CUSUM Charts for Censored Lifetime Data Denisa A. Olteanu

Virginia TechQuality and Productivity

Research ConferenceJune 3rd , 2009

Content

Intro Data Probability Distributions for Lifetimes CUSUM Charts for Lifetimes Conclusions

Introduction

ReliabilityReliability is the ability of a system to perform a required function under stated conditions for a stated period of time.

Quality ControlEarly detection of faults with a monitoring program would allow for repairs to be performed in situations at much less expense.

Life TestsCompanies put n items on a test stand and perform life tests, often under accelerated conditions.

Censoring

Right-Censoring Type I: test stops after a certain time Type II: test stops after a certain number of failures are

recorded

Left-Censoring Item fails before first inspection

Interval Censoring When one records times through periodic inspection

Distributions for Lifetime Data

Typically Non-Normal Most Popular:

Weibull Lognormal Exponential Multinomial

The Weibull Distribution and Relationship to SEV

Probability distribution function for the Weibull distribution:

Then Y=log(T) follows a Smallest Extreme Value (SEV) distribution with:

Log-Normal and Other Distributions

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

General form of the likelihood function for any distribution and including right-censoring:

Maximize it to get parameters’ estimates

Use it to construct likelihood ratio tests

Construction of Likelihood Function forWeibull Data, using the SEV transformation

Log-likelihood function: Uncensored Case

Log-likelihood function: Right-Censored Case

Monitoring Needs

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

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

Underlying CEV approach, independent of the distribution used

Weights replace right-censored data points, weights determined as:

where

and C is the censoring time

CEV Methods: Examples

Steiner and MacKay (2000) developed and recommended the use of the Extreme Value CEV Shewhart-type chart for grouped right-censored data

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

Cases considered:- Uncensored data

- Right-censored

Underlying distribution: Weibull

Positive or negative shifts in the scale parameter, η

General frame: CUSUM chart based on the sequential probability test approach

Samples of n lifetimes are collected from the process

We consider a Weibull distribution for our lifetime data

We use the SEV transformation

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

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

The CUSUM chart plots

where

and y(i) is the i-th sample of n log-lifetimes

The chart signals when S crosses a threshold found through simulations

Uncensored case, Chart for the Scale Parametero The test statistic becomes:

Right-Censored Case, Chart for the Scale Parametero The test statistic becomes:

Properties: Simulation Results 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 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

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