How to Make Data Fit Weibull?

“I call the shots based on aggregation of the data.” Donald J. Trump

Verifying and Validating Alternative Facts

Test or sample data could invalidate the Weibull assumption and could yield undesirable parameter estimates, misrepresent sample uncertainty, and cause unforeseen decision and risk consequences. Wouldn’t it be better to design, conduct tests or collect samples, aggregate, and manage data, so that the Weibull assumptions would not or could not be invalidated? The method, judicious censoring, could be applied to any unimodal probability density function (pdf), and to other pdfs if done carefully. This method could be useful in many contexts:

• Sample or test some subjects or units, some of which fail, and some might survive (right-censored failure times)

• Sequence(s) of failure times TTFF, TBF1, TBF2, etc. (a la Duane-AMSAA-Crow-Nelson “Reliability (MTBF) Growth Testing”) presumed independent and perhaps identically distributed.

• Combination of a and b; e.g. Nigerian refinery centrifugal pumps [********], jet engines [Weckman et al.], emergency diesel generators [Klügel], covered-call durations [Chronim Investments, Inc.]

• Two-sample or multiple reliability comparisons such as in reliability growth

• Sample or test data polluted by sell-through time, return time, reporting delays on most recent data, etc. [ReliaSoft]

The objective is to manage reliability data collection so the data fits the Weibull reliability function as well as possible; i.e., design experiments, plan tests, or take samples accordingly. The Weibull reliability function accommodates more alternatives than constant failure rate-exponential reliability function.

"The advantages of a simple model are:

1. It is economical of time and thought

2. It can be understood readily by the decision maker

3. If necessary, the model can be modified quickly and effectively"

Bierman, Bonini and Housman, Introduction to Quantitative Business Decisions, Addison-Wesley, New York, 1973

The benefits from managing and aggregating reliability data collection to fit the Weibull assumption could be:

• • Reduce strains on brains: yours and your managers’.

• Use Weibull software instead of nonparametric and multivariate statistics, because other people do [ReliaSoft Weibull++, SAS PROC LIFEREG, etc.], and universities teach Weibull [U AZ, U MD, etc.], and standards {Abernathy, ASTM G172, IEC TC56, IEC 62539, IEEE 930, etc.].

• Report credible results within budget and time constraints [Dodson].

• Do homework, get thesis approved, etc. [Nigerian refinery pumps]

• Why not use field reliability data? [George] “Field reliability data is garbage…” [Ralph Evans, IEEE Transactions on Reliability editor]  

Maximum Likelihood Method of Judicious Censoring

If you use maximum likelihood Weibull estimators [Baiko.xls, George, June 1995, Cohen, and many others], then use judicious censoring so that the Weibull assumption remains valid. Assume independence and identical distributions. Suppose two failure times t1 and t2 have been obtained. What to do with a third unit on test or observation?

Suppose you have already estimated Weibull parameters by maximizing the (log) likelihood, L=f(t1)*f(t2), where f(t) is the Weibull pdf. Another failure at t3 might contradict parameter estimates and violate the Weibull assumption, unless ln(t3) lies on or near linear regression line from ln(t1) to ln(t2) on Weibull paper (for example. Terminate testing at t’3, before another failure time t3 could contradict the Weibull assumption, change parameter estimates, and introduce more sample and model uncertainty.

Figure 1. Two observations lie on a straight line on Weibull probability paper

The right-censored likelihood function is L= f(t1)*f(t2)*(1-F(t’3)) instead of L=f(t1)*f(t2)*f(t3). What censoring time t’3 will make (1-F(t’3)) = f(t3), where hypothetical ln(t3) lies on the linear regression line between ln(t1) and ln(t2)? (Notice t’3 is not the same as t3?) 

Sorry. There is no such censoring time t’3 except infinity, because the Weibull pdf is unimodal. OK. Put k more units on test or take k more observations so that (1-F(t’3))k or P(1-F(t’3(j))) = f(t3); j=1,2,…,k, for some k and perhaps different t’3(j) values. When k units have survived to censoring time t’3, terminate testing, plug data into Weibull software, graph data on Weibull paper if you prefer, report Weibull parameter estimates, and quantify sample uncertainty; then go back to whatever you were doing, complacent that you have reinforced beliefs in Weibull reliability statistics.

Figure 2. Weibull pdf and ccdfs

How long should t’3 censoring time be to confirm Weibull assumption (or other unimodal pdf assumption)? It depends on k and on the underlying Weibull shape parameter estimate, b in F(t) = 1–exp[–(t/a)b]. Censoring time, t’3, decreases as b and k increase. Censoring is successful if none of the additional k units fail. Censoring success probability is (1-F(t’3))k or P(1-F(t’3(j))), the probability that k units survive to censoring time(s), assuming independence. Censoring success probability increases as b and k increase. You can see that increasing k or t’3 reduces probability all k survive to censoring time. Sorry about that too.

What if one of the additional k units fails before censoring time? Declare the failed unit to be a non-conforming unit observation and throw it out, if it doesn’t fit the linear regression line on Weibull. Include its failure time t3 in the likelihood function, f(t1)*f(t2)*f(t3)*(1-F(t’4))k-1, and repeat the exercise of adding more units to test until t’4. “If at first you don’t succeed, try, try again.” W. C. Fields. [“How to Cheat With Statistics,” by L. L. George]

Spreadsheet implementations and Mathematica analyses:

• Baiko.xls does maximum likelihood estimation of Weibull parameters.

• WeiCheat.xlsx computes f(t), (1–F(t)), (1–F(t))2,… and f(t)–(1–F(t))k for your choice of Weibull parameters and ages t. Censor at age when f(t)–(1–F(t))k»0.0

• WeiCheat.xlsx also does spreadsheet regression to estimate Weibull parameters from your failure time observations so you could check whether another failure could invalidate Weibull assumption [Dorner].

• WeiFish.nb, Mathematica notebook, solves for maximum likelihood estimate of scale parameter given known shape parameter, when all test units are censored at t’. If shape parameter > 1, then t’ = 0, otherwise k = 0. I.e., if shape parameter > 1, test for t’ = 0; otherwise test no units. Mathematica also solves for Fisher information matrixes, censored or not, for two samples. 

• Censoring could be applied to estimation of parameters of any pdf f(t), although multimodal pdf requires further consideration, because f(t) = (1–F(t’)k has more than one solution.

For example: four Nigerian centrifugal refinery pumps with eight years of failure data including their suspension times. The tests for Weibull fit were eyeball analysis (figure 3) and chi-square goodness-of-fit statistic, comparing empirical and Kaplan-Meier nonparametric reliability estimators with the least-squares Weibull fits. The chi-square goodness-of-fit statistic is S(observed-expected)2/expected, where “observed” is empirical or Kaplan-Meier nonparametric reliability estimate and “expected” is the Weibull reliability function, at observed failure times. The combined distribution, all 4 sets of {TTFF(i), TBF1(i), TBF2(i),…} i=1,2,3,4,  is not Weibull. Individual distributions of {TTFF(1), TBF1(1), TBF2(1)},…,{TTFF(4), TBF1(4), TBF2(4)}, could be Weibull but with different parameters and means, pump 1 with wearout (Weibull shape parameter > 1) and pumps 2-4 with infant mortality (Weibull shape parameter < 1). Distributions of TTFF, TBF1, TBF2, etc. could be Weibull but with different parameters and means, TBF1 had infant mortality and TTFF and other TBFs had wearout although with different means, indicating perhaps that first repair was faulty. Figure 4 shows autocorrelation, so successive pump times-between-failures were dependent, which cast some doubt on the Kaplan-Meier estimates and the goodness-of-fit statistics too.

Figure 3. P-P plots of pump ln(-ln(Kaplan-Meier reliability estimates)

Figure 4. Autocorrelation of successive times-between-failures for pumps

How to get refinery pump failure data or other data to fit Weibull better? Just because the data on Weibull plot seem to lie within the confidence band on does not mean data are Weibull. Weibull confidence limit plots may be single-point limits, not confidence bands, and documentation doesn’t metion whether censoring was accounted for. Estimate Weibull parameters from two adjacent times-between-failures (e.g., TTFF and TBF1) and check whether and how well next time-between-failures fits regression line [Morice and many others]. If it fits, include it; otherwise, go on to the next pair. Did anybody say all data had to fit the same Weibull? Eliminate outliers and try again; there were some outliers. Time data units were months: more precise accounting may fit better. Consider bivariate and multivariate Weibull distributions. Also consider copulas to account for dependence, different parameters, and autocorrelation. Instead of plugging times-between-failures into ReliaSoft Weibull++, plug in median ranks! [********, Dorner, ReliaSoft] “Ignore these restrictions and go ahead with the analysis. Hopefully your thesis advisor or the journal editor falls asleep while reading your paper.” [Chong-Ho Yu]

Recommendations

“It ain't what you don't know that gets you into trouble. It's what you know for sure that just ain't so.” Attributed to Mark Twain. Why does censoring preserve the Weibull assumption vs. uncensored data that may not fit Weibull?

Censoring to preserve the Weibull assumption induces dependence between failure times and censoring times. Censoring increases parameter estimator variance. Compare the Cramer-Rao lower bounds on the variances of parameter estimates for the alternatives: uncensored vs. censored. Their Fisher information matrixes, E[(¶lnL/¶q)2] = –E[(¶2lnL/¶q2)], are: E[(¶Sln(f(t(i))/¶q)2] and E[(¶(Sln(f(t(i))+k*ln(1–F(t’)))/¶q)2] respectively, where the sums are over all observed times-between-failures. The Cramer-Rao lower bounds on the variance (-covariance matrix) of the Weibull parameter estimates, –E[(¶2lnL/¶q2)]-1, for alternative L= f(t1)*(1–F(t2)) is complicated, but I conjecture the ratio of (C-R lower bounds on alternative censored t’2 vs. uncensored t2) parameter variances could be proportional to two (for two units on test). I.e. more uncertainty (more variance) allows more assumptions to be possibly true. [I am still working on this.]

In case of completely censored data, there could be no contradiction to the Weibull assumption or limitation to Weibull parameter claims, depending on censoring time t’. The likelihood of the censored observations is exp[-k(t’/a)b], larger for smaller k and t’; so test fewer units for as little time as you can without failures. Imputing failure times for censored data may be construed as data aggregation. Suppose you knew the shape parameter b pretty well, and suppose you pretend that one out of k units was about to fail at censoring time t’.  Then the scale parameter maximum likelihood estimator is t'^b*k^(1/b), a lower bound on a, without contradicting Weibull assumption! 

Contact me, pstlarry077@gmail.com, if you want Excel workbooks, Mathematica notebooks (WeiFish.nb), or the references. Alternatively, send data and ask questions for other reliability data management problems such as:

• • “What MTBF (prediction) do you want?” [George, Sept. 1995] Use the magic of Weibull mathematics to mix a proportion of infant mortality units with production units to achieve any MTBF prediction you want. 

  • How much to repair or when to replace so that the TBFs in an alternating renewal process TBF1-TTR1-TBF2-TTR2- have a constant failure rate? Subsequent TBF depends on previous TTR; i.e., extent of repair determines subsequent TBF distribution.

  • When to terminate sequential probability ratio tests to select desired null or alternative hypothesis?

  • How to prove any null hypothesis (by restricting space of alternative hypothesis).

  • How to estimate reliability or survivor functions without life data statistically legitimately!

References    

Cohen, A., “Maximum Likelihood Estimation in the Weibull Distribution based on Complete and Censored Samples,” Technometrics, Vol. 7, No 4, 1965

Dodson, Bryan, “Optimization of Reliability Verification Test Strategies,” AQC CIncinnati, vol. 49, no. 0, 1995, http://www.engineeredsoftware.com/papers/test_opt.pdf

Dorner, William, “Using Microsoft Excel for Weibull Analysis,” Quality Progress, Jan. 1, 1999, http://www.qualitydigest.com/magazine/1999/jan/article/using-microsoft-excel-weibull-analysis.html#

George, L. L., “Estimate Weibull Parameters from Warranty Returns,” Reliability Review, ASQC, Vol. 15, No. 2, June 1995

George, L. L., “What MTBF Do You Want?,” Reliability Review, ASQC, Vol. 15, No. 3, Sept. 1995 pp. 23-25

George, L. L., “Would You Believe You Don’t Need Life Data,” https://sites.google.com/site/fieldreliability/ProdSpec

ReliaSoft, “Median Rank based on Mean Order Number,” Reliability HotWire, Issue 104, October, 2009, http://www.weibull.com/hotwire/issue104/relbasics104.htm

Klügel, Jens-Uwe, “Investigation of time-dependent trends in plant-specific data for active components…,” PSA Sept. 2008, Knoxville, TN, http://iet.jrc.ec.europa.eu/apsa/sites/apsa/files/files/documents/Klugel_doc.pdf

Morice, E., “Quelques problèmes d’estimation relatifs à la loi de Weibull,” Revue des statistique applicaèe, tome 16, no. 3, (1968), p.43-68, http://www.numdam.org/item?id=RSA_1968__16_3_43_0

Weckman, G. R., et al., "Modeling the Reliability of Repairable Systems in the Aviation Industry," Computers and Industrial Engineering, 40 2001, 51-63,