Understanding Updated Life Data Results
It is very common for reliability engineers to update
their test results with new data as tests progress, or
to augment their test results with fielded system
information. If the analyst is not careful in
interpreting the results, he or she can easily draw
wrong conclusions about the reliability of the product.
In this article, we present a scenario where updated
life data analysis results can seem confusing to the
analyst. The purpose of the article is to highlight the
impact that new data can have in life data analysis
results and to show one way to handle such a situation in
Weibull++.
The Scenario
An aerospace manufacturer is looking at the
reliability of a new component with an intended mission
duration of 2,100 hours. The reliability engineer has
internal test data for the beta version and has also collected
reliability information from the company’s beta
customers after 2,500 hours of usage, in which no
failures were seen.
The reliability engineer, Lisa, decides to use a 2parameter
Weibull distribution to analyze the data. Since there
are relatively few failure data points and a lot of
suspended data (30 units from customer beta sites), she decides
to use Maximum Likelihood Estimation (MLE) as the
statistical analysis method for the data. MLE uses the actual suspension times,
not just their relative position in terms of where they
occurred in the data set, as rank regression would do.
The life data analysis results are shown in Figure 1.
Figure 1: Folio with Internal Test Data and Suspensions from Beta
Testing After 2,500 Hours of Usage
Lisa then uses the Quick
Calculation Pad (QCP) to obtain the reliability estimate
at the mission duration of 2,100 hours, as shown in Figure 2.
Figure 2: Reliability Estimate at 2,100 Hours with
Suspended Data at 2,500 Hours
Based on the original data set, the reliability at 2,100 hours is
calculated to be 97.65%. She creates a report
distributing this information to her organization.
Later in the process, Lisa is
ready to update the report with new data from the beta
sites. She collects data from the beta sites and sees
that there were still no failures in the field, with the
30 beta units having reached 7,000 cumulative hours of
test time. She is expecting an increase in the
calculated reliability, since there was more accumulated
time and no failures. She wants to keep the analysis
consistent so she again calculates the life data using
a 2parameter Weibull distribution using MLE. Figure 3
shows the updated life data analysis results with the 30
beta site units still operating at 7,000 hours.
Figure 3: Folio with Internal Test Data and Suspensions from Beta
Testing After 7,000 Hours of Usage
She then uses the QCP to obtain the updated
estimate of the reliability at the mission duration of
2,100 hours, as shown in Figure 4.
Figure 4: Reliability Estimate at 2,100 Hours with
Suspended Data at 7,000 Hours
To her surprise, the new reliability estimate is now
95.61%, which is lower than the previous estimate of
97.65%. How is this possible? The beta site units did not
exhibit any failures and had more accumulated hours.
The reliability should have gone up.
Explanation
This is a typical problem of looking at data in
isolation and not understanding the impact of the
selection of the model. Using a 2parameter Weibull
distribution, the model parameters are recalculated both
in terms of the beta parameter, which indicates the
slope of the Weibull probability plot, and the eta
parameter, which is the characteristic life of the
component (i.e. the estimated time by which 63.2%
of the components will have failed).
If you take a closer look at the data, you’ll notice
that both the beta and eta parameters are different in
the original and updated results. The slope has changed
from 5.0151 to 1.5634, and the eta parameter has changed
from 4,427 to 15,280. Figure 5 shows a MultiPlot of the
two Weibull distributions. Before the point at
which the two probability lines cross each other, the
updated analysis (shown in blue) yields lower
reliabilities. After the lines cross, the updated
analysis yields higher reliabilities. Since the mission
duration of 2,100 hours is before the two lines meet,
the updated analysis resulted in a lower reliability
estimate for 2,100 hours.
Figure 5: Weibull Probability Plot
of the Original and Updated Analyses
Solution
There is nothing inherently wrong with the MultiPlot
shown in Figure 5. The analyst
just needs to understand the impact of reestimating a
probability plot based on the data. The number of parameters
that will be used can allow different
"degrees of freedom" for the line to change both in
terms of slope and intercept.
If Lisa wanted to create an
updated report reflecting the progress from using
the suspended data from the beta sites, a possible
solution would be to superimpose the same slope in both
probability plots.
In this case, a good estimate of the actual slope
would involve the original test data and good
engineering judgment in terms of the expected failure
rate behavior. Remember that a high beta can indicate
wearout, while a beta close to 1 indicates more
random failure types, such as insufficient design margins,
external events, etc. A beta less than 1
indicates early life failures, such as those caused by misassembly, manufacturing issues or damage during
shipping or storage.
The original internal test data set, excluding the 30
suspended units from the customer beta sites, is calculated using a 2parameter Weibull distribution and Rank Regression on X
(RRX), since the data are now complete (exact times to
failure) and it is a small data set. For comparison
purposes, the data are also analyzed using MLE, and
Figure 6 shows a
MultiPlot of the two methods, where the black line
represents the RRX analysis and the blue line represents
the MLE analysis.
Figure 6: Weibull
Probability
Plots Including Internal Test Data Only, Using MLE and RXX
Using
engineering judgment about the nature of the expected
failure modes, a beta would be chosen to be used for all
further analyses. In this case, it
would be appropriate to accept the most conservative estimate of beta.
Since the mission duration is 2,100 hours, a smaller
beta would yield more failures in that region. In this case, the RRX beta
of 2.8219 is chosen.
After choosing a beta value, every updated analysis
with new suspended data can be done by superimposing
that value of beta on the data set. In that case, a 1parameter Weibull
distribution would be used and the value of beta
would be provided, as shown in Figure 7 for the data set including
the suspensions at 2,500 hours.
Figure 7: Superimposing a
Beta
Parameter Value onto the Analysis
The same process would be used in the analysis
including the
customer beta site units suspended at 7,000 hours. Using this process
ensures that the slope remains the same in the two data
sets, making it easier to draw comparisons, as shown in
the MultiPlot in Figure 8.
Figure 8: Weibull
Probability
Plot of the Original and Updated Analysis with the Same
Superimposed Beta
Figures 9 and 10 show the new calculated values of reliability
at 2,100 hours. The data set with the 30 customer beta
site units suspended at 2,500 hours exhibits a
reliability of 91.44% and the data set with the customer beta
site units suspended at 7,000 hours
exhibits a reliability of 98.96%. The improvement obtained by adding more hours to the
suspended units is now more evident.
Figure 9: Reliability
Estimate
at 2,100 Hours with Suspended Data at 2,500 Hours and
Superimposed Beta
Figure 10: Reliability
Estimate
at 2,100 Hours with Suspended Data at 7,000 Hours and Superimposed
Beta
Conclusion
When updating life data after a previous analysis, it is
important to understand the impact of the choice of the
model and the statistical analysis method upon the results. A good practice is to look at
probability plots in order to understand the total behavior of
the model. When the analysis and further
updates are performed without examining the overall behavior of
the statistical models, wrong conclusions can be
drawn.
