Mean Cumulative Function (MCF) For Recurrent Events

(Only what I learned so far.)

Outline

• Background

• What is MCF?

• How to estimate MCF?

• Plot of MCF

• Compare two MCFs

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

Coronary Artery Disease

•

Coronary Artery Bypass Graft (CABG) Surgery

•

Percutaneous Coronary Intervention (PCI)

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

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

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.

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)

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.

How to estimate MCF?

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

Transmission data MCF and 95% confidence limits

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.

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

S-plus for MCF

• SPLIDA (S-PLUS Life Data Analysis)

• By W. Q. Meeker

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.