Dr. Bob Replies to E-mail Questions

 

Paul Barringer suggested I share the answers and comments with you under the heading of "The Master Speaks" My ego is not quite large enough to use that heading. I will bold the keywords in the questions and Italicize my reply. The following questions have been asked more than once in E-Mails to me.

  1. Hi Dr. Bob, I was your student last year in the Weibull course at Lohmar, Germany.
    I see that the average and the standard deviation estimates in the SuperSMITH
    software are different from the simple average and standard deviation (as
    calculated by Excell, for example). Please explain why there are differences.
    Thanks a lot for your help.     Dr. Bob’s reply: You cannot use the usual equations for the mean and standard deviation because they do not account for suspensions. Instead we solve for eta and beta by whatever method you have selected and then use the Weibull equations in Appendix G to estimate the mean,m , and standard deviations .



2. With interval and inspection data I use the data shortcut to input points at the same value. For example, 88x9 means there nine failures occurred at time eighty eight. This data appears as the number 9 on the plot. If I change from median rank regression (MRR) to the Inspection Option the plot position of the 9 changes. Why? If you input the 9 failures in separately they would appear as a vertical column of points on the Weibull plot. With MRR the 9 is located in the middle of this column which is approximately where we expect the standard Weibull line to appear. With the Inspection Option the 9 is located at the topmost point of the column, where we expect the Inspection Option line to appear.

3. In the New Weibull Handbook you use median bias for most of the comparison of methods and yet you write about unbiased estimates. I thought an unbiased estimate is one whose expected value equals the true value. I am confused as the expected value is the mean value. Correct? You are correct that an unbiased estimate is one whose expected value equals the true value and
that the mean value is the expected value. However, the mean value is not a good measure, or a typical measure, for skewed distributions and most life distributions are skewed. We recommend the median value instead of the mean value for skewed distributions. Statistical estimates will split 50/50 around the median. For example, we recommend the median rank plotting instead of the mean rank positions for life data analysis. In the case you selected, five failures, B1 and beta are highly skewed. See the simulation results below. For comparing the accuracy of alternative methods like MRR versus MLE, I use the median bias rather than the mean bias. With MLE-RBA Weibull we employ the median bias correction as the standard or default, but we also offer a mean bias correction in our software. These are quite different corrections; the median bias correction is C4^3.5 versus C4^6 for mean bias. The mean bias correction is much larger.

 

MonteCarlo Simulation N=5, True Values eta=1000, beta=1, B1=10….. Median /Mean

 

eta

beta

B1

MRR X on Y

972/1059

1.05/1.179

12.5/37.3

MLE

942/1028

1.25/1.45

23.9/57.2

MLE-RBA

945/1022

1.02/.991

10.4/22.4

MRR Y on X

1068/1124

.912/1.05

6.3/27.9

Note that statisticians prefer mean square error to bias as a measure of accuracy. Engineers prefer bias as a measure because they want to know if the estimate is optimistic or pessimistic. If a failure produces a health, or death, or crash risk, engineers want either an unbiased estimate or a conservative estimate. Optimistic bias estimators like small sample MLE failure forecasts and B life are unacceptable. Our conclusion is to recommend methods with the smallest median bias as our best practice. MRR and MLE-RBA for small and moderate sample sizes are examples.

 

4. When I use a three parameter Weibull the slope, beta, is less than the two parameter beta. Which beta is correct? The 3P Weibull is a much more complex distribution than the two parameter and we have fixed requirements to meet before we adopt the 3P solution. Remember the four hard fixed rules for using 3-parameter:

If you meet all these criteria above the 3p distribution is the best distribution and the 3P beta is the correct beta. The 2p beta is irrelevant. For example, when we have some data sets with missing data and use the 3p and compare it to data sets without missing data from the same source the 2p with all the data fall on top of the 3p with missing data, same beta.

5. For warranty claims forecasting by age you recommend both the Inspection Option and the Kaplan-Meier model. Which should I use? We have research underway comparing all the interval methods, Inspection Option, Probit, K-M, and Interval MLE but it is not completed. Industry seems to prefer the Inspection Option for warranty claims by age but many use K-M. There is a problem with KM. When we use the KM with the actuarial correction, some of your suspensions are eliminated from the data set. This prohibits using the Abernethy Risk forecast as the suspension histogram is wrong. However, this does not effect the estimate of per cent claims at the end of the warranty period which is the usual objective. The Inspection Option does not have this problem. If the probability of repeat warranty claims for the same problem is significant you may want to consider Wayne Nelson’s Graphical Repair method described in Appendix M.

 

6. The failures were produced by two competitive, "dueling ," failure modes. We cannot separate the data into two failure modes. How can we do a failure forecast? The cumulative probability of failure considering both modes is [1-(F(t1))x(F(t2))]. The first step is do a mixture analysis with WSW. If the "p" value supports two Weibulls rather than one Weibull, it is now possible to do an Abernethy risk analysis with the WSW, version 4.0V and later. Alternatively, you could use Monte Carlo Simulation. "RAPTOR" would be a good choice for the simulation software.

7. I want to know if there is more information about how the adjusted rank algorithm (that adjusts the plotting positions for suspensions) was calculated. You reference a Drew Auth as the person that came up with the simplified version of something originally done by Leonard G. Johnson. His book, "The Statistical Treatment of Fatigue Experimentation," has been out of print for some time. Any help you could give in tracking down how this adjusted rank algorithm was derived would be most appreciated. The derivation in Johnson’s book is not too clear; however Charles Mischke’s ASME paper listed in the References is excellent, very clear. I will ask Paul Barringer to add it to his list of significant papers and book in pdf format on his website. You may download it from there.


8. Hi Dr.Bob, I'm lost. I have a Beta of 0.80 and the BetaU is 1.03 and BetaL is 0.70. How does that tell me if the failure is close enough to be "random"/exponential? Your data is not significantly different from an exponential at whatever level of confidence you selected to get the bounds on beta because the value one lies within the interval. If your double sided confidence bounds are at 90% confidence, your data is not significantly different from an exponential at 90% confidence.

9. In Chapter 8 discussing the exponential distribution, you mention the mean time between in-flight shutdowns for commercial engines of 25,000 engine operating hours. As typical commercial flights are 2-4 hours this number seems extraordinary. At the time I wrote that section, 25,000 was the standard; today it is much higher and it is extraordinary. To put that in context, years ago I had lunch with Dr. Von Ohain with the President of Pratt & Whitney Aircraft. Dr. Von Ohain invented the gas turbine and developed the Jumo engine for Germany. He said the mean time between in-flight shutdowns on the Jumo engines in the ME262 was 25 hours. When I expressed shock, he said that was good enough as the mean life of the ME262 was 7 hours and 10 minutes. In the six decades that have followed World War II we have made progress.

10. I have recently been reassigned and need to use statistics in my new position. Could you recommend some introductory statistical texts?

Some recommendations:

The first three are old standards. Suggest AMAZON.COM second hand.

11. My statisticians recommend Cramer-Mise and Kolmogorov-Smirnoff as measures of goodness of fit but you do not. Why? Comparing all the known goodness of fit tests the two best goodness of fit tests are the likelihood ratio test and our "p" value for the correlation coefficient squared (Coefficient of Determination), [as shown in Chi Chao Liu's thesis]. The likelihood ratio test described in Chapter 5, is used with MLE or better with MLE Reduced Bias Adjustment. The p value for r squared presented I Chapter 3, goes with median rank regression, X on Y. Of course, either technique may be used with either method. For testing goodness of fit both are excellent and we recommend using both. For distribution analysis, both work well, but you need at least 21 failures to have enough information to make a credible choice with any method. Less than 21 failures always use the Weibull 2p even if you know the data are log normal. If you do not have Chi Chao Liu's thesis you may download it from Paul Barringers Website.