Volume 4, Issue 2

Reliability Edge Home

Wayne Nelson
Schenectady, NY

Purpose. As a population of repairable systems ages, the systems accumulate repairs and repair costs. The population Mean Cumulative Function (MCF), defined below, for the number or "cost" of repairs as a function of population age yields much useful information. This article and Nelson (2003) show how to estimate and graphically compare the MCFs of two samples of systems. This comparison also applies to recurrence data on recurrent disease episodes, sociological and demographic applications such as childbirth, simulation of production processes and other recurrent events.

Overview. The following sections present:

• Typical repair data on automatic and manual car transmissions.
   
• A population model for such repair data and the population MCF.
   
• A sample estimate and confidence limits for the population MCF.
   
• A graphical comparison and confidence limits for the difference of two sample MCFs.
   
• Further information on the analysis of recurrence data.

Repair Data Example
Transmission data. Table 1A displays a small set of typical repair data. The data are transmission repairs on a sample of 34 cars with an automatic transmission in a preproduction road test. For example, the data on car 24 are a repair at 7,068 miles and its latest mileage 26,744+ miles; here + denotes how long a car has been observed. Table 1B displays similar data on a sample of 14 cars with a manual transmission. Information sought from the data include:

• The mean cumulative number of repairs per car by 24,000 test miles (equivalently 5.5 x 24,000 = 132,000 customer miles).
   
• Whether the population repair rate increases or decreases as a population ages.
   
• Whether the two sample MCFs differ statistically significantly (i.e. convincingly).

Nelson (2003) gives further examples of repair data on blood analyzers, residential heat pumps, power supplies, and other applications.

Table 1A: Automatic transmission repair data

Table 1B: Manual transmission repair data

Censoring. A repairable system's latest observed age is called its "censoring" age, because its future repair experience beyond that age is censored (unknown) at the time of the data analysis. System censoring ages usually differ; this complicates the data analysis and requires the methods here. A system may yet have no repairs; then its data value is just its censoring age. Also, a system may have one, two, three or more repairs before its censoring age.

Age. Here "age" (or "time") means any useful measure of product usage, such as mileage, days, cycles, months, etc.

Population Model and Its Mean Cumulative Function
Model. The following is a general population model for uncensored repair and other recurrent events data. As a function of age t, each system accumulates a total cost (or number) of repairs. Such system "cumulative history functions" are depicted in Figure 1 as smooth curves for easy viewing. Such system functions could potentially extend over any time period of interest. This population of curves is the population model. In practice, the history functions are staircase functions where the rise of each step is the additional cost (or number) or repairs at the corresponding age. But such staircase functions are hard to view in such a plot. In contrast, data from a sample of such systems consist of just the observed early portions of their history functions, subsequent history being censored.

Figure 1: Population cumulative cost histories (uncensored), distribution at age t, and MCF M(t)

MCF. At a particular age t, there is a population distribution of cumulative cost (or number) of repairs, which is depicted vertically in Figure 1 as a continuous density. At age t, the distribution has a mean M(t). In Figure 1, the heavy curve depicts M(t) as a function of t. M(t) is called the Mean Cumulative Function (MCF) for the cost (or number) of repairs. It is simply the mean curve. It provides most information sought from repair and other recurrence data.

Number of repairs. In Figure 1, the cost distribution is depicted as continuous but may be discrete. For a count of the number of repairs, the corresponding vertical distribution for the cumulative numbers of repairs at age t is discrete and has the integers values 0, 1, 2, 3, etc. M(t) is the mean of that integer-valued distribution.

Repair rate. When M(t) is for the number of repairs (or recurrences), the derivative

is assumed to exist and is called the population "instantaneous repair rate." For other types of recurrences, it is called the "recurrence rate" or "intensity function." It is expressed in repairs per usage time per system, for example, transmission repairs per 1000 miles per car. Some mistakenly call m(t) the "failure rate." This causes confusion with the quite different failure rate (hazard function) of a life distribution for non-repaired units, usually components that fail once. That failure (hazard) rate has an entirely different definition, meaning and use, as expounded by Ascher and Feingold (1984).

MCF Estimate and Confidence Limits
Estimate. Nelson (2002, 2003) presents an unbiased nonparametric estimate M*(t) for the population M(t). ReliaSoft's RDA Utility calculates and plots such MCF estimates with approximate 95% confidence limits. For the transmission data, the RDA Utility was used to obtain nonparametric MCF estimates (red staircases) and 95% limits (green staircases) that appear in Figures 2A and 2B. Each jump in a plot corresponds to a repair age in the sample. In practice, one knows that the population MCF is a smooth curve. Thus, one usually imagines a smooth curve drawn through the staircase estimate.

Figure 2A: Automatic transmission MCF and 95% confidence limits

Figure 2B: Manual transmission MCF and 95% confidence limits

Interpretation. The plots have the following interpretations.

• A smooth curve through Figure 2A or 2B has a derivative m*(t) that decreases as the population ages. That is, the repair rate decreases as each population ages. This is typical of products with manufacturing defects.
   
• Such a smooth curve extended to 24,000 test miles in Figure 2A yields an estimate of an average of 0.3 automatic transmission repairs per car during design life. Figure 2B yields a higher estimate 0.5 for manual transmissions.

Thus, these plots answer two of the questions above. It remains only to compare the automatic and manual transmission MCF estimates, as described below.

Confidence limits. Figures 2A and 2B display two-sided approximate 95% confidence limits (green staircases) for the corresponding population MCF. These pointwise limits enclose the population MCF with 95% confidence at any one age t. For example, the 95% confidence limits for the mean cumulative number of automatic transmission repairs at 24,000 test miles (read from Figure 2A) are 0.2 and 0.5 repairs per car. Such limits do not simultaneously enclose the entire MCF curve over the range of the data with 95% confidence.

Nelson (2003) presents the laborious calculation and theory of these pointwise limits. ReliaSoft's RDA Utility calculates and plots them.

Assumptions. The nonparametric MCF estimate and approximate confidence limits above entail minimal assumptions described by Nelson (2003). In contrast, the parametric models and estimates presented in Rigdon and Basu (2000) and in Ascher and Feingold (1984) apply to a single system (not a sample of systems) and require more assumptions that may not be valid in applications.

Comparison of Two MCFs
Background. This section shows how to compare two sample MCFs to assess whether they differ statistically significantly, that is, convincingly. The comparison is like that for two sample means. However, here we are comparing two mean curves. To do this, we look at the difference between the two sample curves at age t and the two-sided 95% confidence limits for that difference. At an age t where those pointwise confidence limits do not enclose zero, the difference is statistically significant. Otherwise, the two sample MCFs at that age do not differ statistically significantly. Nelson (2003) presents the theory and calculations for the limits for this difference.

Transmission comparison. Figure 3 displays the difference between the sample MCF functions and 95% confidence limits for the difference, which were calculated and plotted by ReliaSoft's RDA Utility. At most every age, the limits enclose zero. Thus, the sample MCFs do not differ statistically significantly (convincingly) over the range of the data. Of course, the two samples contain few cars and few repairs, and consequently the confidence limits are wide and the comparison is insensitive. In view of no significant difference, one might pool the two data sets to estimate a common MCF with greater accuracy, if that would be consistent with engineering knowledge.

Figure 3: Plot of the difference of the MCFs (automatic minus manual)

Further Information
Nelson (2003) provides other applications and information on:

• Predicting future numbers or costs of repairs for a fleet.
   
• Analyzing history functions that are continuous and that increase and decrease.
   
• Analyzing such data on cost, downtime and other measures of repair severity.
   
• Analyzing availability data, which includes downtime for repairs.
   
• Analyzing data with more complex censoring, such as system histories with gaps.
   
• Analyzing data with a mix of types of repairs.
   
• The minimal assumptions on which the nonparametric estimate M*(t) and confidence limits depend.
   
• Estimating relationships, such as the Cox model, for the dependence of M(t) on engineering variables.

Acknowledgements
This revised version of Nelson (2000) 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
Ascher, Harry and Feingold, Harry (1984), Repairable Systems Reliability, Marcel Dekker, New
     York.

Nelson, Wayne (2000), "Graphical Comparison of Sets of Repair Data," Quality and Reliability
     Engineering International 16, 235-241.

Nelson, Wayne (2002), "Graphical Analysis of Repair Data," Reliability Edge, 4th Quarter, 16
     18. Available at http://www.ReliaSoft.com/newsletter/4q2002/index.htm.

Nelson, Wayne (2003), Recurrent-Events Data Analysis for Repairs, Disease Episodes, and
     Other Applications, ASASIAM Series on Statistics and Applied Probability, SIAM,
     Philadelphia, PA. http://www.siam.org/books/sa10.

Rigdon, S.E. and Basu, A.P. (2000), Statistical Methods for the Reliability of Repairable
     Systems, Wiley, New York.