Progress in Reliability ASQ Silicon Valley Statistics and
Progress in Reliability? ASQ Silicon Valley Statistics and Reliability Discussion Group January 13, 2021 “Giving statistical software to engineers is like giving guns to children” Dick Mensing, LLNL 1981 “…major reliability improvements are made possible, primarily, by radically new technology either in the product itself or in manufacturing methods. ” C. Knight, IEEE, 1989
Summary of Progress? • Would you like age-specific field reliability estimates for all products and service parts. It’s easy, if you track everything by name and serial number. • What if there’s no lifetime data? GAAP requires statistically sufficient ships and returns data in sales, Bo. M, and service costs • What if there’s life limits? E. g. , Epipen epinephrine 2 -year limit • What if there’s cycles? E. g. , COVID-19 cases and deaths; survival function = reliability function • Why not compare sample vs. population? Pfizer’s placebo has 86% efficacy compared with US population. Use neutrosophic statistics for life tests. • Progress in Artificial Stupidity: Cognitive Reli-Ability Test sample questions.
Outline of Progress in Reliability? • Why not estimate reliability and survival functions, without life data? • GAAP requires statistically sufficient data, including repairable or renewal processes! • Why not estimate reliability, with life limits, without life data? • Life limits remove cohorts from risk • COVID-19 survival analysis, with cyclic cases, without life data [CA, US, …] • Livermore protest, Trump Tulsa Rally, Sturgis HD, seasons, … cause cycles • Vaccine efficacy using population risk • “Why Kill Controls? ” Neutrosophic Statistics? ! • Pfizer COVID-19 placebo has 88% efficacy! Convenience sample vs. population • Progress in Artificial Stupidity: “Cognitive Reli-Ability Test” • CRE test is a minimal qualification. CRA test is high qualification.
Use available data! Data required by GAAP! “Failure to exploit innovation has become almost a guarantee of mediocrity. ” [C. Bruce Tarter, LLNL Director] • BIOSTATISTICS: Case and death counts are statistically sufficient to make nonparametric estimates of survival functions • Observe M(t)/G/∞ inputs and outputs, ergodic theory, max. likelihood Ĝ(t) • Compare treated sample survival function estimate with untreated population! • Multivariate: COVID-19 death or recovery, network tomography, origin-destination and travel times, renewal or recurrent processes, without or with AI [Bin Yu et al. ] • RELIABILITY: Ships and returns counts are statistically sufficient to make nonparametric estimates of reliability functions • GAAP requires: revenue=>sales=>ships, and warranty and service costs =>returns • returns are parts, part-installed-base(t) = product-installed-base(t)*(I N) 1 where N is gozinto matrix of parts per next-assembly (Bo. M) • Products, service parts, humans, covered calls, trout, droughts, breast implants, …
Counts are not lifetime data • Lifetime data requires tracking individuals, products or units by name, serial number, or other unique identifier, from birth to death (“endpoint”) • T 1, T 2, T 3, … subjects 1, 2, 3…, perhaps censored • Ships (Cases) n 1, n 2, n 3, … and returns (deaths or recoveries) R 1, R 2, R 3, …contain lifetime information • Deaths could have come from current or any previous cohort! Deaths are not identified by case cohort. • Complaints, repairs, replacements, spare parts’ sales may not be identified by cohort
People Say… • “It can’t be done. ” [Nonparametric estimation without life data] • …estimation requires detailed individual patient data on the time from admission (or illness onset) to death or full recovery. ” [Yu et al. , SARS] • “For device failures, the year of device implant was not known. ” “An individual patient’s risk…could not be sufficiently evaluated in this study, such as device model and years since implant. ” [Maisel, JAMA. ICDs] • “Accurate assessment of actual device performance is not possible based on limitations of the post-marketing surveillance system for medical devices. ” [Estes, JAMA, AEDs] • "The data required to provide an expected replacement profile are not automatically available at this time; thus, the expansion of this theory [actuarial forecast] into full practical application. " [James Krupp, automotive] • “…as well as the failure to maintain life cycle part and labor histories at the end item level, makes it difficult to apply standard “economic useful life” models…” [RAND, M 1 Tanks] • • “Field data are garbage. ” [Ralph Evans, IEEE Trans. on Reliability editor] “It takes too much work. ” [Apple geologist acting as reliability engineer] “Where can I get software for repairable systems? ” [Wayne Nelson] “I knew it was a good idea all along. ”
M(t)/G/∞ service system I/O counts: unlimited vs. life limit = 1 period Ships Returns n 1 R 1 n 2 R 2 n 3 R 3 n 4 Time R 4
With Life Limits: Unobserved Truncations vs. Observed expirations, X 1, X 2, … Ships Returns n 1 R 1 n 2 R 2 n 3 R 3 n 3 X 1 R 3 X 2 n 4 Time R 4
What if you count expirations? • If you don’t know X 1, X 2, … counts of expirations, likelihood is • PPoisson[Rj, Sl(i)*g(j i)], Si=1, 2, …, min(j, life limit)], and Pj = 1, 2, …number of cohorts [George, Mirasol, Kotkin, Eick et al. (1993)] • Rj = returns in period j and Poisson[Rj, Sl(i)*g(j i)] = (Sl(i)*g(j i)])Ri. Exp[ Sl(i)g(j i)]/Ri! • If you know X 1, X 2, … counts of expirations, likelihood is • PPoisson[Rj, Sl(i)*g(j i)]*(1 G(life limit))Xi • I haven’t done this; ask and send data if you want reliability estimate with expiration counts
Nonparametric Reliability Estimation, With Life Limits but Without Life Data Epi. Pen epinephrine dose is life limited Flight Safety-Critical systems’ parts with life limits not tracked by serial number Auto parts with recommended replacement mileage or age Rx. Bandz. com Mini. Ject could use Epi. Pen reliability as Bayes prior distribution RCM includes life limits depending on failure rate function Renewal or recurrent processes without renewal counts
Epi. Pen: Assuming life limit = 4 years • Sales from https: //www. sec. gov/Archives/edgar/data/1623613/0001193125167 19397/d 265624 dex 991. htm and FORECAST() • Return counts from Drug. Watch. com and news Returns Year Sums Sales 2015 2016 2017 2018 2019 2020 2015 8, 300, 000 120 ? ? ? Limited 2016 8, 000 ? ? Limited 2017 9, 322, 290 ? ? 2018 9, 880, 237 ? ? ? 2019 10, 438, 186 ? ? 2020 10. 966, 133 ? 120 360 720 840 1080
Least-Squares Reliability Estimates • Carl Harris and Ed Rattner (AIDS cases) and Patric Oscarsson and Oerjan Hallberg (Ericsson) used least squares • Epi. Pen sales from 2008; early returns estimated 120/year, Drug. Watch. com says 2880 since 1993. “R(t) limited” based on 20152020
Max. Likelihood Reliability Estimates R(t) • George and Agrawal, NRLQ, 1973, https: //sites. google. com/site/fieldreliability/ or • Sales from 2008; early returns 120/year, Drug. Watch. com says 2880 returns since 1993. R(t) limited based on 2015 -2020. See similar shapes?
Methods for Life Limits • For max. likelihood, restrict ratios of Sreturns/Sships to ships cohorts still within life limits and “pool adjacent violators” (PAV) • For least squares, restrict actuarial hindcasts Sg(s)n(t s) to ships cohorts n(t s) within life limits: g(s) is pdf of service life. Sum from s = 0 or 1 to t = life limit. • Trying to automate spreadsheets for any life limit • Deal with operating hours per calendar hour. Life limits could be operating hours. Ships and returns counts are in calendar intervals. • Deal with random conformity with life limits • Program in R package (using PAVA)?
Uncertainty? • Not sample uncertainty, because cases and death counts (or ships and returns counts) are population data! • Uncertainty of M/G/∞ < uncertainty of M(t)/G/∞ • https: //sites. google. com/site/fieldreliability/random-tandem-queues-andreliability-estimation-without-life-data • Fisher information and Cramer-Rao lower bound for variance of max. likelihood estimator • Confidence limits on actuarial forecast (expected demand) • Prediction limits on demand random variable for fill rate • Bootstrap https: //en. wikipedia. org/wiki/Bootstrapping_(statistics)/ • Two-pack?
WIP • Randomness around life limit • Simultaneously estimate reliability and life limit distributions • Life limit is same as truncation, except you don’t know time-to-failure if before life limit • What if proportion a used beyond life limit? • Combine with Optimal Opportunistic Maintenance for life limited parts • “Build that sucker so it doesn’t come back to the shop for 600 hours!” Kelly AFB engine manager • “What else shall we replace as long as we’ve got this torn apart? ” • Kits: customers who bought A also bought ? ? ? • What else shall we stock to go with life-limited parts’ stocks? • Extend life limit? (d. P[Life ≤ t]/dt)*(dt/d$$$) for t > life limit?
References Billingsley, Patrick, Ergodic Theory and Information, Wiley, 1965 Chan, Kuen Chuen Gary, “Survival analysis without survival data: connecting length-biased and case-control data, ” Biometrika. 2013; 100(3): 10. 1093/biomet/ast 008. , doi: 10. 1093/biomet/ast 008 Eick, Stephen G. , William A. Massey, and Ward Whitt, “Mt/G/∞ Queues with Sinusoidal Arrival Rates, ” Mgt. Sci. , vol. 39 no. 2, Feb. 1999, pp. 241 -253 S. G. Eick, W. A. Massey and W. Whitt, “The Physics of the Mt/G/∞ Queue, ” Ops. Res. , 41, 731 -742, (1993) George, L. L. , Please see https: //sites. google. com/site/fieldreliability/ or https: //sites. google. com/view/fieldreliability/ Harris, Carl M. and Edward Rattner, 1997, “Estimating and projecting regional HIV/AIDS cases and costs, 1990 2000: A case study, ” Interfaces, Vol. 27, No. 5, pp. 38 53 Meyer Kotkin, “The Output of the M(t)/G(t)/∞ Queues, ” AMSAA, AD-A 120545, July 1982 Mirasol, N. M. , The Output of an M/G/∞ Queuing System is Poisson, " Operations Research, 11, 282 -284, (1963) Oscarsson, Patric and Örjan Hallberg, “ERIVIEW 2000 – A Tool for the Analysis of Field Statistics, ” Ericsson Telecom AB, (2000) Bruce W. Turnbull, “Nonparametric estimation of a survivorship function with doubly censored data, ” J. Amer. Statist. Assn. , Vol. 69, No. 345, pp 169 -174, March 1974, https: //sites. google. com/site/fieldreliability/home/turnbull J. S. Usher and T. J. Hodgson, "Maximum likelihood analysis of component reliability using masked system life-test data, " IEEE Transactions on Reliability, vol. 37, no. 5, pp. 550 -555, Dec. 1988, doi: 10. 1109/24. 9880. Wilson, S. , Joyce, T. and Lisay, E. Reliability estimation from field return data. Lifetime Data Anal 15, 397– 410 (2009). https: //doi. org/10. 1007/s 10985 -009 -9118 -4 Wolf, Michael and Dan Wunderli, “Bootstrap Joint Prediction Regions, ” Working paper 64, ISSN 1664 -705, May 2014, University of Zurich, Economics
https: //sites. google. com/site/fieldreliability/c orona-virus-survival-analysis % daily increase in infected cases in Tulsa area 0, 2 • Max. likelihood and least squares nonparametric estimators quibble • Cycles: Livermore protest in March, Tulsa Trump Rally, Sturgis, etc. 0, 18 0, 16 0, 14 0, 12 • What do cycles do to estimators? Do survival functions change? Definitions change from symptomatic, Xray, to PCR, antibody • Vaccination efficacy and survival analyses. 0, 1 0, 08 0, 06 0, 04 0, 02 0 1. 17. 2020 3. 7. 2020 4. 26. 2020 6. 15. 2020 8. 4. 2020 9. 23. 2020 11. 12. 2020
Nonparametric Methods Agree! • Estimate pdf ĝ(s) for M(t)/G/∞ system from input and output counts • It’s pretty hard to dispute M(t) ~Poisson[l(t)] =>output ~Poisson[l(t-S)] where S is lifetime with distribution G(s) • Least squares: min S(observed expected)2/expected or S(observed expected)2 (SSE) • Observed R(t) = deaths, failures, Returns, spares sales, etc. • Expected R(t) = Sn(t s)ĝ(s), s=1, 2, …, t • Max. Likelihood = PPoisson[n(t); Sl(t s)ĝ(s)], s=1, 2, …, t • “Pool adjacent violators” algorithm E[l(t S)]=Sl(t s)ĝ(s) to make ĝ(s) nondecreasing in t
Compare Interval Ships 0, 99 0, 98 Returns 1 2 3 4 5 6 100 100 100 Interval 1 Ships 1 2 3 4 5 6 0, 97 1 1 3 5 1 5 0, 96 0, 95 0, 94 Ships 104 87. 5 80. 6 106. 6 94. 8 98. 1 Returns 1 2 3 Poisson R(t) Returns 100 110 120 130 140 150 0 1 1 3 5 1 5 1 0, 995 0, 99 0, 985 0, 98 0, 975 0, 97 0, 965 0, 96 0, 955 0 1 2 Poisson R(t) 4 5 npmle R(t) 3 5 nplse R(t) 6 7 nplse R(t) actuarial 1 0, 99 0, 98 0, 97 0, 96 0, 95 0, 94 0 1 2 Poisson R(t) 3 4 npmle R(t) 7 nplse R(t) 4 npmle R(t) 6 5 nplse R(t) 6 7
CFR and US Moving Average (7 days) Cases US COVID-19 Cases and Case Failure Rate 80000 70000 CFR 7, 00% 6, 00% Cases 60000 5, 00% 50000 4, 00% 40000 3, 00% 7 -day 30000 2, 00% 20000 1, 00% 10000 Daily Cases CFR Staggered 7 -day CFR 20 20 10 . 1 9. 20 20 2. . 1 10 10 . 5. 2 0 20 20. 2 0 9. 28 9. 21 . 2 0 20 20. 2 0 9. 14 9. 7. 20 20 20. 2 0 8. 24 7 day CFR 8. 31 20 20. 2 0 8. 17 20 8. 10 8. 3. 20 . 2 0 20 20. 2 0 7. 27 7. 20 . 2 0 20 20. 2 0 7. 13 7. 6. 20 20 20. 2 0 6. 29 20. 2 0 6. 22 20. 2 0 6. 15 8. 20 20 0, 00% 6. 1. 20 20 0
California
Cycles: 7 and 14 day Cross- and Auto-Correlations 1 0, 8 0, 6 0, 4 0, 2 0 0 2 4 6 8 10 -0, 2 -0, 4 -0, 6 -0, 8 Cross Auto cases Auto deaths 12 14 16
California Survival Function, Oct. 3 -Dec. 5 • CFR = 0. 73%; unbiased it’s 1. 24%. Steps occur weekly! California Survival Function from 10/3/2020 1 0, 995 0, 99 0, 985 0, 98 0, 975 0, 97 0 1 2 3 4 5 6 7 8 9 10111213141516171819202122232425262728293031323334353637383940414243444546474849505152535455565758
Actuarial Death Forecasts Sn(t s)ĝ(s) • n(t s) is case count and ĝ(s) is estimate of pdf of time from infection to death • Var[Sn(t s)ĝ(s)] = Sn(t s)2 Var[ĝ(s)]+2 SSn(t)n(s)Cov[ĝ(s), ĝ(t)] • Multivariate lognormal fits better than normal Observed 12/6/05 755 Hindcast, Forecast 1561 Standard Dev. 1273 12/13/05 1159 12/20/05 1696 1219 12/27/05 1785 1275
What is Vaccine Efficacy? • Defined as 1 Risk(vacc)/Risk(unvacc) or 1–Cases(vacc)/N(vacc)/Cases(unvacc)/N(unvacc) N Vacc Unvacc cases 21, 720 21, 728 Rate/day Efficacy 8 0. 000368 95. 06% 162 0. 003913 Placebo efficacy is 86. 25% compared with US! Pfizer sample is not representative. It’s a “convenience” sample n Unvacc. US unvacc. Vacc. cases 21, 728 382. 2 M 21, 720 Rate/day 162 0. 003913 17. 8 M 0. 054235 8 0. 000368 86. 25% 99. 3%
“Why Kill Controls? ” Ronald Fisher said so • “Real-world data and real-world evidence are playing an increasing role in health care decisions, ” FDA • Ho: survival function of treated same as untreated vs. • Ha: survival function of treated sample is stochastically better P[Life>t|treated] > P[Life>t|untreated] for some t • Why not compare treated sample vs. untreated population survival functions? I. e. Kaplan-Meier (treated sample) vs. npmle (untreated population) • Allowed by FDA for phase II clinical trials • Likelihood ratio, Kolmgorov-Smirnov (K-S), Ed Gehan, Kullback-Leibler divergence (K-L), … tests: most require lifetime data!
“Neutrosophic” Phase II life tests? K-S Test • Sample of grouped lifetime data Period 1 2 Period Sums Cases 98 100 198 Deaths Period 1 2 Deaths Period 2 3 2 5 2 • Neutrosophic samples with same period sums Period 1 2 Sums 1 2 2 3 5 1 2 2 2 1 4 5 1 2 2 2 0 5 5 1 2 2 2 4 1 5 • Run neutrosophic K-S test, LR test, K-L, or rank tests 1 2 2 2 5 0 5
“Neutrosophic” Kolmgorov-Smirnov test • Ho: Same treated sample and untreated population survival functions vs. Ha: Different for some t • Simulate population life data with the same column sums or event counts as in the sample data. • Compute the K-M estimator from the simulated population life data and its K-S distance from the sample K-M survival function estimate. • If the sample K-M estimator K-S distance is less than some percentile of the simulated |population sample| K-S distance, do not reject the null hypothesis
“Why Kill Controls? ” References • Aslam, Muhammad, “Introducing Kolmogorov−Smirnov Tests under Uncertainty: An Application to Radioactive Data, ” http: //pubs. acs. org/journal/acsodf, ACS Omega 2020, 5, 914− 917 • Belin, Lisa, Yann De Rycke, and Phillippe Broët, “A two-stage design for phase II trials with time-to-event endpoint using restricted follow-up, ” Contemporary Clinical Trials Communications, Volume 8, December 2017, Pages 127 -134, https: //doi. org/10. 1016/j. conctc. 2017. 09. 010 • Chan, Kwun Chuen Gary. (2013) Survival analysis without survival data: connecting length-biased and case-control data. Biometrika 100 (3): 764 -770 • Dean, N. , Gsell, P. S. , Brookmeyer, R. , Crawford, F. , Donnelly, C. , Ellenberg, S. , Fleming, T. , Halloran, M. E. , Horby, P. , Jaki, T. , Krause, P. , Longini, I. , Mulangu, S. , Muyembe-Tamfum, J. J. , Nason, M. , Smith, P. , Wang, R. , Henao-Restrepo, A. , and De Gruttola, V. (2020). “Creating a Framework for Conducting Randomized Clinical Trials During Disease Outbreaks. ” The New England Journal of Medicine, 382, 1366 -1369 • FDA, “Submitting Documents Using Real-World Data and Real-World Evidence to FDA for Drugs and Biologics Guidance for Industry, ” May 2019 • Fleming, Thomas R. and David P. Harrington, “A Class Of Hypothesis Tests For One and Two Sample Censored Survival Data, ” Technical Report Series, No. 9, August 1980 • [7] Fleming, T. R. and D. P. Harrington, Counting Processes and Survival Analysis, Wiley-Interscience, New York, 1991 • Gehan, E. A. (1965). “A generalized Wilcoxon test for comparing arbitrarily singly-censored samples. ” Biometrika 52, 203 -223 • [4] George, L. L. , and A. C. Agrawal, “Estimation of a hidden service distribution of an M/G/∞ system, ” Naval Research Logistics, 20: 549– 555. doi: 10. 1002/nav. 3800200314, https: //sites. google. com/site/fieldreliability/home/m-g-infinity-service-distribution • George, L. L. “Ergodic Theory, Nyquist Samples, and Field Reliability, “ Triad Systems Corp. , March 1996 (Ergodeny. doc) • George, L. L. "Product Reliability Comparison with Censored Data, ” or “To the Man With a Hammer, Everything Looks Like a Nail, " ASQ Reliability Review, Vol. 17, No. 1, March 1997 (KSCentst. doc) • George, L. L. , “Compare Population and Customer Reliability, ” Quality and Productivity Research Conference, ASQ and UC Berkeley, Santa Rosa, CA May 1998 (QPRC 98. doc) • George, L. L. , “Why Kill Controls? R. A. Fisher says so, ” 1999 (Clin. Tril. doc) • https: //sites. google. com/site/fieldreliability/home/why-kill-controls/
More References • Gnedenko, B. V. , Yu. K. Belyayev, and A. D. Solovyev, Mathematical Methods of Reliability Theory, Academic Press, New York, pp. 274 -276, 1969 • Grover, N. B. , “Two-sample Kolmogorov-Smirnov test for truncated data, ” https: //doi. org/10. 1016/0010468 X(77)90039 -3 • Joyner, Michael, et al. , “Effect of Convalescent Plasma on Mortality among Hospitalized Patients with COVID-19: Initial Three Month Experience, ” Med. Rxiv preprint, Aug. 2020, https: //doi. org/10. 1101/2020. 08. 12. 20169359 • Koziol, James A. and David P. Byar, “Percentage Points of the Asymptotic Distributions of One and Two Sample K-S Statistics for Truncated or Censored Data, ” Technometrics, Vol. 17, No. 4, pp. 507 -510, 1975, doi = 10. 1080/00401706. 1975. 10489380, https: //www. tandfonline. com/doi/abs/10. 1080/00401706. 1975. 10489380 • Le. Blanc, Michael and Catherine Tangen, “Choosing Phase II Endpoints and Designs: Evaluating the possibilities, ” Clin. Cancer Res. 2012 Apr 15; 18(8): 2130– 2132. Published online 2012 Mar 8. doi: 10. 1158/1078 -0432. CCR-12 -0454 • [5] Nikiforov, A. M. “Algorithm AS 288, Exact Smirnov Two-sample Tests for Arbitrary Distributions, ” Appl. Statist, v. 43, No. 1, pp 265 -284, 1994 • [6] ibid, “Subroutine GSMIRN, ” statlib@lib. stat. cmu. edu • Smarandache, Florentin, Introduction to Neutrosophic Statistics, Sitech & Education Publishing, Columbus, Ohio, 2014
Progress in Artificial Stupidity • Calling. Bullshit. org, [Bergstrom and West] about AI, p-values, … • “Dump training data with known and unknown answers and do some linear algebra. ” • I will upload to https: //sites. google. com/site/fieldreliability/: • “Artificial Reliability: Is There Reality in Reliability? ” Unfinished. Maybe never finished as stupidity progresses faster than I do • 20 -question “Cognitive Reli-Ability Test” • All Cognitive Reli-Ability Test questions, purported answers, scoring, and test reliability • Sample questions from “Cognitive Reli-Ability Test. ”
Cognitive Reli-Ability Test 1. Name these animals [Norsk: nose-horn, flood-horse] 2. Draw a clock showing ten minutes after 11 3. Draw a cube
What is a Cognitive Reli-Ability Test? • Based on cognitive ability assessment tests: Alzheimers? Dementia? • Assesses level of economic, statistical, and evidence-based knowledge of reliability practices • Adapted to the needs of reliability information consumers • and to the needs of biostatisticians’ and epidemiologists’ customers for survival analyses and their applications • Test score indicates gap between state of traditional reliability practices and what could be done if knowledge was used • Prioritize potential improvements • Brain exercise improves health
Reliability meaning depends… What is the relevant definition of reliability? • A. The state or quality of being reliable [Merriam-Webster] • B. The ability to yield the same or compatible results in several tests or trials [Psychology] • C. The ability to successfully function for a specified time under specified conditions [Wikipedia] • D. The probability of successful function [according to customers] for a specified time [age, operating hours, calendar time] under specified conditions [customers’ conditions, not lab tests] [O’Connor], [bracket comments are mine] • E. The probability that a product, system, or service will perform its intended function adequately for a specified period of time, or will operate in a defined environment without failure [ASQ] • F. A, B, and C • G. D or E
Risk means different things. What is reliability-related risk? • A. The “effect of uncertainty on an expected result. ” [ISO 9000: 2015] • B. The “effect of uncertainty on objectives. ” [ISO 31000: 2009] • C. “…reliability risk is a major component of the risks facing an organization. ” [Fred Schenkelberg] • D. [Economic] risk is expected cost, SP[Failure(j)]*Cost of failure(j) • E. Risk is expected, discounted cost, S P[Failure(j)at age t]*(Cost of failure(j) at age t)*exp[ r*t]dt; where r is discount rate, the sum is over possible failures j, and the integral is over nonnegative times-to-failures t.
Recognize these reliability functions: exp[ lt], exp[ (t/ )b], 1 exp[ (s 2/(2 s 2)ds/ 2 ps 2]? • A. Exponential, Tweedie, chi-square • B. Exponential, Weibull, Rayleigh • C. Cumulative distribution functions • D. Complimentary cumulative distribution functions • E. Exponential survival function, Weibull survival function, chi-square survival function • F. Exponential, Weibull, normal (Note; partial credit given for exponential, extreme-value (type 1) or Gumbel distribution
From the CRE Exam, ASQ RRD Dec. Newsletter • Which of the following is true if all the subsystems in a series system have a constant failure rate? • • A. The failure rate of the system is constant B. The failure rate of the system will increase as more subsystems are added C. The failure rate of the system is the sum of the subsystem failure rates D. All of the above • The reliability of a system consisting of two units in parallel is 0. 96. If the reliability of each component is increased by 10%, what is the percentage increase in the reliability of the system? • A. 10% B. 5% C. 3. 33% D. 2. 66% • What is missing from these questions?
Cognitive ability includes math! Reliability improved 5%: what is the change in the failure rate? • A. Increased 5% • B. Decreased 5% • C. Depends on reliability • D. Depends on the reliability function • E. Depends on the failure rate function • F. none of the above
People bought Weibull software. When is reliability Weibull? • A. “n components connected in series, are independent, and a failed unit is replaced immediately. Then the distribution of the times between failures tends to the Weibull as time increases. ” [Kececioglu paraphrased] • B. Failure strength of large concrete columns in compression [Çinlar] • C. Failure strength of a chain of iid links in tension • D. Max. entropy distribution for a real random variable with expected value Xk equal to lk and ln(Xk) equal to ln(lk) g (Euler-Mascheroni constant) • E. Duration of covered calls (no kidding!) [Chronim Investments] • F. Service life of gas turbine engines [Weckman] • G. Service life of emergency diesel power plants [Klügel]
Must you have lifetime data to estimate reliability? Minimum data you really need for nonparametric field reliability estimation is…? • A. According to experts, lifetime data, times to failures tracked by name or serial number, is necessary. • B. The Kaplan-Meier or Nelson-Aalen nonparametric estimator requires grouped, perhaps censored, lifetime data. • C. Generally accepted accounting principles require installed base and returns counts data that are statistically sufficient to make nonparametric reliability estimates, even if failures are renewal process counts. • D. Total time and number of failures is sufficient, if failure rate is constant. • E. Two random samples of total time on test (or observation) and number of failures is sufficient if reliability is Weibull.
Distribution of time between events vs. distribution of event counts: exponential is to Poisson as inverse Gauss is to ? ? ? • A. Normal distribution • B. Binomial process • C. Brownian motion with drift • D. Tweedie distribution • E. Distribution of Phillips-Lumi. LEDs L 70 lumens deterioration
RCM believes these models fit all failure rates. Which failure rate(s) requires no maintenance? • • • A. Wear-out curves B. Top curve second column C. Middle curve second column D. Bottom curve second column E. B, C, and D F. You can do no maintenance if you want
Did you think reliability was just for one product or part? What about dependence? • A. Data is available to estimate the joint reliability function of Tesla Model S battery and charger, battery and drive unit, drive unit and charger. [www. pluginamerica. org] • B. Was Walter Shewhart’s rule #1, “Preserve all relevant information in data”? • C. Software is available to estimate joint distributions even from censored lifetimes, even without life data: Surv. Corr R-package, Mathematica, … • D. Joint distribution of corona-virus conditional time to death and time to recovery. Equivalent to joint distribution of times to failure in alternative failure modes.
What is the reliability of the Cognitive Reli. Ability Test? • A. Better than ASQ CRE (Certified Reliability Engineer) exam, because CRE is a minimal qualification? • B. As good as the Alzheimer’s disease cognitive ability tests? • C. Take this test again. Do the answers match? • D. Give this test to an associate and ask them to answer the questions as you would have! Do the answers match?
References • Montreal Cognitive Assessment (Mo. CA), https: //www. verywellhealth. com/alzheimers-and-montreal-cognitive-assessment -moca-98617 • Automatic Cognitive Assessment Delivery (ACAD), https: //acad. tchpc. tcd. ie/ • Bergstrom, Carl T. and Jevin D. West, Calling Bullshit, Random House, 2020 • Klügel, Jens-Uwe, “Investigation of time-dependent trends in plant-specific data for active components…, ” PSA Sept. 2008, Knoxville, TN http: //iet. jrc. europa. eu/apsa/sites/apsa/files/documents/Klugel_doc. pdf • Weckman, G. R. , et al. , "Modeling the Reliability of Repairable Systems in the Aviation Industry, " Computers and Industrial Engineering, 40 2001, 51 -63, • O’Connor, Patrick D. T. , Practical Reliability Engineering, 4 th edition, 2002, Wiley
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