Larry George — Active Contributor

Originally published in the ASQC Reliability Review, Vol. 15, No. 3, Sept. 1995 pp. 23-25 

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. 

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???

“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.

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…” 

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?

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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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.

Originally published in the ASQ Reliability Review, Vol. 12, No 3, June 1992

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.

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!

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.

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!

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?