Transcript Slide 1

CUSUM Charts for Censored
Lifetime Data
Denisa A. Olteanu
Virginia Tech
Quality and Productivity Research
Conference
June 3rd , 2009
Content
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Intro
Data
Probability Distributions for Lifetimes
CUSUM Charts for Lifetimes
Conclusions
Introduction
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Reliability
Reliability is the ability of a system to perform a required
function under stated conditions for a stated period of
time.
Quality Control
Early detection of faults with a monitoring program would
allow for repairs to be performed in situations at much
less expense.
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Life Tests
Companies put n items on a test stand and perform life
tests, often under accelerated conditions.
Censoring
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Right-Censoring
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Left-Censoring
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Type I: test stops after a certain time
Type II: test stops after a certain number of failures
are recorded
Item fails before first inspection
Interval Censoring
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When one records times through periodic inspection
Distributions for Lifetime Data
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Typically Non-Normal
Most Popular:
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Weibull
Lognormal
Exponential
Multinomial
Weibull Distribution
Effect of the Shape Parameter for η = 100
0.03
0.025
0.02
f(t)
β = .5 (Early Failure)
0.015
β = 1 (Random Failure)
β = 3 (Wear Out)
0.01
β = 5 (Rapid Wear Out)
0.005
0
The Weibull Distribution and
Relationship to SEV
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Probability distribution function for the Weibull
distribution:
Then Y=log(T) follows a Smallest Extreme Value
(SEV) distribution with:
Log-Normal and Other
Distributions
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If T has a log-normal distribution with parameters μ
and σ, then Y=log(T) is normally distributed with
mean μ and standard deviation σ, and the normal
theory applies
For interval censoring, the counts of failures in each
interval have a multinomial distribution
Other distributions: exponential as a particular case
of the Weibull with shape parameter 1
The Likelihood Function
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General form of the likelihood function for any
distribution and including right-censoring:
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Maximize it to get parameters’ estimates
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Use it to construct likelihood ratio tests
Construction of Likelihood Function for
Weibull Data, using the SEV transformation
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Log-likelihood function: Uncensored Case
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Log-likelihood function: Right-Censored Case
Monitoring Needs
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Interest in monitoring for changes in the parameters
of the usually non-normal distributions used in
Reliability (focus on Weibull)
Different types of censoring patterns present (focus
on right-censoring)
Searched literature for monitoring methods of
interest
In the Literature:
Monitoring Lifetimes
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Approaches:
 Conditional Expected Value (CEV) methods
 Monitoring for changes in small percentiles of
interest
 Methods based on likelihood ratio tests
 Other methods
Shewhart-type charts for uncensored data, with only
one parameter changing and CEV-based methods
monitoring for shifts in mean are predominant
In the Literature:
CEV Methods
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Underlying CEV approach, independent of the
distribution used
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Weights replace right-censored data points, weights
determined as:
where
and C is the censoring time
CEV Methods: Examples
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Steiner and MacKay (2000) developed and recommended
the use of the Extreme Value CEV Shewhart-type chart for
grouped right-censored data
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They monitor for decreases in the mean of the Weibull
distribution, that models lifetimes; the shape parameter is
fixed
They use the SEV transformation and plot the sample
averages of the transformed data, with censored points
replaced by the CEV weights
Zhang and Chen (2004) constructed a EWMA chart for
monitoring the mean of censored Weibull lifetimes using
CEV approach
CUSUM chart development
for Lifetimes
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Cases considered:
- Uncensored data
- Right-censored
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Underlying distribution: Weibull
Positive or negative shifts in the scale parameter, η
General frame: CUSUM chart based
on the sequential probability test approach
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Samples of n lifetimes are collected from the process
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We consider a Weibull distribution for our lifetime data
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We use the SEV transformation
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The in-control values (under the null hypothesis that the
process is in control) for our parameters of interest are
given, or estimated from in-control historical data using
MLE
The shift to an out-of-control situation in the parameter of
interest is defined by giving an out-of-control (alternative
hypothesis) value for the parameter
General frame: CUSUM chart based
on the sequential probability test approach
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Cumulative Sum (CUSUM) charts:
 Generally superior to traditional Shewhart charts
Likelihood Ratio Tests:
 Prominence as measure of statistical evidence in
hypothesis testing, sequential sampling, and
development of CUSUM charts
 Accommodate different underlying distributions
 Accommodate censoring
General frame: CUSUM chart based
on the sequential probability test approach
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The CUSUM chart plots
where
and y(i) is the i-th sample of n log-lifetimes
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The chart signals when S crosses a threshold found
through simulations
Uncensored case, Chart for
the Scale Parameter
o
The test statistic becomes:
Right-Censored Case, Chart
for the Scale Parameter
o
The test statistic becomes:
Properties: Simulation Results
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CUSUM chart for monitoring the scale parameter
eta, beta fixed, uncensored case:
- Sample size=Number of failures=20
- Beta=0.5
- In-control eta=1
- Shift d=0.5, out-of-control eta=0.5
- Number of simulation replications=1000
- Number of generated samples=1000
- Chart threshold=4.56
- Out-of-control ARL=4.88, simulation error=0.005
- In-control ARL=378, simulation error=7
Properties: Simulation Results
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CUSUM chart for monitoring the Scale parameter
eta, beta fixed, right - censored case:
- Sample size=20
- Number of failures=15
- Beta=0.5
- In-control eta=1
- Shift d=0.5, out-of-control eta=0.5
- Number of simulation replications=1000
- Number of generated samples=1000
- Chart threshold=1.22
- Out-of-control ARL=11.2, simulation error = 0.03
- In-control ARL=385, simulation error=7
CUSUM Chart
Conclusions
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SPRT-based CUSUM Charts for Non-normal distributions
and Censored data should bridge the gap between
Reliability and Quality Control fields
The existing methods in the literature for monitoring
lifetimes predominantly focus on uncensored data,
Shewhart-type charts, and monitor for the mean, while
reliability professionals usually focus on individual
parameters