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.

1. How to Graph, Analyze
and Compare Sets of
Repair Data
Wayne Nelson
©2011 ASQ & Presentation Wayne Nelson
Presented live on Jun 09th, 2011
http://reliabilitycalendar.org/The_Reli
ability_Calendar/Webinars_‐
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2. ASQ Reliability Division
English Webinar Series
One of the monthly webinars
on topics of interest to
reliability engineers.
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Division members only) visit asq.org/reliability
To sign up for the free and available to anyone live
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Webinars to find links to register for upcoming events
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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.com
PURPOSE: To survey new nonparametric models,
analyses, and informative plots for recurrent events data
with associated values (costs, time in hospital, running
hours, and other quantities). Previous theory (often
parametric, e.g., NHPP) handles just counts of recurrent
events. 7.`6 '10
1

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. TYPICAL EXACT AGE DATA
Automatic Transmission Repair Data (+ obs'd miles)
CAR M I L E A G E . CAR M I L E A G E
We use age (or usage) of each unit rather than calendar date.
Multiple censoring times are typical.
3

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. 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. UNIT MODEL: Each unit has an uncensored Cumulative
History Function for the cumulative number of events.
4 - Cum.
No.
3 -
2 -
1 -
0 -----+x-x-+-x--+----+----+----+
THOUSAND MILES
6

9. Cumulative History Function for "cost", a new advance:
400 - Cum.$
Cost
300 -
200 -
100 -
0 -----+----+----+----+----+----+----+----+
0 1000 2000 3000 4000
A G E I N D A Y S
Costs (or other values) may be negative, a new advance,
e.g., scrap value and bank account withdrawal.
7

10. NONPARAMETRIC POPULATION MODEL consists of all
uncensored 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. Recurrence Rate for the number of recurrences
per population unit
m(t) ≡ dM(t)/dt
is often of interest. m(t) is the mean number of
transmission 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” and
confuse it with the hazard rate of a life distribution.
Using multivariate distributions of times to and
between events is complicated and less informative.
For cost, m(t) is the mean cost rate (average $ per
month per population unit).
9

12. 1.0
0.90
PLOT OF NONPARAMETRIC MCF ESTIMATE M*(t) &
0.8
CONF. 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 Miles
Decreasing repair rate. Here M*(24,000) = 0.31 repairs/car.
10

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
obs'd no. 1/r obs'd 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+ 13
13809+ 33 21762+ 12
14235+ 32 21876+ 11
14281+ 31 21888+ 10
17844+ 30 21974+ 9
17955+ 29 22149+ 8
18228+ 28 22486+ 7
18676+ 27 22637+ 6
19175+ 26 22854+ 5
19250 26 1/26=0.04 0.31 23520+ 4
19321+ 25 24177+ 3
19403+ 24 25660+ 2
19507+ 23 26744+ 1
19607+ 22 29834+ 0
11

14. Bladder Tumor Treatment MCFs (Placebo and Thiotepa)
Compare treatments; understand the course of the disease
and when to schedule exams. SAS plots.
12

15. Cost Data and the MCF Estimate
Days 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 + 104
1057 + 103
1170 + 102
1200 + 101
1232 + 100
1269 130.20 100 130.20 /100= 1.30 2.70 0.010 0.028
1355 + 99
1381 150.40 99 150.40 /99 = 1.52 4.22 0.010 0.038
1471 113.40 99 113.40 /99 = 1.15 5.37 0.010 0.048
1567 151.90 99 151.90 /99 = 1.53 6.90 0.010 0.058
1642 191.20 99 191.20 /99 = 1.93 8.83 0.010 0.068
13

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. MIX OF EVENTS Subway Car Traction Motors. Design
Failures 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. COMPARISON OF MCFs OF DATA SETS
1.0 1.0
(A) (B)
0.8 0.8
0.6 0.6
MCF 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
AGE
Do sample MCFs differ statistically significantly?
16

19. Manual Transmission Repair Data (+ obs'd 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. Automatic Transmission Manual Transmission
1.0
1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6 0.6
M M
0.5 C 0.5
C
F 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 Mileage
ReliaSoft RDA plots.
18

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
Mileage
Limits enclose 0 => no convincing difference. ReliaSoft plot.
19

22. Herpes Episodes -- Comparison of
Episodic and Suppressive Valtrex Treatments
Provided by Richard Cook with permission of GSK. S-Plus plot.
20

23. Amazon.com Orders MCF for Two Promotions , SAS plot
Promo. 1
Promo. 2
21

24. Babies Born to Statisticians (♦ Men, □ Women). Excel plot
1.6
1.4
1.2
1
MCF
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. SOFTWARE for calculating and plotting the MCF
estimate and limits and the difference of two sample
MCFs for count and "cost" data:
• SAS Reliability Procedure.
• SAS JMP Package.
• Meeker's (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. 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. 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. 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. CONCLUDING REMARKS
These new nonparametric methods, plots, and
software for cost or other values of recurrent
events are useful for many applications.
Extensions of the methods and corresponding
software are needed to handle more
complicated applications.
27

30. REFERENCES
Cook, 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. MORE APPLICATIONS
Bladder Tumor Treatment MCFs (Placebo & Thiotepa, SAS plots)
29

32. MCF difference (Placebo−Thiotepa) & 95% limits, SAS plot.
30

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
- 5
MCF - 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. Defrost Control MCF (data grouped by month, SAS plot)
32

35. Defrost Control Log-Log Plot (Excel plot)
1000
100
10
MCF%
1
0.1
1 2 5 10 20 50 100 200
MONTHS
33

36. 2
0
Blood Analyzer Burn-in -- Sample MCF
MCF
34

37. Cumulative Hours in the Workforce
800 - Cum. Hours Worked
600 -
400 -
200 -
0 +----+----+----+----+----+----+
0 10 20 30
W E E K S
35

38. Unemployment Contributions and Payments
Cum. $ Contributed
200 -
0 -
-200 -
-400 -
0 +----+----+----+----+----+----+
0 10 20 30
W E E K S
36

39. NEGATIVE VALUES OF EVENTS (BANK ACCT.)
3000 - $ in Acct
2000 -
1000 -
0 -----+----+----+----+----+----+
0 200 400 600 800 1000 1200
D A Y S
37

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. Difference of the Two MCFs
39

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 45
Essentially constant repair rate (4.8% per year) over all
vintages.
40