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Objectives

To provide an introduction to the statistical analysis of

failure time data

To discuss the impact of data censoring on data analysis

To demonstrate software tools for reliability data analysis

Organization

Reliability definition

Characteristics of reliability data

Statistical analysis of censored reliability data

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- 1. Impact of Censored Data on Reliability Analysis (截尾数据对于可靠性估计和 寿命试验计划的影响) Rong Pan ©2014 ASQ http://www.asqrd.org
- 2. Rong Pan Associate Professor Arizona State University Email: rong.pan@asu.edu Impacts of Censored Data on Reliability Analysis
- 3. Outlines 1/11/2014Webinar for ASQ Reliability Division3 Objectives To provide an introduction to the statistical analysis of failure time data To discuss the impact of data censoring on data analysis To demonstrate software tools for reliability data analysis Organization Reliability definition Characteristics of reliability data Statistical analysis of censored reliability data
- 4. Reliability 1/11/2014Webinar for ASQ Reliability Division4 Meeker and Escobar (1998) ‒ “Reliability is often defined as the probability that a system, vehicle, machine, device, and so on will perform its intended function under operating conditions, for a specified period of time.” Condra (2001) ‒ “Reliability is quality over time.” Leemis (1995) ‒ “The reliability of an item is the probability that it will adequately perform its specified purpose for a specified period of time under specified environmental conditions.
- 5. Reliability Function 1/11/2014Webinar for ASQ Reliability Division5 The reliability function is the probability that an item performs its function for a fixed period of time: The time at which an item fails to perform its intended function is called its failure time. The failure time of an item is a continuous nonnegative random variable, often denoted T 𝑹 𝒕 = 𝑷𝒓𝒐𝒃(𝐢𝐭𝐞𝐦 𝐩𝐞𝐫𝐟𝐨𝐫𝐦𝐬 𝐢𝐧𝐭𝐞𝐧𝐝𝐞𝐝 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧 𝐮𝐧𝐝𝐞𝐫 𝐢𝐧𝐭𝐞𝐧𝐝𝐭𝐞𝐝 𝐜𝐨𝐧𝐝𝐢𝐭𝐢𝐨𝐧 𝐟𝐨𝐫 𝐚𝐭 𝐥𝐞𝐚𝐬𝐭 𝐭 𝐭𝐢𝐦𝐞 𝐮𝐧𝐢𝐭𝐬) )Pr()( tTtR )Pr()(1)( tTtRtF
- 6. Understanding Hazard Function 1/11/2014Webinar for ASQ Reliability Division6 Reliability function Define a hazard function Instantaneous failure Is a function of time Only exponential distribution has constant hazard (failure rate) Relationships between reliability function and hazard function t tTttTt th t )|(Pr lim)( 0 )( )( )( tR tf th t dxxhtH 0 )()( )( )( tH etR )(1)( tRtF dt tdF tf )( )( )Pr()( tTtR
- 7. Characteristics of Reliability Data 1/11/2014Webinar for ASQ Reliability Division7 Failure time censoring Right censoring Left censoring Interval censoring Data from reliability tests Type-I censoring (time censoring) Type-II censoring (failure censoring) Read-outs (multiple censoring)
- 8. Right Censoring 1/11/2014Webinar for ASQ Reliability Division8 Actual failure time exceeds observation In life tests Type-I censoring (time censoring) Type-II censoring (failure censoring)
- 9. Example 1/11/2014Webinar for ASQ Reliability Division9 Low-cycle fatigue test of nickel super alloy (Meeker & Escobar (1998), p. 638, attr. Nelson (1990), p. 272) kCycles Censor 211.626 0 200.027 0 57.923 1 155 0 13.949 0 112.968 1 152.68 0 156.725 0 138.114 1 56.723 0 121.075 0 122.372 1 112.002 0 43.331 0 12.076 0 13.181 0 18.067 0 21.3 0 15.616 0 13.03 0 8.489 0 12.434 0 9.75 0 11.865 0 6.705 0 5.733 0
- 10. Left Censoring 1/11/2014Webinar for ASQ Reliability Division10 Actual life time less than observation May occur when the item is inspected at a fixed time point Less often than right censoring
- 11. Example 1/11/2014Webinar for ASQ Reliability Division11 In the nickel super alloy example, suppose that the observation starts only after 10,000 cycles. kCycles start kCycles end 211.626 211.626 200.027 200.027 57.923 * 155 155 13.949 13.949 112.968 * 152.68 152.68 156.725 156.725 138.114 * 56.723 56.723 121.075 121.075 122.372 * 112.002 112.002 43.331 43.331 12.076 12.076 13.181 13.181 18.067 18.067 21.3 21.3 15.616 15.616 13.03 13.03 8.489 8.489 12.434 12.434 * 10 11.865 11.865 * 10 * 10
- 12. Interval Censoring 1/11/2014Webinar for ASQ Reliability Division12 Observation gives upper and lower bound on failure time Occurs often with scheduled inspections Right censoring and left censoring are special cases Read-outs Grouped data
- 13. Example 1/11/2014Webinar for ASQ Reliability Division13 In the nickel super alloy example, suppose that the test units are inspected at 25, 50, 100, 200 kCycles. kCycles start kCycles end read-outs 0 25 12 25 50 2 50 100 2 100 200 8 200 * 2
- 14. Multiple Censored Data 1/11/2014Webinar for ASQ Reliability Division14 More than one censoring mechanisms are employed Exact failure times, right censoring times and interval censoring times are very common in practice May not be easily recognized Calendar time vs. lifetime
- 15. Example 1/11/2014Webinar for ASQ Reliability Division15 Adapted from an example in Meeker & Escobar (1998), p. 8. Nuclear power plant use heat exchangers to transfer energy from the reactor to stream turbines. A typical heat exchanger contains thousands of tubes. With age, heat exchanger tubes develop cracks. Suppose there are three plants. Plant 1 had been in operation for 3 years, Plant 2 for 2 years, and Plant 3 for only 1 year. All heat exchangers are of the same design and operated under similar conditions. At the beginning, each plant has 100 new tubes. Failed tubes will be removed from the heat exchanger during operation. Plant 1: 1 failure in the first year, 2 failures in the second year, and 2 failures in the third year. Plant 2: 2 failures in the first year, 3 failures in the second year. Plant 3: 1 failure in the first year.
- 16. Example (cont.) 1/11/2014Webinar for ASQ Reliability Division16 Year 1: 300 tubes are tested, 4 failed (left censored), 99 removed (right censored). Year 2: 197 tested, 5 failed (interval censored), 95 removed (right censored). Year 3: 97 tested, 2 failed (interval censored), 95 survived (right censored). Y1 Y2 Y3 P1 P2 P3 (95 survived) (95 survived) (99 survived)
- 17. Example (cont.) 1/11/2014Webinar for ASQ Reliability Division17 initial year 1 year 2 year 3 Plant 1 100 1 2 2 Plant 2 100 2 3 Plant 3 100 1 start year end year at risk removed failed 0 1 300 99 4 1 2 197 95 5 2 3 97 2 start year end year readouts * 1 4 1 * 99 1 2 5 2 * 95 2 3 2 3 * 95 Data by group Data by age Data by censoring type
- 18. Simulation 1/11/2014Webinar for ASQ Reliability Division18 Monte Carlo method Assume a probabilistic model Generate random numbers Compute the statistics of interest Repeat it many times Estimate confidence intervals Useful for evaluating and comparing data analysis methods Useful for evaluating and comparing life test plans
- 19. Simulate Censored Failure Time 1/11/2014Webinar for ASQ Reliability Division19 In MS Excel® Many build-in functions for generated random numbers from specific distribution, such as NORMINV(p, mu, sigma) Utilize GAMMAINV(p, alpha, beta) to generate exponentially distributed failure times Set p=rand(), alpha=1, beta=mean failure time Utilize NORMAINV(p, mu, sigma) to generate the failure times with lognormal distribution Set p=rand(), mu and sigma are the parameters of the lognormal distribution Compute exp(normal random number) No build-in inverse function for Weibull distribution Use function [(-ln(1-rand()))/a]^(1/b) Where a is the intrinsic failure rate of Weibull distribution, b is a shape parameter Use If() function to create censored failure times
- 20. Features of Lifetime Distribution 1/11/2014Webinar for ASQ Reliability Division20 Failure data from electrical appliance test (Lawless, p.7. Attr. Nelson (1970)) Variable: cycles to failure (exact failure time) Nonnegative Right (positively) skewed Some long life observations Normal distribution may not be a good idea!
- 21. Exponential Distribution 1/11/2014Webinar for ASQ Reliability Division21 The simplest lifetime distribution One parameter or Constant failure rate (constant mean-time-to- failure, MTTF) Memoryless property Regardless of past experience, the chance of failure in future is the same. Closure property System’s failure time is still exponential, if its components’ failure times are exponential and they are in a series configuration. )exp()|( ttf )/exp(/1)|( ttf
- 22. Weibull Distribution 1/11/2014Webinar for ASQ Reliability Division22 When the hazard function is a power function of time Two common forms Two parameters Either characteristic life or intrinsic failure rate and shape parameter Relationship with exponential distribution When the shape parameter is known 1 )( t th tt tf exp)( 1 t tR exp)( 1 )( tth ttR exp)( tttf exp)( 1 /1
- 23. Rectification Plot failure probability on a complementary log-log scale Plot time on a log scale Some important features on the plot Slope is the shape parameter Characteristic life can be found at where the failure probability is (1-1/e)=0.632 Reliability at a given lifetime depends on distribution parameters, except at the characteristic life 1/11/2014Webinar for ASQ Reliability Division23 Weibull Plot )log(log))](1log(log[ ttF
- 24. 1/11/2014Webinar for ASQ Reliability Division24 Weibull Plot of Electrical Appliance Data
- 25. Lognormal Distribution 1/11/2014Webinar for ASQ Reliability Division25 From normal to lognormal and vice versa If T has a lognormal distribution, then log(T) has a normal distribution If X has a normal distribution, then exp(X) has a lognormal distribution Median failure time Log(t50) is a robust estimate of the scale parameter of lognormal distribution
- 26. Parametric Distribution Models 1/11/2014Webinar for ASQ Reliability Division26 Maximum likelihood estimation (MLE) Likelihood function Find the parameter estimate such that the chance of having such failure time data is maximized Contribution from each observation to likelihood function Exact failure time Failure density function Right censored observation Reliability function Left censored observation Failure function Interval censored observation Difference of failure functions )(tR )(tF )()( tFtF )(tf
- 27. Exponential Distribution 1/11/2014Webinar for ASQ Reliability Division27 Exact failure times Failure density function Likelihood function Failure rate estimate Type-I censoring Reliability function Likelihood function Failure rate estimate it i etf )( n i itn n etttL 1 ),...,,;( 21 n i it n 1 ˆ ct c etR )( c r i i trntr ccr etttttL )( 21 1 ),...,,...,,;( r i ci trnt r 1 )( ˆ
- 28. Exponential Distribution (cont.) 1/11/2014Webinar for ASQ Reliability Division28 Type-II censoring Likelihood function Failure rate estimate General formula for Exponential failure times r r i i trntr rrr etttttL )( 121 1 ),...,,...,,;( r i ri trnt r 1 )( ˆ timetestingtotal failuresofnumber RateFailure failuresofnumber timetestingtotal MTTF
- 29. Precision of Failure Rate Estimate 1/11/2014Webinar for ASQ Reliability Division29 Sum of exponential distributions becomes gamma distribution Independently identical distributed (i.i.d.) Exact confidence intervals for the cases of exact failure time and type-II censoring An approximated confidence interval for the case of type- I censoring n i i ngammaT 1 ),(~ TTTTTT rr 2 , 2 2 2/1,2 2 2/,2 )ˆ.(.ˆ),ˆ.(.ˆ 2/2/ eszesz r es 2ˆ )ˆ.(.
- 30. Effect of Censoring 1/11/2014Webinar for ASQ Reliability Division30 Widened confidence bounds Use the electrical appliance data Right censoring at 10000, 5000, 4000, 3000, 2000, 1000 # of censored 0 1 4 5 11 23 28 % censoring 0 2.78% 11.11% 13.89% 30.56% 63.89% 77.78% MTTF 2756.806 2738.343 2645.406 2591.097 2956.92 4336.846 3830.125 lower bound 2038.868 2056.893 1964.68 1916.454 2124.224 2809.555 2262.377 upper bound 3936.114 4095.027 4047.949 3998.769 4863.355 9502.421 12474.54 difference 1897.246 2038.134 2083.27 2082.315 2739.131 6692.866 10212.16
- 31. Final Remarks 1/11/2014Webinar for ASQ Reliability Division31 Be aware of censoring when analyzing reliability data Ignoring censored data will bias failure(reliability) estimates Often underestimate reliability The amount of information of censored data depends on the censoring type Nonparametric methods are based on ranks Often utilize the ratio of number of failures and number of items at risk Parametric methods are based on likelihood functions Maximum likelihood estimation Computation becomes complicated Use software Simulation is a very useful tool for studying the effect of sample size or censoring

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