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Mean Cumulative Function (MCF) For Recurrent Events
(Only what I learned so far.)
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Outline
• Background
• What is MCF?
• How to estimate MCF?
• Plot of MCF
• Compare two MCFs
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Years from Procedure (PCI/CABG)
Pro
ba
bili
ty
0.5
0.6
0.7
0.8
0.9
1.0
0 1 2 3 4 5 6 7 8
CABG (10111)PCI (19532)
Logrank test: p < 0.0001
Survival Free From Revascularization
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Coronary Artery Disease
•
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Coronary Artery Bypass Graft (CABG) Surgery
•
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Percutaneous Coronary Intervention (PCI)
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Years from Procedure
%
0
20
40
60
80
100
0 1 2 3 4 5 6 7 8
Logrank trend test (PCI): p <0.0001
19951996
19971998
19992000
2001CABG
(1995-2001)
PCI
Survival Free From Revascularization
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Limitations associated with PCI: Restenosis
• The treated vessel becomes blocked again. Usually occurs within 6 months
after the initial procedure Balloon angioplasty alone: 40% Stenting: 25%
• In-stent restenosis Scar tissue overgrow and obstruct
the blood flow Typically within 3 to 6 months Brachytherapy Drug elute stent
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Recurrent events• A sample units can undergo repeated events, such as repairs of products,
recurrences of tumors, and restenosis of coronary artery in our case.
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Analysis of recurrent events
• Time-to-first event simple and easy to interpret. conventional survival analysis method ignores information hence inefficient
• Wei, Lin Weissfeld (WLW) marginal modelevent number is used as a stratification variable; separate model per
stratum
• Prentice, Williams and Peterson (PWP) conditional method: ‘at-risk process’ for jth event only becomes 1 after the (j - 1)th event
• Andersen and Gill (AG) method: ‘at-risk process’ remains at 1 until unit is censored
• Wayne Nelson: Mean cumulative function (MCF)
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What is MCF?
• Product reliability analysis
• When a repairable system fails, it is repaired and placed back in service. As a repairable system ages, it accumulates a history of repairs and costs of repairs.
• At a particular age t, there is a population distribution of cumulative cost (or number) of repairs; the distribution has a mean M(t), called the Mean Cumulative Function (MCF) for the cost (or number) of
repairs.
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How to estimate MCF?
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How to estimate MCF?
Unit (Age in Months, Cost in $100)
6 (5,$3) (12,$1) (12,+)
5 (16,+)
4 (2,$1) (8,$1) (16,$2) (20,+)
3 (18,$3) (29,+)
2 (8,$2) (14,$1) (26,$1) (33,+)
1 (19,$2) (39,$2) (42,+)
System Repair Histories for Artificial data
Event (Age,Cost)Number in
Service Mean Cost MCF
1 (2,$1) 6 $1/6=0.17 0.17
2 (5,$3) 6 $3/6=0.50 0.67
3 (8,$2) 6 $2/6=0.33 1
4 (8,$1) 6 $1/6=0.17 1.17
5 (12,$1) 6 $1/6=0.17 1.33
6 (12,+) 5
7 (14,$1) 5 $1/5=0.20 1.53
8 (16,$2) 5 $2/5=0.40 1.93
9 (16,+) 4
10 (18,$3) 4 $3/4=0.75 2.68
11 (19,$2) 4 $2/4=0.50 3.18
12 (20,+) 3
13 (26,$1) 3 $1/3=0.33 3.52
14 (29,+) 2
15 (33,+) 1
16 (39,$2) 1 $2/1=2.00 5.52
17 (42,+) 0
Calculation of MCF for Artificial Data
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Transmission data MCF and 95% confidence limits
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Using SAS for MCF estimation
• RELIABILITY procedurenonparametric estimates of population MCF
and its 95% confidence interval plot the estimated MCF for the number of
repairs or the cost of repairs estimates of the difference of two MCFs and
confidence intervals plot the difference of two MCFs and confidence
intervals.
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Using SAS for MCF estimationsymbol c=blue v=plus; proc reliability data=valve; unitid id; mcfplot days*value(-1) / cframe = ligr
ccensor = megr; inset / cfill = ywh ; run;
Obs id days value1 251 761 -1
2 252 759 -1
3 327 98 1
4 327 667 -1
5 328 326 1
6 328 653 1
…
89 422 582 -1
Repair Data Analysis
Age
Sample
MCF
Standard
Error
95% Confidence
Limits
Unit ID
Lower
Upper
61 0.024 0.024 -0.023 0.072 393
76 0.049 0.034 -0.018 0.116 395
84 0.073 0.041 -0.008 0.154 330
87 0.098 0.047 0.006 0.19 331
92 0.122 0.052 0.021 0.223 390
98 0.146 0.056 0.037 0.256 327
…
761 . . . . 251
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S-plus for MCF
• SPLIDA (S-PLUS Life Data Analysis)
• By W. Q. Meeker
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References
• Nelson, Wayne (2003), Recurrent-Events Data Analysis for Repairs, Disease Episodes, and Other Applications, ASA SIAM Series on Statistics and Applied Probability, SIAM, Philadelphia, PA.
• Nelson, W. (1995), "Confidence Limits for Recurrence Data--Applied to Cost or Number of Product Repairs,"
Technometrics, 37, 147 -157.