Introduction to Non-Parametric Recurrent Event 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 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 .
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
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 MCFi 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.
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:
3rd Step - Calculate MCF: For each sample recurrence age i, the mean cumulative function is calculated as follows:
at the first observed recurrence age.
1. Nelson, Wayne, Recurrent Events Data
Analysis for Product Repairs, Disease Recurrences, and Other Applications,
Copyright 2005 ReliaSoft Corporation, ALL RIGHTS RESERVED