Accelerated life tests (ALTs) are employed to generate failure time data at higher-than-normal-use stress levels. ALT planning is critical for achieving statistical efficiency and reducing experimental cost through design of experiments (DOE). In this talk, I will describe a real world example of ALT planning and its impact on decision making. I will present models for regression with failure time data, including exponential and Weibull regression. Censoring, which is present in many life testing experiments, and its effect on regression models is discussed. Graphical methods for data analysis of life testing experiments are discussed, as well as the software for ALT planning and data analysis.
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3. An Introduction to ALT Planning
g
Rong Pan, Ph.D.
Schools of Computing, Informatics, Decision Systems Engineering
Arizona State University
4. Outline
• Topic 1: Statistical Inferences in ALT
• T i 2: Experimental Design in ALT
Topic 2 E i lD i i
• Topic 3: Software
p
• Q&A
• References:
• Wayne B. N l
W B Nelson (1990) Accelerated Testing: Statistical Models, Test
A l t d T ti St ti ti l M d l T t
Plans, and Data Analysis, John Wiley & Sons, Inc., Hoboken, NJ.
• William Q M k and L i A E
Willi Q. Meeker d Luis A. Escobar (1998) Statistical Methods for
b St ti ti l M th d f
Reliability Data, John Wiley & Sons, Inc., New York, NY.
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5. Topic 1: Statistical Inferences in ALT
• Backgrounds of topics: ALT and SSALT
• E
Exponential and W ib ll regression
i l d Weibull i
• Statistical inference methods
• Parameter estimation
• Conclusions
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6. Accelerated Life Testing (ALT)
• The need for highly reliable components and materials are widely
required for long-term performance
– unacceptable length of time and cost of product life testing experiments
under use condition
• Units are tested under more severe conditions (or stresses) than the
use condition
Stress
Stressed condition
Use-condition
Failure Time
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7. Step-Stress Accelerated Life Testing (SSALT)
• SSALT is an advanced case of ALT
• Under SSALT, test units are run at different stress levels over time
(
(usually increased stress levels) while ALT is conducted at a constant
y )
stress level
Stress
x2
x1
Failure Time
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8. An Example
• Nelson (1980) data, obtained from a SSALT of cryogenic cable
insulation
- consists of four different test plans (groups)
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9. Things to Remember
• Failure time data are often censored
• Right censoring
• Interval censoring
• By test plan: type I censoring, type II censoring
type-I censoring type-II
• Failure time distribution cannot be normal distribution
• E ponential
Exponential
• Weibull
• L
Lognormal
l
• Data need to be extrapolated
• Use condition is outside of experimental region
• Extrapolation model is needed
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10. Exponential Regression
• Failure time distribution is assumed to be exponential distribution
p
1
f (t ) = e −t / α , t>0
α
• Mean failure time (or mean time to failure, MTTF) and failure rate (or
hazard function)
MTTF = α = 1 / λ
• Relationship with covariates (stresses)
log MTTF = β 0 + β1 x1 + β 2 x2 + ....
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11. Weibull Regression
• Weibull distribution generalizes exponential distribution
g p
γ γ −1 −( t / α )γ
f (t ) = γ t e , t>0
α
• Characteristic life α
• Relationship with covariates (stresses)
log α = β 0 + β1 x1 + β 2 x2 + ....
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12. Acceleration Model
• Acceleration Factor (AF): the acceleration constant relating times
( ) g
to fail at the two stresses
(time to fail at lower stress) = AF x (time to fail at higher stress)
- through AF, we can project the results obtained from experiments
to the use condition
- even if information on AF is unknown, the results at higher stress
levels can b extrapolated t th use-condition b an appropriate
l l be t l t d to the diti by i t
physical acceleration model
e.g., Arrhenius, inverse power, Eyring models, etc.
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13. Statistical Inference Methods in Reliability
• The statistical inference technique for ALT/SSALT
q
– Estimate model parameters
– Predict failure behavior at the use condition
• Two main statistical approaches:
– i) Classical approach (based on MLE)
– ii) Bayesian approach
• (-) the limitation due to its inherent model complexity and
computational intractability
• (+) the advent of Markov chain Monte Carlo (MCMC)
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14. Dept. of Industrial Engineering
Classical Approach
• Find the contributions of each observation to the total likelihood
function
• Failure time – probability of failure (probability density function)
• Right censoring time – probability of survival (reliability function)
• Interval censoring – probability of failure in the interval (difference of two failure
functions)
• Loglikelihood
• Maximize loglikelihood
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15. Example - likelihood
• Total likelihood over l stress levels:
l ki
L = ∏∏ f ( yij ) R ki : risk set at stress level i
cij 1− cij
( yij )
i =1 j =1
l ki
= ∏∏ λ exp(−λi yij ) cij: indicator variable
cij
i
i =1 j =1 for censoring
Using μij = λi yij and K = ∑ i =1 ki
l
l ki
L = ∏∏ μijij exp(− μij ) × yij
c − cij
i =1 j =1
K
= ∏ μ kck exp(− μ k ) × yk ck
−
k =1
ck ~ Poisson (μk) Offset: does not depend on λk 13
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16. Bayesian Approach
• Assume some prior distribution for parameters
• Prior information is subjective
• Noninformative priors reduce the subjectivity of Bayesian analysis and minimize the
impact of priors on posterior distributions
• Combine likelihood function with prior distributions
posterior ∝ likelihood × prior
• Inferences are made based on posterior distributions
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17. The Example - Analysis
• Parameter Estimation
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18. Conclusions
• ALT/SSALT is an designed experiment for test-to-failure
• Pay attention to failure data type
• Select an appropriate regression model
• Data analysis is not difficult …
• If you know how to use a computer tool
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19. Topic 2: Experimental Design in ALT
Problem Statement
Model & Model Parameters
Use Condition & Simulation
Design Region constraints & Feasibility Region
Parameter Estimation comparison between different experimental
designs
Conclusions
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20. Things to Remember
• How to plan an ALT
• E i
Engineering concerns – material, equipment, h
i t i l i t human resources, b d t
budgetary and ti
d time
constraints
• Statistical concerns – sample size, q
p quality of inference
y
• Standard DOE
• Unsuitable to failure time data – non-normal distribution, non linear regression,
non normal non-linear
censoring
• Optimal test plans
• Difficult to obtain
• Could be sensitive to model assumption
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21. A Real-World Example
Situation:
– ALT plan for evaluating solder joint reliability
Problem:
– Minimize uncertainty about the model parameter estimates
– Equipment, Materials, and Time to Market constraints
– Use of “industry standard” test conditions lead to sub-optimal model
parameter estimates
Scope:
– Eyring based acceleration model
– Weibull life distributions simulated based on “known” model parameters
– Interval and right censored data
Objective:
– Compare the design matrix influence to other design factors (n censoring)
(n,
– Identify designs that reduce parameter estimation variance (D-optimality
criteria)
See more, Monroe and Pan, Journal of Quality Technology, 2008.
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22. Model Parameters
Eyring based model
TTF = A ⋅ (ΔT )− a (t −b exp ⎡− c⎛ ⎞⎤
dwell )
1
⎢ ⎜ ⎜
⎟⎥
⎢
⎣ ⎝ Tmax ⎟⎥
⎠⎦
t dwell
Tmax
T
ΔT = Temp
Amplitude
A lit d
Tmin
1 cycle
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23. Test Instrument
Temperature Cycle Chamber
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24. Log-Linear Transformation
Eyring based model
⎡ ⎛ 1 ⎞⎤
TTF = A ⋅ (ΔT ) (tdwell )
−a −b
exp ⎢ − c⎜
p ⎜ T ⎟⎥
⎟
⎣ ⎝ max ⎠⎦
⎛ 1 ⎞
ln (TTF ) = ln ( A) − a ⋅ ln (ΔT ) − b ⋅ ln (t dwell ) − c⎜
⎜T
⎟
⎟
⎝ max ⎠
Log-linear
Log linear function
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25. Case Study Via Simulation
Assume Product lifetime ~ Weibull (αuse=10,000; β=2)
Assume Use environment
– t-dwell = 10 minutes; ΔT = 65°C; Tmax = 85°C
“Known” parameter estimates [1]
– a= 2.65;
; b= 0.136;
; c= 2185
Used to generate expected lifetime distributions for each test condition
– Characteristic life α = αuse /AF
life,
Compare various censoring conditions
– None (exact cycles to failure)
– Right censoring at characteristic life
– Interval censoring (every 250 cycles)
[1] = “An Acceleration Model for Sn-Ag-Cu Solder Joint Reliability under Various Thermal Cycle Conditions”. Hewlett Packard Company.
Surface Mount Technology Association International (SMTA). September 25, 2005.
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26. Constraints
Materials:
– T-max ≤ 125°C
T max 125 C (test boards melt)
– tdwell ≥ 3 minutes (stress relaxation threshold)
Equipment: Temperature Cycle Chamber
– T min ≥ -55°C
T-min 55°C (condenser limit)
– T-max≤ 150°C (pressure vessel limit)
– tdwell ≤ 24 minutes (availability)
Time to Market
– ΔT ≥ +80°C
– AF ≥ 3.5x use condition (
(test time limit of 6 months)
)
Unique Test Conditions (N=4)
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27. Design Region
View
– td ll=24 minute plane
dwell
Constraints
– Material constraint (blue)
– Equipment (green)
– Time to market ΔT (red)
– Time to market AF (orange)
Design region
– In plane: between 5 vertices
– Out of plane: between dwell
time of 3 and 24 minutes
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28. Standards Based Testing
Originated prior to the implementation of “mechanism based” or “use
mechanism based use
condition” based testing strategies
Goal: simply meet the performance set by the previous product
These conditions were not selected in a Design of Experiment context
However, customers are very familiar with these benchmarks and often
request these tests from their suppliers
Example: Temp Cycle “B”
B
– Temperature range: 180°C [-55°C, +125°C]
– t-dwell ~10 minutes (total cycle time specified)
– Tmax 125
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29. Designs Considered
Standards based Orthogonal (23-1) Recommended
1.90%
1 90% 24.87%
24 87% 70.71%
70 71%
= in plane (tdwell = 24 minutes)
= out of plane (tdwell=8 minutes)
D-efficiency scores are percentages in red
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30. Censoring Options Considered
No censoring – exact cycles to failure over entire lifetime
Right censoring at characteristic life (63.2%)
Interval censoring – readouts taken every 250 cycles
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31. Results: How to read the graph
True parameter value
Sample sizes
Data censoring
Experimental D i
E i t l Design M t i
Matrix
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32. Results: Parameter b
D-optimal design converges to true estimate much faster
Is robustness to both right and interval censoring
Is efficient with minimal sample sizes required
Instability of estimates for Standard design with small sample size
y g p
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33. Conclusions
D-Optimal based experimental designs:
Work well with constrained regions
Improved precision on parameter estimates
– Recommended design w/ n=25 outperformed Standards design w/ n=500.
– Slightly better results than orthogonal design (fractional factorial)
Test planning is an influential step
– O t i h b th sample size and censoring effects in terms of influence
Outweigh both l i d i ff t i t f i fl
– Yet they are not often considered in practice
Enable model form to be validated without masking of variables
Assume that the model form is known
A th t th d lf i k
– Orthogonal designs may be a preferred choice for robustness when
model form is unknown a priori
– Optimality solution in model specific
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34. Topic 3: Software
Data
D t analysis
l i
– Most statistical software can handle it, e.g., SAS, MiniTab, S-plus, R
– Some reliability engineering software dedicated to failure time data
analysis: Weibull++, ALTA
ALT planning
– Not many tools available, so may need special codes for specific task
– A few of them: JMP, SPLIDA, Minitab
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35. SAS & JMP
SAS Proc for failure time data analysis
P f f il ti d t l i
– Proc LIFEREG fits Weibull, lognormal, loglogistic regression models for
censored data
– Proc PHREG fit the proportional hazard regression model
– Bayesian data analysis can be requested in these two procedures
JMP
– JMP is a business unit of SAS Inc., specializing design of experiments
– JMP9 has enhanced ALT test planning
– Helpful tutorial website: www.jmp.com/applications/reliability
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36. R & SPLIDA
R is f
i free
– Supported by the statistics community
– Survival package
– library(survival)
– Surv() defines a survival data objects
– survreg() fits ALT regression model
SPLIDA
– Developed by Dr. Meeker
p y
– Originally a free add-on program to S-Plus, recently converted it to R
– Provides the functions for single variable ALT data analysis, multiple
regression ALT data analysis, residual diagnosis, and ALT planning
analysis diagnosis
– Simulating and evaluating ALT experiments
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37. Weibull++ & ALTA
Reliability
R li bilit engineering software f
i i ft from R li S ft
ReliaSoft
Developed for solving engineering problems
– Interactive user interface
– Spreadsheet format
– Graphical displays
Weibull++ can fit most lifetime distributions for censored data
ALTA is for ALT and ADT data analysis
– Physical acceleration model is explicitly defined
– Can handle more complicated tests, such as SSALT.
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38. Minitab
Reliability functions are li it d b t enough f most engineering
R li bilit f ti limited, but h for t i i
applications
– Reliability data analysis Stat->Reliability/Survival->Accelerated life
testing… or Regression with life data…
– ALT test planning Stat->Reliability/Survival->Accelerated life test plans…
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39. Summary
Topics discussed
T i di d
– Statistical inference concerns with how to estimate model parameters
– Design of experiments concerns with how to plan experiments
efficiently
ALT data analysis and test planning require advanced
statistical methods
Following techniques introduced: Weibull regression, MLE,
Bayesian inference D-optimal experimental design
inference,
Appreciation of DOE
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