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How to graph, analyze and compare sets of repair data

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Many products accumulate repeated repairs and repair costs over time. Analysis of such recurrence data requires special statistical models and methods not covered in basic reliability books. This tutorial webinar presents a simple and informative model and plot for analyzing data on numbers or costs of repeated repairs of a sample of units. The plot is illustrated with transmission repair data from preproduction cars on a track test. This article also presents a method for comparing two such data sets, illustrated with automatic and manual transmissions. Computer programs that calculate and make the plots and comparisons with confidence limits are surveyed.

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How to graph, analyze and compare sets of repair data

  1. 1. How to Graph, Analyze  and Compare Sets of  Repair Data Wayne Nelson ©2011 ASQ & Presentation Wayne Nelson Presented live on Jun 09th, 2011http://reliabilitycalendar.org/The_Reliability_Calendar/Webinars_‐_English/Webinars_‐_English.html
  2. 2. ASQ Reliability Division  English Webinar Series One of the monthly webinars  on topics of interest to  reliability engineers. To view recorded webinar (available to ASQ Reliability  Division members only) visit asq.org/reliability To sign up for the free and available to anyone live  webinars visit reliabilitycalendar.org and select English  Webinars to find links to register for upcoming eventshttp://reliabilitycalendar.org/The_Reliability_Calendar/Webinars_‐_English/Webinars_‐_English.html
  3. 3. Tutorial for RAMS 2011. Copyright (C) 2011 Wayne Nelson. HOW TO GRAPH, ANALYZE, AND COMPARE SETS OF REPAIR DATA Wayne Nelson, consultant, Schenectady, NY, WNconsult@aol.comPURPOSE: To survey new nonparametric models,analyses, and informative plots for recurrent events datawith associated values (costs, time in hospital, runninghours, and other quantities). Previous theory (oftenparametric, e.g., NHPP) handles just counts of recurrentevents. 7.`6 10 1
  4. 4. OVERVIEW• RECURRENCE DATA AND INFORMATION SOUGHT• NONPARAMETRIC POPULATION MODEL• MCF -- MEAN CUMULATIVE (INTENSITY) FUNCTION• RECURRENCE RATE FOR COUNT DATA• MCF ESTIMATE AND CONFIDENCE LIMITS● MCF ESTIMATE FOR COST• COMPARISON OF DATA SETS• ASSUMPTIONS• SOFTWARE• EXTENSIONS• NEEDED WORK• CONCLUDING REMARKS● MORE APPLICATIONS 2
  5. 5. TYPICAL EXACT AGE DATAAutomatic Transmission Repair Data (+ obsd miles) CAR M I L E A G E . CAR M I L E A G EWe use age (or usage) of each unit rather than calendar date.Multiple censoring times are typical. 3
  6. 6. Display of Automatic Transmission Repair Data 0 5 10 15 20 25 30 CAR +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ THOUS. MILES 024 | X | 026 X | 027 X X | 029 |X | 031 | | 032 | | 034 | X | 035 | | 098 | | 107 | X | 108 | | 109 | | 110 | | 111 | | 112 | | 113 | | 114 | | 115 | | 116 | | 117 | | 118 | | 119 | | 120 | | 121 X | 122 | | 123 | | 124 | | 125 | | 126 | | 129 | | 130 | | 131 | | 132 | X | 133 | X | +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ THOUS. MILES 0 5 10 15 20 25 30 4
  7. 7. Information Sought• Average number of repairs per transmission at 24,000 test miles ( x 5.5 = 132,000 customer miles = design life).• How do automatic and manual transmissions compare?• The behavior of the repair rate (increasing or decreasing?). 5
  8. 8. UNIT MODEL: Each unit has an uncensored CumulativeHistory Function for the cumulative number of events. 4 - Cum. No. 3 - 2 - 1 - 0 -----+x-x-+-x--+----+----+----+ THOUSAND MILES 6
  9. 9. Cumulative History Function for "cost", a new advance:400 - Cum.$ Cost300 -200 -100 - 0 -----+----+----+----+----+----+----+----+ 0 1000 2000 3000 4000 A G E I N D A Y SCosts (or other values) may be negative, a new advance,e.g., scrap value and bank account withdrawal. 7
  10. 10. NONPARAMETRIC POPULATION MODEL consists of alluncensored cumulative history functions. No process assumed.The population Mean Cumulative Function (MCF) denoted M(t)(usually Λ(t) for counts) contains most of the sought information. 8
  11. 11. Recurrence Rate for the number of recurrencesper population unit m(t) ≡ dM(t)/dtis often of interest. m(t) is the mean number oftransmission repairs per car per 1000 miles at(mile)age t. λ(t) is the notation for NHPPs. Some wrongly call m(t) the “failure rate” andconfuse it with the hazard rate of a life distribution. Using multivariate distributions of times to andbetween events is complicated and less informative. For cost, m(t) is the mean cost rate (average $ permonth per population unit). 9
  12. 12. 1.0 0.90PLOT OF NONPARAMETRIC MCF ESTIMATE M*(t) & 0.8CONF. LIMITS Transmissions (Nelson 1988, 1995, 2003) 0.7 0.6 M C 0.5 F 0.4 0.3 0.2 0.1 0.0 0 5 10 15 20 25 30 1000 MilesDecreasing repair rate. Here M*(24,000) = 0.31 repairs/car. 10
  13. 13. * Calculate the MCF Estimate M (t) (1) (2) (3) (4) (1) (2) (3) (4)Mileage No. r mean MCF Mileage No. r mean MCF obsd no. 1/r obsd no. 1/r 28 34 1/34=0.03 0.03 20425+ 21 48 34 1/34=0.03 0.06 20890+ 20 375 34 0.03 0.09 20997+ 19 530 34 0.03 0.12 21133+ 18 1388 34 0.03 0.15 21144+ 17 1440 34 0.03 0.18 21237+ 16 5094 34 0.03 0.21 21401+ 15 7068 34 0.03 0.24 21585+ 14 8250 34 1/34=0.03 0.27 21691+ 1313809+ 33 21762+ 1214235+ 32 21876+ 1114281+ 31 21888+ 1017844+ 30 21974+ 917955+ 29 22149+ 818228+ 28 22486+ 718676+ 27 22637+ 619175+ 26 22854+ 519250 26 1/26=0.04 0.31 23520+ 419321+ 25 24177+ 319403+ 24 25660+ 219507+ 23 26744+ 119607+ 22 29834+ 0 11
  14. 14. Bladder Tumor Treatment MCFs (Placebo and Thiotepa)Compare treatments; understand the course of the diseaseand when to schedule exams. SAS plots. 12
  15. 15. Cost Data and the MCF EstimateDays Cost$ No. r Mean Cost MCF for Rate m MCF at risk Cost / r Cost = 1/r for No. 141 44.20 119 44.20 /119= 0.37 0.37 0.008 0.008 252 + 118 288 + 117 358 + 116 365 + 115 376 + 114 376 + 113 381 + 112 444 + 111 651 + 110 699 + 109 820 + 108 831 + 107 843 110.20 107 110.20 /107= 1.03 1.40 0.009 0.018 880 + 106 966 + 105 973 + 1041057 + 1031170 + 1021200 + 1011232 + 1001269 130.20 100 130.20 /100= 1.30 2.70 0.010 0.0281355 + 991381 150.40 99 150.40 /99 = 1.52 4.22 0.010 0.0381471 113.40 99 113.40 /99 = 1.15 5.37 0.010 0.0481567 151.90 99 151.90 /99 = 1.53 6.90 0.010 0.0581642 191.20 99 191.20 /99 = 1.93 8.83 0.010 0.068 13
  16. 16. MCF for Cost for Fan Motor Repairs (Excel Plots) 60 50 40 MCF$ 30 20 10 0 0 1000 2000 3000 4000 DAYS MCF for Number of Fan Motor Repairs 0.30 0.25 0.20 MCF# 0.15 0.10 0.05 0.00 0 1000 2000 3000 4000 D  A  Y  S 14
  17. 17. MIX OF EVENTS Subway Car Traction Motors. DesignFailures with and without Modes A, B, C: MAll(t) = M1(t) + ⋅⋅⋅ +MK(t). Modes need not be statistically independent. Excel plot. 50 All Design Modes 40 30 Design w/o ABC MCF% 20 ABC 10 0 0 12 MONTHS 24 36 15
  18. 18. COMPARISON OF MCFs OF DATA SETS 1.0 1.0 (A) (B) 0.8 0.8 0.6 0.6MCF MCF 0.4 0.4 0.2 0.2 0.0 0.0 0 20 40A G E 60 80 100 0 20 40 60 80 100 AGEDo sample MCFs differ statistically significantly? 16
  19. 19. Manual Transmission Repair Data (+ obsd miles) CAR _____M I L E A G E______ 025 27099+ 028 21999+ 030 11891 27583+ 097 19966+ 099 26146+ 100 3648 13957 23193+ 101 19823+ 102 2890 22707+ 103 2714 19275+ 104 19803+ 105 19630+ 106 22056+ 127 22940+ 128 3240 7690 18965+ 17
  20. 20. Automatic Transmission Manual Transmission 1.0 1.0 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6M M 0.5 C 0.5CF 0.4 F 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0.0 0.0 0 5000 10000 15000 20000 25000 30000 0 5000 10000 15000 20000 25000 30000 Mileage MileageReliaSoft RDA plots. 18
  21. 21. POINTWISE COMPARISON AT A SINGLE AGE t Var[M*1(t)−M*2(t)] = Var[M*1(t)] + Var[M*2(t)] 0.4 0.3 (Automatic − Manual) 0.2 0.1 0.0 M C -0.1 F -0.2 -0.3 -0.4 -0.5 -0.6 0 5000 10000 15000 20000 25000 30000 MileageLimits enclose 0 => no convincing difference. ReliaSoft plot. 19
  22. 22. Herpes Episodes -- Comparison of Episodic and Suppressive Valtrex TreatmentsProvided by Richard Cook with permission of GSK. S-Plus plot. 20
  23. 23. Amazon.com Orders MCF for Two Promotions , SAS plot Promo. 1 Promo. 2 21
  24. 24. Babies Born to Statisticians (♦ Men, □ Women). Excel plot 1.6 1.4 1.2 1MCF 0.8 0.6 0.4 0.2 0 0 10 20 30 40 50 60 A G E (Y E A R S) 22
  25. 25. SOFTWARE for calculating and plotting the MCFestimate and limits and the difference of two sampleMCFs for count and "cost" data:• SAS Reliability Procedure.• SAS JMP Package.• Meekers (1999) SPLIDA routines for S-PLUS.• ReliaSoft RDA Utility (Repair Data Analysis). This has naïve (too short) confidence limits.● SuperSmith Visual "Nelson Recurrent Event Plot."• Nelson & Doganaksoy (1989) Fortran PC program.● Minitab "Nonparametric Growth Curves". 23
  26. 26. ASSUMPTIONS FOR THE MCF ESTIMATE M*(t):1> Simple random sample from the population.2> Random (uninformative) censoring.3> M(t) is finite. Clearly so for finite populations.Then the nonparametric estimator M*(t) is unbiased.ASSUMPTIONS FOR APPROX. CONF. LIMITS:4> M*(t) is approximately normally distributed.5> Variances and covariances in Var[M*(t)] are finite.6> The population is infinite, at least 10× the sample size.NOT ASSUMED (usually false in practice)• Counting process such as NHPP, renewal, parametric, etc.• Independent increments, a common dubious simplifying assumption for counts. 24
  27. 27. AVAILABLE EXTENSIONS• Continuous cumulative history functions, e.g., - cumulative energy output of a power plant, - cumulative up-time of locomotives (availability).• A mixture of types of events (e.g. failure modes).• Predictions of future numbers and costs of recurrences.• Estimates, plots, and conf. limits for interval age data.• Other sampling plans (stratified, cluster, etc.).• Left censoring and gaps in histories.• Multivariate event values (cost and downtime).• Regression models (Cox proportional hazards, etc.).• Informative censoring (frailties).• Parametric models, Rigdon & Basu (2000). 25
  28. 28. NEEDED WORK• A hypothesis test for independent increments of a NHPP.• Prediction limits for a future number or cost of recurrences.• Confidence limits and software for data with left censoring and gaps.• Theory and commercial software for comparison of entire MCFs. - Cook, R.J., Lawless, J.F., and Nadeau, C. (1996), "Robust Tests for Treatment Comparisons Based on Recurrent Event Responses, Biometrics 52, 557-571.• Methodology for terminated histories.• Better confidence limits for interval age data.• Efficient computation of confidence limits for large data sets. Current computations are too intensive.• More regression models (Cox model is often poor) for count and cost/value data.• Better parametric models (without independent increments). 26
  29. 29. CONCLUDING REMARKSThese new nonparametric methods, plots, andsoftware for cost or other values of recurrentevents are useful for many applications.Extensions of the methods and correspondingsoftware are needed to handle morecomplicated applications. 27
  30. 30. REFERENCESCook, R.J. and Lawless, J.F. (2007), The Statistical Analysis of Recurrent Events, Springer, New York.Nelson, Wayne (1988), "Graphical Analysis of System Repair Data," J. of Quality Technology 20, 24-35.Nelson, Wayne (1995), "Confidence Limits for Recurrence Data -- Applied to Cost or Number of Product Repairs," Technometrics 37, 147-157.Nelson, Wayne B. (2003), Recurrent Events Data Analysis for Product Repairs, Disease Recurrences, and Other Applications, SIAM, Philadelphia, ASA/SIAM. www.siam.org/books/sa10/.Rigdon, S.E., and Basu, A.P. (2000), Statistical Methods for the Reliability of Repairable Systems, Wiley, New York. 28
  31. 31. MORE APPLICATIONSBladder Tumor Treatment MCFs (Placebo & Thiotepa, SAS plots) 29
  32. 32. MCF difference (Placebo−Thiotepa) & 95% limits, SAS plot. 30
  33. 33. Naval Turbines MCF Censoring Ages 1211 2 1 2 1 1 11 1 1 1 11 11 ‫׀ ׀ ׀ ׀‬ ‫׀ ׀ ׀ ׀ ׀ ׀ ׀ ׀ ׀ ׀ ׀ ׀ ׀ ׀ ׀ ׀‬ - - 1 20- 1 - 21 - 3 - 4 - 2 11 15- 31 - 1121 - 5MCF - 12 2 - 42 10- 221 21 - 9 - 1 234 - 21332 - 32 43 5- 23252 1 - 1F1 - F3 - 3H - L 0- K ‫׀ ׀ ׀ ׀ ׀ ׀ ׀ ׀ ׀ ׀ ׀ ׀ ׀ ׀ ׀ ׀ ׀ ׀ ׀ ׀‬ 0 5 10 15 20 25 THOUSANDS OF HOURS 31
  34. 34. Defrost Control MCF (data grouped by month, SAS plot) 32
  35. 35. Defrost Control Log-Log Plot (Excel plot)1000 100 10MCF% 1 0.1 1 2 5 10 20 50 100 200 MONTHS 33
  36. 36. 2 0Blood Analyzer Burn-in -- Sample MCF MCF 34
  37. 37. Cumulative Hours in the Workforce800 - Cum. Hours Worked600 -400 -200 - 0 +----+----+----+----+----+----+ 0 10 20 30 W E E K S 35
  38. 38. Unemployment Contributions and Payments Cum. $ Contributed 200 - 0 --200 --400 - 0 +----+----+----+----+----+----+ 0 10 20 30 W E E K S 36
  39. 39. NEGATIVE VALUES OF EVENTS (BANK ACCT.)3000 - $ in Acct2000 -1000 - 0 -----+----+----+----+----+----+ 0 200 400 600 800 1000 1200 D A Y S 37
  40. 40. Doses of a Concomitant Medication under Two Treatments Chris Barker (2009), "Exploratory method for summarizing concommitant medication data – the mean cumulative function," Pharmaceutical Statistics. 38
  41. 41. Difference of the Two MCFs 39
  42. 42. Repairs of Large Power Transformers (Left Censored) Correct MCF 250 200 150 MCF% 100 50 0 0 5 10 15 20YEARS25 30 35 40 45Essentially constant repair rate (4.8% per year) over allvintages. 40

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