Non-Parametric Recurrence Data Analysis

Non-parametric recurrence data analysis provides a nonparametric graphical estimate of the mean cumulative function (MCF) for the number or cost of recurrences per unit versus age. In the reliability field, the Mean Cumulative Function (MCF) can be used to: [31]

• Evaluate whether the population repair (or cost) rate increases or decreases with age (this is useful for product retirement and burn-in decisions).

• Estimate the average number or cost of repairs per unit during warranty or some time period.

• Compare two or more sets of data from different designs, production periods, maintenance policies, environments, operating conditions, etc.

• Predict future numbers and costs of repairs, such as the next month, quarter or year..

• Reveal unexpected information and insight.

This section includes the following subsections:

• The Mean Cumulative Function (MCF) for Recurrence Data

• Confidence Limits for the MCF

The Mean Cumulative Function (MCF) for Recurrence Data

In non-parametric analysis of recurrent events data analysis, each population unit can be described by a cumulative history function for the cumulative number of recurrences. It is a staircase function that depicts the cumulative number of recurrences of a particular event, such as repairs, over time. Figure 11.1 depicts a unit's cumulative history function:

Figure 11.1: 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. At age t, the units have a distribution of their cumulative number of events. That is, a fraction of the population has accumulated 0 recurrences, another fraction has accumulated 1 recurrence, another fraction has accumulated 2 recurrences, etc. This distribution differs at different ages t and has a mean M(t) called the mean cumulative function (MCF) . The M(t) is the pointwise average of all population cumulative history functions (see Fig. 11.2).

Figure 11.2: Illustration of MCF and population distribution at age t.

For the case of uncensored data, the mean cumulative function M(ti ) values at different recurrence ages ti are estimated by calculating the average of the cumulative number of recurrences of events for each unit in the population at ti.

When the sample history functions are censored, the following estimate is appropriate:

1st Step - Order All Ages: Order all recurrence and censoring ages 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 be put in a certain order or be sorted randomly.

2nd Step - Calculate the Number, ri , of Units that Passed Through Age ti:

(1)

N is the total number of units and r1 = N at the first observed age, which could be a recurrence or suspension.

3rd Step - Calculate MCF Estimate, M*(t): For each sample recurrence age ti, calculate the mean cumulative function estimate as follows:

where:

at the earliest observed recurrence age t1.

Recurrent Events Data Analysis Example 1

A health care company maintains five identical pieces of equipment used by a hospital. When a piece of equipment fails, the company sends a crew to repair it. The following table gives the failure and censoring ages for each machine, where the + sign indicates a censoring age.

Estimate the MCF values and the 95% confidence limits ignoring the repair duration.

Solution to Recurrent Events Data Analysis Example 1

The MCF estimate is obtained as follows:

Top

Confidence Limits for the MCF

Upper and lower confidence limits for M(ti ) are:

  (2)

  (3)

where α (50%<α<100% ) is (100 - confidence level), Kα is the α standard normal fractile and Vari[M*(ti)] is the variance of the MCR estimate at recurrence age ti. The variance is calculated as follows:

(4)

where ri is defined in Eqn. (1), Ri is the set of the units that have not been suspended by ti and dji is defined as follows:

dji = 1 if the jth unit had an event recurrence at age ti
dji = 0 if the jth unit did not have an event reoccur at age ti

Recurrent Events Data Analysis Example 2

Using the data in Example 1, estimate the 95% confidence bounds.

Solution to Recurrent Events Data Analysis Example 2

Using Eqn. (4), the following table of variance values can be obtained:

Using Eqn. (2) and K5 = 1.644 for a 95% confidence level, the confidence bounds can be obtained as follows:

The analysis in Examples 1 and 2 can be obtained automatically in Weibull++ using the Non-Parametric RDA Specialized Folio, as shown next.

Note: In the above Folio, the F refers to failures (or events) and E refers to suspensions (or censoring ages).

The results with calculated MCF values and upper and lower 95% confidence limits are shown next along with the graphical plot.

Recurrent Events Data Analysis Example 3

The following table displays transmission repairs on a sample of 14 cars with manual transmission in a preproduction road test [31]. Here + denotes the censoring age (how long a car has been observed).

The car manufacturer seeks to estimate the mean cumulative number of repairs per car by 24,000 test miles (equivalently 5.5 x 24,000 = 132,000 customer miles) and to observe whether the population repair rate increases or decreases as a population ages.

Solution to Recurrent Events Data Analysis Example 3

The data is entered into a Non-Parametric RDA Specialized Folio in Weibull++, as shown next.

The results are as follows:

The results indicate that after 13,957 miles of testing, the estimated mean cumulative number of repairs per car is 0.5. Therefore, by 24,000 test miles, the estimated mean cumulative number of repairs per car is 0.5.

The MCF plot is shown next.

A smooth curve through the MCF plot has a derivative that decreases as the population ages. That is, the repair rate decreases as each population ages. This is typical of products with manufacturing defects.

See Also:
Recurrent Events Data Analysis

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