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Median Ranks Calculation for Fractional Failures

 

 

In the September 2015 issue of Reliability HotWire, we discussed fractional failure analysis and provided an example on how to perform the analysis in Weibull++. The analysis takes into account the effectiveness of an assumed corrective action or product redesign. If the corrective action or the improvement in the redesign is not 100% effective on the failure mode being analyzed, a certain percentage of the failure mode is removed by changing it to a partial failure and a partial suspension. In this issue, we will describe how the unreliability estimates are calculated in the analysis.

The median rank method, which is used in Weibull++, estimates unreliability values based on the failure order number and the cumulative binomial distribution. When the data set contains only failures, calculation of the failure order number and the corresponding median rank value is straightforward. However, to accommodate the fractional failures in the data (partial failure and partial suspension), it is necessary to adjust the order of each failure. We will first present a method for calculating the adjusted failure order number (mean order number) for a data set that contains fractional failures, and then we will look into multiple ways to calculate the median ranks using the mean order number.

Mean Order Number (MON) Calculation

Prior to computing MON, the data points are arranged in ascending order. The MON for each failure is calculated as:

Equation 1

where, j is the jth failure item and Ij is the increment.

The increment, I, is defined by:

Equation 2

where:

N is the sample size, or total number of items in the test

PMON is the previous mean order number

NIBPSS is the number of items beyond the present suspended set

The increments are calculated for the first data point and for each suspension in the data set. For all other data points, the increment is the same as the increment for the previous data point. For example, for the data set in Table 1, the increment for the last two failures stays constant at 1.33.

Table 1. Mean order number calculation for a data set that contains a suspension
State Time to Failure Increment MON
F T1 1 1
S T2 1.333333 -
F T3   2.333333
F T4   3.666667

To use the MON calculation for fractional failures, the equation should be adjusted to include the partial failure of an item. The adjusted equation can be expressed as:

Equation 3

where Fj is the failure fraction of the jth failure item.

Median Ranks Calculation

The rank can be found for any percentage point, P, greater than zero and less than 1, by solving the cumulative binomial equation for Z. This represents the rank, or unreliability estimate, for the jth failure in the following equation for the cumulative binomial:

Equation 4

where N is the sample size and j is the order number.

The median rank is the value that the true probability of failure, Q(Tj), has for the jth failure out of a sample of N units at the 50% confidence level. The median rank is obtained by solving this equation for Z at P = 0.5:

Equation 5

For example, if N = 4 and we have four failures, we would solve the median rank equation for the value of Z four times; once for each failure with j = 1, 2, 3 and 4, respectively. However, when there are fractional failures in the data set (or when all suspensions are not at the end of the ordered data set, as in Table 1), the j values will not necessarily be integers and we will need to use linear interpolation between the median ranks values calculated for the integers.

There are other methods for estimating median ranks. We will cover some of them briefly in the following sections.

Beta and F Distributions Approach

One of the methods for estimating median ranks is by applying two transformations to the cumulative binomial equation, first to the beta distribution and then to the F distribution. Then, the median ranks value for the jth failure is given as:

Equation 6

where Equation 8 is the F distribution at the 0.50 point, with 2(N - MONj + 1) and 2MONj degrees of freedom, for failure j out of N units.

Benard's Approximation

Another quick but less accurate approximation of the median ranks is given by Benard's approximation:

Equation 8

where MRj is the median ranks, MONj is the mean order number for jth failure and N is the total number of items tested.

Johnson's Approximation

Leonard Johnson defined the equations for computing the approximate median rank values as:

Equation 9

for N > 20, and:

Equation 10

for N ≤ 20, where MRj is the median ranks, MONj is the mean order number for jth failure and N is the total number of items tested.

Example

Suppose we have the testing failure data for a product, as shown in Table 2. We want to know the median rank for each failure. The first step is to calculate the MON for each failure in the data set.

Table 2. Fractional failure data
Number in State Time to Failure
1 59
0.4 71
1 95
1 108
1 137
0.7 154
1 177
1 199
1 219

The fractional failure at 71 hours is considered as 40% failure and 60% suspension, and the one at 154 hours is considered as 70% failure and 30% suspension. Then, the mean order numbers (MON) for each failure can be calculated via the increment method. The results are given in Table 3.

Table 3. Mean order number calculation via increment method
Number in State Time to Failures/Suspensions Increment (I) MONj
Failures Suspensions
1   59 1 1
0.4   71 1.4
  0.6 71 1.075 -
1   95 2.475
1   108 3.550
1   137 4.625
0.7   154 5.3775
  0.3 154 1.15563 -
1   177 6.533125
1   199 7.68875
1   219 8.84438

The increment and the MON for the first failure will be 1. Then the second failure, which is a fractional failure, will have a MON of 1.4 because it only had 40% failure.

When we encounter the first suspension data point, we need to calculate the increment. For example, the first increment is calculated as:

Equation 11

Then, the MON for the third failure can be calculated as:

Equation 12

We need to pay attention to the MON calculation for the sixth failure because it is also a fractional failure and is calculated as:

Equation 13

After calculating the MON for each failure in the data set, we can calculate the median rank values via the interpolation method, beta and F distribution approach, Benard's approximation and Johnson's approximation. The results are given in Table 4.

Table 4. Median rank values calculated via 4 different approaches
MON Via Interpolation Beta and F Distribution Approach Benard's Approximation Johnson's Approximation
1 7.413 7.413 7.447 7.413
1.4 11.632 10.520 11.702 11.671
2.475 23.026 22.413 23.138 23.117
3.55 34.5 34.557 34.574 34.562
4.625 45.991 46.058 46.011 46.007
5.3775 54.036 53.967 54.016 54.019
6.533125 66.388 66.736 66.310 66.323
7.68875 78.720 79.113 78.604 78.627
8.844375 90.946 91.377 90.898 90.931

All the results, other than the second and third values calculated via beta and F distribution approach, are within 0.5% or less error margin. Weibull++ uses a more complex algorithm to calculate the median rank values that encompasses the beta and F distribution and interpolation approaches. The values are presented in the following figure.

Probability Plot

Conclusion

In this article we presented the increment method for calculating the mean order number, adjusting for fractional failures. Then we provided four different approaches for calculating the median ranks based on the MON. No significant difference is observed between the methods for the given data set. However, it should be noted that the size of the difference between the methods is based on the data set being analyzed.

References

  1. Kececioglu, Dimitri. Reliability & Life Testing Handbook. 2 vols. Englewood Cliffs, New Jersey: Prentice Hall, Inc., 1993-94.
  2. "Life Data Analysis Reference Book," ReliaWiki.org, accessed September 8, 2016, http://reliawiki.org/index.php/Life_Data_Analysis_Reference_Book

 

 
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