Wayne Nelson Purpose. This
article presents a simple and informative plot for analyzing data on numbers
or costs of repeated repairs of a sample of products. The plotting method
provides a nonparametric graphical estimate of the mean cumulative number or
cost of repairs per unit versus age. The mean cumulative function (MCF)
presented below can be used to (2) estimate the average number or cost of
repairs per unit on warranty or over the design (3)
compare two or more sets of data from different designs, production periods, (4) predict future numbers and costs of repairs, (5) reveal unexpected information and insight, an important advantage of plots. This article presents the basic population model for such repeated events data and its mean cumulative function (MCF). Using a typical data set for repair data (transmission repair data from cars on a preproduction road test), this article shows how to calculate and plot a sample estimate of the MCF for data from products with a mix of ages. This is followed by an explanation of how to use and interpret such plots. ReliaSoft's RDA Utility, a free software tool available from www.weibull.com, can be used to automate the analysis and plotting. The author presents such analyses and plots in more detail in his new book, RecurrentEvents Data Analysis for Product Repairs, Disease Episodes, and Other Applications, published in 2003 in the ASASIAM Series on Statistics and Applied Probability. Repair Data Transmission data. Table 1 displays typical repair data for 34 cars in a preproduction road test. Information sought from the data includes (1) the mean cumulative number of repairs per car by 24,000 miles (the "design life" since 1 test mile equals 5.5 customer miles) and (2) whether the population repair rate increases or decreases as the population ages. For each car, the data set consists of the car mileage at each transmission repair and the latest mileage. For example, car 24 had a repair at 7,068 miles, and it was observed until 26,774 miles. In this table, a + indicates the latest mileage observed for a car, called its "censoring age." Table
1: Transmission repair data Censoring. A unit’s latest observed age is called its "censoring age," because its repair history beyond that age is unknown. Usually, unit censoring ages differ. The different censoring ages complicate the data analysis and require the methods here. A unit may have no failures; then the censoring age is the only data value. Other units may have one, two, three or more repairs. Age. Here "age" (or "time") means any useful measure of product usage, such as mileage, days, cycles, months, etc. The Population and
Its Mean Cumulative Function
Repair rate. When M(t) is for the number of repairs, the derivative
is called the population "instantaneous repair rate." It is also called the "recurrence rate" or "intensity function" when some other repeating occurrence is observed. It is expressed in repairs per unit time per product, e.g. transmission repairs per 1000 miles per car. Some mistakenly call m(t) the "failure rate," which causes confusion with the quite different failure rate (hazard function) of a life distribution for nonrepaired units (usually components). The failure rate for a life distribution has an entirely different definition, meaning and use, as noted by Ascher and Feingold (1984). Estimate and Plot of
the MCF Steps. The following steps yield a nonparametric estimate M*(t) of the population MCF M(t) for the number of repairs from a sample of N units.
Table
2: MCF calculations
Plot. Figure 2 was plotted by ReliaSoft's RDA Utility, which does the calculations above. The program also calculated and plotted Nelson's (2003) nonparametric approximate 95% confidence limits for M(t), shown above and below each data point with green lines.
How
to Interpret and Use the Plot Mean cumulative number. An estimate of the population MCF by a specified age is read directly from the staircase or a curve through the plotted points. For example, from Figure 2, the estimate of the MCF at 24,000 test miles is 0.31 repairs per car during design life, an answer to a basic question. Repair rate. The derivative of such a curve (imagined or fitted) estimates the repair rate m(t). If the derivative increases with age, the population repair rate increases as products age. If the derivative decreases, the population repair rate decreases with age. The behavior of the rate is used to determine burnin, overhaul and retirement policies, as described by Nelson (2003). In Figure 2, the repair rate (derivative) decreases as the transmission population ages, the answer to another basic question. Other information. Nelson (2003) gives other applications and information on
Conclusion
and Acknowledgements Acknowledgments. This revised version of Nelson (1998) appears here with the kind permission of Wiley, publisher of Quality and Reliability Engineering International. The author gratefully thanks Mr. Richard J. Rudy of Daimler Chrysler, who generously granted permission to use the transmission data here. The author is pleased to acknowledge Mr. Pantelis Vassiliou, Mr. Adamantios Mettas, and Ms. Lisa Hacker for their valuable contributions to this article. References Nelson, Wayne
(1998), "An Application of Graphical Analysis of Repair Data," Quality
and Nelson, Wayne
(2003), RecurrentEvents Data Analysis for Repairs, Disease Episodes, and
