Progress in Field Reliability?

What if you don’t have lifetime data??

I searched the Internet for “Why Use Weibull Reliability?” and got 196 hits “similar sites omitted”. Some hits looked like they were generated by Chat GPT. None showed data. None told how to test or use Weibull reliability estimates. Many sell Weibull software. All require lifetime data.

Assuming a continuous, parametric distribution such as Weibull conveniently extrapolates limited, censored lifetimes to estimate MTBF, even though MTBF may be far in the future. Https://en.wikipedia.org/wiki/Weibull_distribution lists many uses for Weibull distributions. A client had me use Weibull to model durations of covered calls!

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Ongoing Reliability Testing (ORT) is a way to estimate reliability function, as a function of independent variables, primarily age. This is because reliability is “the probability of successful function to specified ages under specified conditions” [O’Connor et al.]. 

Think of planning ORT as a design of an experiment (DoE). A DoE objective is to quantify effects of factors (independent variables) on the dependent variable, the reliability function. (The reliability and failure rate functions contain the same information, because reliability = P[Life > t] = exp[–∫a(u)du], where a(u) denotes the failure rate function and the integral runs from 0 to t.)

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Imagine product or part lifetime as the service time in a self-service system. Suppose you could observe periodic input and output counts of that self-service system, without identifying individual service times. How would you estimate the service time distribution without following individual products or parts from input to output? The maximum likelihood reliability estimator for an M/G/Infinity self-service-time distribution function from “ships” (inputs) and “returns” (outputs) counts also works for nonstationary arrival process M(t)/G/Infinity self-service systems, under a condition! A constant or increasing arrival (inputs or ships) rate satisfies the condition. If you identify returns by failure mode then you could estimate make nonparametric estimates of field reliability by failure mode and quantify reliability growth, without life data, if the condition is satisfied.

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I learned actuarial methods for forecasting and spares planning while working for the US Air Force Logistics Command in the 1970s [AFLCM 66-17 and AFM 400-1]. I am grateful for the education, and I am sorry to report that the USAF has reverted to MTBF management.

The US AFLC actuarial methods were developed for engine management in the 1960s by RAND Corp. [Giesler]  They estimated age-interval failure rates and made actuarial forecasts of engine demands depending on the flying-hour program plan. An actuarial forecast is ∑a(s)n(t-s), s=1,2,…,t, where a(s) is actuarial failure rate conditional on survival to age s and n(t-s) is the installed base of age t-s. Periodic meetings consolidated engine lifetime and failure data into agreements on actuarial failure rates, for forecasting engine demands and for war readiness spares requirements.

The USAF actuarial methods assume constant failure rates within short age intervals, Poisson demands, and ignore variance induced by variable flying hours per aircraft in the flying hour program. I later figured out how to estimate actuarial failure rates for all engines, major modules, and their service parts, with or without life-limits and without lifetime data; I computed the distribution of demand forecasts, not Poisson. I offered to show AFIT faculty, AFOSR, AFRL, and RAND how to extend actuarial methods to all service parts [George, 1993].

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ASQC Reliability Review, Vol. 13, March 1993

This paper shows how to estimate field reliability functions from ships and returns. It offers to estimate field reliability functions from your data. It suggests how you can use these estimates to improve service and inventory management. [Links to Google Sheet and user guide follow the paper.]

You can estimate field reliability functions without life data. You don’t need to know each part’s time to failure. In fact, if you don’t know times to failures, you have to estimate field reliability functions from ships and returns, unless you have a sample of times to failures, and some of the sample failed.

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Submitted to ASQC Reliability Review, Nov. 1996

A client wanted to compare the Kaplan-Meier (nonparametric maximum likelihood) estimators of the reliabilities of the old and new products. That is, he wanted me to test reliability functions, H_o: R₁(t)=R₂(t) for all nonnegative t vs. H_a: R₁(t)≠R2(t) for some nonnegative t.

Because I’m lazy and fixed in my ways and because I thought it would be easier to explain, I chose the Kolmgorov-Smirnov (K-S) test [Gnendenko]. It’s convenient, practically every statistics text has the tables, and I can program tables and the test statistic easily. The test uses the maximum absolute difference between the Kaplan-Meier estimates of the two reliability functions. Reject Ho if maximum absolute difference, D_mn=max|R₁(t)-R₂(t)| exceeds a critical value, where m and n are the two sample sizes.

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 This article appeared in 1983 in the ORSA Applied Probability SIG under the pseudonym “Anonymous” 

In another country far away, there were power plants that generated cheap electricity by unclear means. Since operation of the plants involved some risk, the Unclear Regulatory Commission was established to license power plants for operations. The URC decreed that any plant could have a license if its probability of an unclear accident was smaller than 0.000001 per year. The plant operators said they were uncertain. 

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SAS, JMP, R-”Survival”, Minitab, ReliaSoft, XLStat, and perhaps other statistics programs offer the Kaplan-Meier nonparametric reliability estimator as a default. Take credit for using nonparametric reliability estimation and avoiding unwarranted assumptions. What could go wrong using the Kaplan-Meier estimator?

• Cohorts could be non-stationary, random processes! 
• Failures could be recurrent process counts, not dead-forever! 
• Lifetime data depends on the censoring process(es); e.g., competing risks!
• Greenwood’s variance estimator errs! Covariances are missing!
• Alternative estimators could be more efficient than Kaplan-Meier!

Are you using all the information in data available from population data required by GAAP? If you don’t have lifetime data, use periodic failure counts. This article describes an example where the Kaplan-Meier estimator from grouped lifetime data is less efficient than using periodic failure counts, even though you don’t know which cohort they came from! 

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What is the efficiency of your reliability estimators and failure forecasts? Compare naïve forecasts: guess, AFR, MTBF, or times series extrapolations such as ARIMA, GMDH, and AI, vs. forecasts based on reliability estimators. With grouped lifetime data, the Kaplan-Meier estimator is statistically efficient, under a condition often ignored! With lifetime data you could use maximum likelihood estimators like Weibull or Kaplan-Meier from (a sample of?) grouped lifetimes. Without lifetime data you could estimate reliability from population ships and returns counts from data required by GAAP! This article compares efficiency and cost of forecasts based on alternative data and estimators.

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Originally published in the ASQC Reliability Review, Vol. 15, No. 3, Sept. 1995 

This article shows reductio ad absurdum in action. Yes you can achieve any MTBF you want, by mixing products with Weibull life distributions, but you won’t want the consequences. The article also shows the absurdity of specifying MTBF, alone. 

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How to distinguish a renewal process from a “generalized” renewal process? Compare observed monthly returns vs. actuarial returns forecasts using actuarial return rate estimates of TTFF and TBF (Time To First Failure and Time Between Failures). A geophysicist masquerading as an Apple reliability engineer said, “It’s too hard to figure out the probability that a return came from a computer made in an earlier year.”  It’s harder if returns could be second, third, or???

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“It is the policy of my Administration to respond to the coronavirus disease 2019 (COVID-19) pandemic through effective approaches guided by the best available science and data” [Biden Executive order, 2021]. That epidemic inspired the simultaneous nonparametric estimation of survival functions from case to recovery and case to death, without lifetime data (figure 1)!

Why not do the same for multiple-failure-mode data? This article shows nonparametric, multiple-failure-mode, maximum likelihood reliability estimation in a spreadsheet. Data are system first-failure times and the corresponding failure modes that caused the first system failures (table 1). However those data are dependent. I will explain the likelihood function, lnL, and how to find the maximum likelihood reliability estimates for all failure modes simultaneously.

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When a system fails for the first failure in one mode at time t, this data is right censored data for other failure modes! How to estimate reliability functions for all failure modes from first failure data?

Google AI says, “’Competing risks’ refers to a statistical scenario where a subject can experience failure from multiple possible causes, but once one failure occurs, it prevents the observation of any other potential failures, essentially creating “multiple failure modes” that compete with each other to be the first event observed; this means analyzing the probability of a specific failure type needs to account for the possibility of other competing failures happening first.” “Use appropriate statistical methods: Employ statistical models specifically designed for competing risks analysis…” 

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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?

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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].

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