1 )
θ k y ~ N (θˆ k , Σ
k
k
k
where
2
ˆI =  l (θ k ; y )
k
θ k 2 θ
(observed Fisher information at θˆ k )
k
θˆ k
0.0529 
ˆ  Iˆ 1   0.1142
θˆ k  [7.35, 0.90] Σ
k
k
symmetric 0.0489 


Illustration
• The quality of the approximation needs to be checked
e.g. Kolmogorov-Smirnov (K-S) test (Martz et.al 1988, Technometrics).
• The posterior normality needs to be checked
e.g. Kass and Slate 1994 Ann. Statist. ).
Step 2
Computation of the Expected Number of
Failures at Low Stress Levels
Construction of Prior Distributions
  k , 
  i ,  for xi  0,1
  H , 
Information on
the value of 1
 i   k  1 xi

 is a constant
Density Function of the Constructed Prior
1
  i ,      ( i  1 xi ,  ) 
1
 ( i  1 xi )
 ( 1 )d 1
i
 (  ˆ ) 2 
1
 3/ 2
 exp  
  erf ( i )  erf ( i ) 

1/ 2


2 ( var(ˆ )) (i  i )
 2 var(ˆ ) 
i  1,..., k  1
where  ( 1 ) is a uniform distribution defined on an interval [1 ,1 ]
erf is the error function given by the definite integral erf ( z )  2
1/ 2

z
0
e  t dt
i  ˆ k  1 xi
i  ˆ k  1 xi , i  ˆ k  1 xi ,   cov( ˆ k , ˆ ) / var1/ 2 ( ˆ k ) var1/ 2 (ˆ )
 i var1/ 2 (ˆ )  i var1/ 2 (ˆ )   (  ˆ ) var1/ 2 ( ˆ k )
 
(2 var( ˆ k ) var(ˆ )(1   2 ))1/ 2

i
 i var1/ 2 (ˆ )  i var1/ 2 (ˆ )   (  ˆ ) var1/ 2 ( ˆ k )
 
(2 var( ˆ k ) var(ˆ )(1   2 ))1/ 2

i
2
Illustration of the Constructed Priors at
65⁰C and 45⁰C
Let Ea 0.8,1.2, i.e 1 ~ Uniform0.8,1.2
Uncertainty over  becomes larger for lower testing temperature
Expected Number of Failures at Low Stress
In order to see 5 failures, the
temperature is almost on the middle
of the testing region !!
Another Point of View:
Prior Information v.s Information To Be Obtained

det I  θi 
det Ii
  2l (θi ) 
 2 log  (θi )

where I  θi  =E  
and I  θi  = 
2 
θi 2
 θi 
Information to be
obtained by performing
a test at stress level i
Little
Information
obtained from
low temp
“Information”
contained in the
prior knowledge
Step 3
Planning at the Lower Temperature Level
With Auxiliary Stress
•The choice of AS
•The loading of AS
•The integration of AS in test planning
The Choice of AS
Assumpotions:
1). The effect of AS is well understood
2). The failure mode does NOT change after an AS is introduced
Auxiliary Stress: Humidity
Hallberg-Peck Model (Livingston, 2000):
 RH j 
   0  1s  p log 

RH
0 

RH 0 : use humidity level, 60%
RH : humidity level in test (  90%)
The Choice of Loading Profile for AS
ConstantStress Loading
Step-Stress
Loading
A 2-step step-stress loading profile is preferred due to the following
reasons:
• The initial loading will not be too harsh
• The stress can be dynamically monitored given a target time
compression factor (only amplify the failure as needed)
• The verification of the effect of the selected AS is possible
Setting a Target Acceleration Factor
equivalent test duration, ci(e)
Time Compression Factor:  i 
actual test duration, ci
LCEM Cumulative Exposure Model
(Yeo and Tang 1999, Tang 2003)
A Bayesian Planning Problem
Min E1 (var( y p (1)))  1( XT Λ 1X) 11T
s.t.
target time compression  i , for i  1,..., k  1
( x1 , x2 ,..., xk 1 )  [0,1]k 1
Stress levels
(1 , 2 ,..., k 1 )  [0,1]k 1 :  i 1 i  1  k
Sample allocation
(h1 , h2 ,..., hk 1 )  [0,1]k 1
Initial level of AS
( 1 , 2 ,..., k 1 )  [0, c]k 1
Stress changing time for AS
k 1
where
T
1
1 1
X

x
x
x
k 
 1 2
0
 E1 (var( y p ( x1 )))

0
E1 (var( y p ( x2 )))

Λ


0
0

0
0
0


0



var( y p ( xk )) 
0
Planning Results
Planning Information:
 Sample size
n1  120  n2  76
 Test duration
c1  1800  720  1080
 Posterior density at xH

  θ 2  ~ N θˆ 2 , Iˆ 1  θ2 
 1 ~ Uniform  0.8,1.2
 p3
 Maximum RH = 90%
Use RH = 60%
  3

Testing
Condition
Temp
(C)
RH
(%)
Use
45
60
Low
53
High
85
Testing
Duration
Sample
Size
See
Profile
1080hrs
76
60
720hrs
44
Humidity Loading Profile at Low Temperature
Low Humidity Level: 60%
High Humidity Level: 90%
Holding Time: 170.5 hrs
Expected Failures:
Interval [0, 170.5] : No
failure
Interval [170.5,1080]: 5
failures
Interval [1080, ): 71 censored
Illustration: ALT without AS
Relative
Humidity
Point B: (53, 60%)
Failure Probability < 0.01
Point A: (85, 60%)
Failure Probability = 0.32
60%
Temperature
53
85
Illustration: ALT with AS
Relative
Humidity
Point D: (53, 90%)
Failure Probability = 0.08
90%
Point C: (53, 60%)
Failure Probability < 0.01
Point A: (85, 60%)
Failure Probability = 0.32
60%
Temperature
53
85
Sensitivity of the Optimal Plan to p
RHT:
Relative change
of low humidity
holding time
RT/RH
Relative change of
low humidity/low
temperature
RSD:
Relative change of
Asymptotic SD
Sensitivity of the Plan to
the Activation Energy
RHT:
Relative change
of low humidity
holding time
RT/RH
Relative change of
low humidity/low
temperature
RSD:
Relative change of
Asymptotic SD
Evaluation of the Developed ALT Plan
References of Part II
• Liu X and Tang LC (2010), “Planning sequential constantstress accelerated life tests with stepwise loaded auxiliary
acceleration factor”, Journal of Statistical Planning and
Inference, 140, 1968-1985.