Issue 57, November 2005

Introduction to Non-Parametric Recurrence Data Analysis

In Life Data Analysis (LDA), it is assumed that events (failures) are independent and identically distributed (iid). However, there are many cases where events are dependent and not identically distributed (such as repairable system data) or where the analyst is interested in modeling the number of occurrences of events over time rather than the length of time prior to the first event, as in LDA.

Weibull++ provides two approaches to analyze such data:

   • The non-parametric approach:

The focus of this article, this approach is based on the well-known Mean Cumulative Function (MCF). Weibull++'s Non-Parametric RDA module for this type of analysis builds upon the work of Dr. Wayne Nelson, who has written extensively on the calculation and applications of MCF [1].

  • The parametric approach:

Weibull++'s Parametric RDA module for this type of analysis is based on the General Renewal Process (GRP) model, which is particularly useful in understanding the effects of the repairs on the age of a system. The parametric approach will be discussed in future issues.

The Mean Cumulative Function (MCF)

In non-parametric recurrent events data analysis, every unit of the population can be described by a cumulative history function for the number of recurrences. It is a staircase function that tracks the accumulated number of occurrences of a particular event over time. The following is an example of the cumulative history function:

Figure 1: Cumulative number of failures

The nonparametric model for a population of units is described as the population of cumulative history functions (curves). It is the population of all staircase functions of every unit in the population. The intersection of the staircase function with the vertical line passing through an age value i creates a distribution for the values of cumulative number of events, which has values 0, 1, 2, 3...etc.

At age i, a fraction of the population has accumulated zero recurrences, another fraction has accumulated one recurrence, another fraction has accumulated two recurrences, etc. This distribution differs at different ages and has a mean MCF_i called the mean cumulative function (MCF). The mean cumulative function is the pointwise average of all population curves passing through the vertical line at each age i as can be seen in the next figure.

Figure 2. Illustration of MCF plot.

For the case of uncensored histories, the mean cumulative function MCF_i values at different recurrence ages i are estimated by calculating the average of the cumulative number of recurrences of events for each unit in the population at i. When the histories are censored, the following steps are applied.

1st Step - Order Ages: All recurrence and censored ages are sorted from smallest to largest. If a recurrence age for a unit is the same as its censoring (suspension) age, the recurrence age goes first. If multiple units have a common recurrence or censoring age, then these units could follow a certain order or be sorted randomly.

2nd Step - Calculate the Number r Observed:

where i is the ages of all events (recurrences, suspensions), N is the total number of units and r₀ = N at the first observed age (could be an occurrence or suspension).
 

3rd Step - Calculate MCF: For each sample recurrence age i, the mean cumulative function is calculated as follows:

where:

at the first observed recurrence age.

Confidence Bounds
Assuming that the values of cumulative number of events at any recurrence age i follow a lognormal distribution, the upper and lower confidence bounds for the mean cumulative function are:

where α (50% < α < 100%) is the confidence level, K_α is the α standard normal percentile and Var_i is the MCF variance at recurrence age i. The variance is calculated as follows:

where R_i is the set of the units that have not been suspended by i and d_ji is defined as follows:

References

1. Nelson, Wayne, Recurrent Events Data Analysis for Product Repairs, Disease Recurrences, and Other Applications, ASA-SIAM, 2003.
 

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