Your Reliability Engineering Professional Development Site
The Kaplan-Meier reliability estimator errs on Fred’s bicycle ships and failure data! The Kaplan-Meier estimate was computed from Fred’s bicycles’ grouped failure data in the body of a “Nevada” table. It disagrees with the reliability estimate from ships cohorts and monthly failures (without knowing which cohort the failures came from). It disagrees with least squares nonparametric reliability estimates. All but the Kaplan-Meier estimate agree! Which would you prefer?
[Read more…]
The Kaplan-Meier reliability estimator is the nonparametric, maximum likelihood estimator from right-censored, grouped lifetime data. It has been used since publication, most statistics programs do it, and it has been taught since I was in school. I give away a spreadsheet version.
Lifetime data requires tracking individual subjects or units from their start to failure, death, or censoring. Data may be collected periodically grouped by cohorts: monthly sales, ships, or other collections of individuals, subjects, or units and each cohort’s lifetimes. Data could be displayed in a “Nevada” table with random cohorts in one column, and each cohort’s lifetimes grouped in periodic age-at-failure intervals in columns to the right [Schenkelberg].
People claim poor correlation of predicted and observed MTBFs. That is understandable because handbook failure rates and fudge factors for quality and environment were derived from unknown populations or samples. People also claim there is no basis for applying statistics or probability to MTBF predictions. MTBF predictions use failure rate averages that lack statistical causation. Why not incorporate Paretos in MTBF predictions?
Paretos are fractions of equipment failures caused by each type of part or subsystem. They represent what really happens. Incorporating Paretos requires statistics to adjust MTBF predictions. That causes Paretos in MTBF predictions to match field Paretos. A 1992 ASQ Reliability Review article “MIL-HDBK-217G” proposed using observed Paretos to adjust handbook MTBF predictions with a “Reality” factor.
[Read more…]
Insert these pages into your copy of MIL-HDBK-217. The boldface text is changed to MIL-HDBK-217E [1], section 5.2, on parts count reliability prediction. The changes explain how to use “Paretos,” proportions of parts failing in the field, to compute a reality factor that makes predicted Paretos match field Paretos. You can use field Paretos to calibrate predictions for new equipment. You probably have field Paretos on related parts used in your other equipment, which is now in the field. Remember, the field determines reliability.
[Read more…]
The Kaplan-Meier estimator is the maximum likelihood, nonparametric reliability estimator for censored, grouped lifetime data. It’s traditional. It’s in statistical software. Greenwood’s variance formula is well known. Could Kaplan-Meier be improved: smaller variance, better actuarial forecasts, seasonality, separate cohort variability from reliability? Could you estimate reliability without life data and preserve privacy?
[Read more…]
I estimated actuarial failure rates, made actuarial forecasts, and recommended stock levels for automotive aftermarket stores. I wondered how to account for seasonality in their sales? Time series forecasts account for seasonality but not for age, the force of mortality accounted for by actuarial forecasts. I finally figured out how to seasonally adjust actuarial forecasts. It’s the same method, David Cox’ “Proportional Hazards” model, used to make “Semi-Parametric” estimates and “Credible Reliability Predictions”.
[Read more…]
Lifetime data is nice to have, but lifetime data is not necessary! Generally Accepted Accounting Principles require statistically sufficient data to estimate nonparametric reliability and failure rate functions. Some work is required!
ISO 14224 “Petroleum, Petrochemical and Natural Gas Industries—Collection and Exchange of Reliability and Maintenance Data for Equipment” requires lifetime data to estimate exponential or Weibull reliability functions! Sales or ships and returns or failure counts are statistically sufficient to make nonparametric estimates of reliability and failure rate functions, without unwarranted distribution assumptions or lifetime data!
[Read more…]
Fred asked me to explain why use nonparametric statistics? The answer is reality. Reality trumps opinion, mathematical convenience, and tradition. Reality is more interesting, but quantifying reality takes work, especially if you track lifetimes. Using field reliability reality provides credibility and could reduce uncertainty due to tradition and unwarranted, unverified assumptions.
Data is inherently nonparametric. Cardinal numbers are used for period counts: cohorts, cases, failures, etc. Accounting data is numerical; it is derived from data or from dollars required by GAAP (Generally Accepted Accounting Principles); e.g., revenue = price*(products sold), service cost = (Cost per service)*(Number of services), or numbers of spare parts sold. Why not do nonparametric reliability estimation, with or without lifetime data?
[Read more…]
Time series forecasts are easy to make and data are available. They’re like driving while looking in the rear-view mirror. A survey listed 31 forecasting software programs: none actuarial [Yurkewicz]. Actuarial failure forecasts are less biased and are more precise than time series failure forecasts, because actuarial failure forecasts use age-specific failure rates. How much better?
The example in this article shows the 5% to 95% time series confidence interval width is 44.78 vs. the nonparametric actuarial Kaplan-Meier actuarial forecast width of 12.63, from grouped failure data, and actuarial forecast width of 15.45, from ships and returns counts.
[Read more…]
AFRs are periodic ratios of failure counts divided by installed base. Have you seen meeting rooms wallpapered with AFR charts (Annualized Failure Rate)? Have you sat through debates about the wiggles in AFR charts? Fred Schenkelberg wondered if reliability could be estimated from AFRs and their input data? How about age-specific reliability and actuarial failure rate functions? Actuarial forecasts? MTBFs? Wonder no more!
[Read more…]
The Kaplan-Meier reliability estimator is for dead-forever products or parts, given individual lifetime data or a “Nevada” table of periodic ships cohorts and their grouped failure counts. This estimator presumes that ships cohorts are NOT random. Production, sales, installed base, and cohort case counts are random! What does that do to Kaplan-Meier reliability estimates? What is the nonparametric reliability function estimator if ships cohorts are random?
[Read more…]
The New-Products manager asked me, “Your actuarial failure rate estimates (from vehicle registrations, bills-of-materials, and automotive aftermarket store sales) are for dead-forever parts with at most one failure. What if auto parts could be renewed or replaced more than once?” Chagrined, I wrote a spreadsheet program to estimate actuarial rates for renewal processes, without life data. But what is the corresponding estimator from grouped, cohort renewal counts like the Kaplan-Meier estimator for grouped, cohort failure counts?
[Read more…]
Fred wrote, “I would like to suggest that you continue writing articles – make them more tutorial in nature as if teaching someone the stats from scratch that you use in your articles. Instead of loads of references to papers and procedures, explain the concepts and math involved.” OK, I’ll try.
Nonparametric field reliability estimators require no unwarranted distribution assumptions and they preserve all information in data. Here’s how to compute them, without life data, while preserving all relevant information in ships and returns counts.
[Read more…]
Azmat Siddiqi suggested a certification in reliability statistics in 2022. Azmat believes in knowing and using the reliability statistical information in test, installed base, failures, and service data. Thanks Azmat.
I propose Certification in Reliability Statistics to recognize statistics knowledge, work experience, and applications. Certification in Reliability Statistics should provide assurance to employers, contractors, and collaborators that reliability statistics are estimated and used to the best extent with available data, including uncertainty quantification, with or without life data.
[Read more…]
The LinkedIn ASQ RRD group published this question from a reliability manager. Replies included:
