Design and Analysis of Accelerated Reliability Tests

I asked ASQ if I could reproduce these articles here. They referred me to www.asqrd.org, and Marc Banghart said go ahead; I am working on them; please be patient or contact me. Original articles: 

http://asq.org/reliability/2008/05/design-of-accelerated-reliability-tests-part-1-piecewise-linear-failure-rate-models-en.pdf

http://asq.org/reliability/2004/09/reliability-review-v24-i03-full-issue.pdf

ASQ memberships login isrequired to access those URLs

Design of Accelerated Reliability Tests, Part 1: 

Piecewise Linear Failure Rate Models

ASQ Reliability Review, Vol. 24, No. 2, June 2004

Uploaded here Sept. 8, 2017, needs checking

Sudden Increase in Gravity Ends Accelerated Life Test

Professor Richard E. Barlow said in an interview, “Spherical pressure vessels are used on the Space Shuttle. They are also used in very highly classified situations. [deuterium-tritium pressure vessels] Anyway, these pressure vessels were being tested as well as strands that were used to construct these special vessels. There were accelerated tests that went on for 7 years. We analyzed the data at the end of a seven-year period. As a result of this analysis, we could see for one thing that failure distributions were changing with stress levels. This meant that you should not use, for example, an exponential life distribution model with the usual power law as a function of stress level for the failure rate. The analysis that I did using computer facilities which were extremely good at that time at Lawrence Livermore Laboratory, analyzed data over 7 years.”

“A result of the analysis was that the shape parameter of the Weibull life distribution changes with stress level. You cannot use an exponential life distribution where the failure rate changes with different stress levels. We analyzed the data independent of the scale parameter using total time on test plots. Then we used posterior distributions based on a Weibull-type model. It becomes absolutely clear that the Weibull shape parameter changes with stress level. As the stress level goes down, mean life goes up and the Weibull shape parameter also goes up. That means that the failure rate is increasing more and more rapidly with age.”

In the 1970s in a Livermore “string farm,” Kevlarâ strands with various weights hung in accelerated life tests. Pressure vessel reliability can be approximated from strand reliability, with a Taylor series. If the Kevlar strands failed the test, certain pressure vessels would be replaced every five years. The experiment was designed nicely, based on the results of pull tests, so that strand reliability at the working stress would be estimatable from observations at several, higher stress levels. Every now and then at the string farm, there’d be a “plop,” and a sound recording device noted the age at which the strand broke and the weight hit the ground. Then came the Livermore earthquake of 1980. Ploppity-plop went the weights.

Ordinary DoE Doesn’t Work for Reliability Tests

The National Institute of Standards and Technology (NIST) says one purpose of an accelerated test is “to obtain enough failure data at high stress to accurately project (extrapolate) what the CDF at use will be.” [CDF stands for cumulative distribution function = 1-reliability function.]

Statistical design of experiments (DoE) tacitly presumes that every component will fail. In reliability tests, few may fail, even under acceleration. Such “censored” observations can’t be ignored, but some failures must occur for meaningful conclusions.

Management wants to demonstrate product MTBF ASAP. Demonstration tests are too short to show old-age reliability and MTBF. So let’s learn as much as we can about reliability during the infancy, youth, and useful life while providing a credible MTBF estimate. To do these things, we have to assume reliability and acceleration models, defend the assumptions, and extrapolate.

Part 1 of this article proposes piecewise linear failure rate functions and estimates their reliability, infant mortality, and MTBF. It proposes acceleration alternatives, including one that accelerates testing greatly. Part 2 gives designs and statistical analysis of test data, assuming a piecewise linear failure rate function and power law acceleration.

Piecewise Linear Failure Rate Function

Regression and DoE texts start with linear models, Y = a + bX, because they’re simple to analyze and explain and because the linearity assumption can be tested. In the reliability context, the piecewise linear failure rate function resembles the infant mortality and constant portion of the bathtub curve [George, 2002].

The piecewise linear failure rate function begins with a decreasing linear segment splined to a linear constant or increasing failure rate line segment thereafter. The knot point (intersection) of the line segments occurs at age to, the end of infant mortality. This formula is

a(t) = a+b(to-t) for t < to and

a(t) = a for t ³ to.

A positive coefficient b represents infant mortality. A negative coefficient b could indicate there is random delay before the failures are reported, perhaps due to sell-through time. Figure 1 plots the piecewise linear failure rate function

a(t) = 0.0001+0.0001(7-t) for t<7 and

a(t) = .00001 for t ³ 7.

Figure 1. A piecewise linear failure rate function with a = b = 0.0001 and to = 7. Age is on horizontal axis

The piecewise linear model is used in test Conconi to determine your aerobic threshold. Dr. Francesco Conconi observed that the square of heart rate was a piecewise linear function of exercise stress level, measured in speed or revolutions. Above the aerobic threshold heart rate, your body uses anaerobic metabolism to maintain the exercise stress level without consuming oxygen. Therefore, your heart rate increases less per unit of increase of speed, because it isn’t pushing oxygen around in your bloodstream. A spreadsheet to estimate your aerobic threshold is available from the author.

Kalbfleisch and Prentice [pp 97-98] propose the failure rate function exp[a(t)] with piecewise linear a(t), for postoperative risk, to represent infant mortality. They assume that the age to at which postoperative risk ends is known. In biostatistical survival analysis, the objective is to estimate the effects of factors on the failure rate, and there are usually many failures. In reliability, we need to estimate the failure rate function itself, the age at which infant mortality ends may be unknown, and failures are rare.

The reliability function R(t) is related to the failure rate function by

R(t) = P[Life > t] = exp[-òa(s)ds],

where the integral runs from 0 to age t. Figure 2 shows the reliability function for the piecewise linear failure rate function in figure 1,

R(t) = Exp[(bt2)/2-t(a+bto)] for t < to and

R(t) = Exp[-at-(bto2)/2] for t ³ to.

Figure 2. Reliability function of a piecewise linear failure rate function with a = b = 0.0001 and to = 7

The probability of failure due to infant mortality is

P[Life £ to|not a random failure] = [1-R(to)] = 1-exp[-bto2/2].

For small b and to values, this is approximately bto2/2, the area under the failure rate function. Use this probability to decide whether a process improvement for reducing infant mortality is worthwhile. It’s worthwhile if the risk (expected cost) per unit before improvement is greater than after, i.e.,

[bto2/2][per unit cost of infant mortality failure] > [-b'to'2/2][per unit cost of the fix],

where b' and to' are the infant mortality parameters after improvement. If the process improvement incurs a fixed cost, revise the risk computation to total cost.

The MTBF formula for a piecewise linear failure rate function involves the imaginary error function. For small a, b, and to, the first-order Taylor series expansion for MTBF, (1-to2b)/2+to2b/6-ato4b/24, converges rapidly. The MTBF corresponding to the parameter values in figure 2 is 9975.5. Figure 3 shows the MTBF for various a and b values.

Figure 3. MTBF for some a and b values with to = 7

A feature of a piecewise linear failure rate function is that, if you could eliminate infant mortality, then your product would have constant failure rate, MTBF = 1/a, and chi-square MTBF confidence limits. For example, if to is one age unit, then the MTBF is exp[-b/2]/a » [1-b/2]/a, compared with MTBF = 1/a for constant failure rate. Infant mortality can be eliminated by burn-in, so estimating to is important.

Which Acceleration Model?

Accelerated testing may have several effects on a piecewise linear failure rate function, effects that can be used to estimate MTBF and reliability under normal use and help reduce infant mortality, increase MTBF, and improve reliability.

Arrhenius, Eyring, and power-function models are often used to model acceleration [Nelson], and the reliability function is often assumed to be exponential, lognormal, or Weibull, with a shape parameter independent of acceleration. These are strong assumptions, violated by Kevlar strands.

Arrhenius’ law describes chemical reaction kinetics as a function of temperature. It is usually applied to the (constant) failure rate or scale parameter. Are your product failures due to chemical reaction kinetics? Is the scale parameter the only parameter being accelerated? The indifference about which parameter Arrhenius’ law applies to should make you nervous. Barlow says that the shape parameter b of the Weibull reliability function exp[-(t/a)b] is not constant under acceleration Kevlar, and the Weibull standard deviation, a[G(1+2/b)-(G(1+1/b))2]1/2, increases linearly with acceleration. The string farm data showed that the standard deviation decreased at a rate greater than linear. Meeker and Hahn recommend testing at several levels of acceleration; then you can simultaneously estimate the acceleration model and the reliability at working stress [Viertl, Bagdonavicius and Nikulin].

Unfortunately, systems don’t have the same acceleration model as their parts. For a counterexample, suppose a series system of two independent parts with constant failure rates will be tested under acceleration. Suppose Arrhenius’ law accelerates the failure rates of both parts, aj = exp[-Cjexp[Eaj/(kT)]], where Cj is a constant and Eaj is the activation energy for part j, k is Boltzmann’s constant, and T is temperature in degrees Kelvin. The system failure rate function is

exp[-C1exp[Ea1/(kT)]-C2exp[Ea2/(kT)]].

This expression is not Arrhenius’ law unless C1 = C2 and Ea1 = Ea2.

Accelerating the Piecewise Linear Failure Rate Function

Stress can accelerate a piecewise linear failure rate in any combination of the following ways:

·         a, the constant failure rate, increases,

·         b, the slope, increases, and perhaps to, the age at the end of infant mortality, decreases, or

·         the constant segment becomes a linearly increasing function.

These effects can be attributed to:

·         The constant failure rate increase indicates that acceleration causes more random failures.

·         The slope b increase, decreasing to, or both indicate that acceleration exacerbates process defects.

·         Acceleration may cause premature wearout, causing the constant failure rate to change to an increasing failure rate function.

Each of these possible failure causes is actionable; then accelerated test data and the parameter estimates help determine whether improvements to process, random effects, and design are justified.

Figure 4 plots accelerated failure rate functions:

·         a(t) = 0.0002+0.0001(7-t) for t<7 and a(t) = .00002 for t ³ 7

·         a(t) = 0.0001+0.0002(4.95-t) for t<4.95 and a(t) = .00001 for t ³ 4.95

·         a(t) = 0.0001+0.00005t+0.0001(7-t) for t<7 and a(t) = 0.0001+0.00005t for t ³ 7

The first failure rate function has a constant failure rate of 0.0002 instead of 0.0001. The second has the knot point at 4.95 instead of 7; the age 4.95 is the square root of 49/2, to keep bto2/2 constant when b increases to 0.0002. The third has an increasing failure rate after age 7.

Figure 4. Effects of acceleration on the piecewise linear failure rate function

Acceleration can affect reliability demonstration tests in several ways. Define the reliability acceleration factor as

RAF(t) = (1-Rnormal(t)/(1-Racc(t)).

This factor quantifies reliability relative to that under normal conditions for some age t, such as at the end of a test. After 60 time units, the reliability acceleration factors are, for the parameter values listed above figure 3:

·         RAF(60) = 1.705 for increased constant failure rate, a

·         RAF(60) = 1.288 for increased infant mortality, b

·         RAF(60) = 11.350 for changing from constant to linearly increasing failure rate, a+ct

This last RAF value shows a substantial increase induced by changing from a constant to an increasing failure rate. This apparently unrealistic change in failure modes is justified, because you can still estimate the reliability under normal conditions from accelerated tests. For instance, you might simply accelerate the rate at which random events occur, linearly, not just from one constant rate to a higher constant rate. Step stress increases have been recommended [Bagdonavicius and Nikulin], but not linearly increasing stress. Food for thought.

Because many accelerated life tests are of systems with different parts, why not use a simple model of acceleration and test hypotheses about the acceleration model in the same way we test hypotheses about regression models? Model the parameters of the piecewise linear failure rate function as power functions of stress. The power law,

Accelerated failure rate = a(t|unaccelerated)stressp,

subsumes several acceleration models, including Arrhenuis’ [Shaked]. This power law requires only estimation of the power p in the formula.

Conclusions and Recommendations

Why assume complex reliability and acceleration models? You’ll just have to defend them, probably without sufficient sample size, test time, data. Statisticians start with simple models, because they are simpler to defend and can be estimated and tested with less failure data than elaborate models.

The simplest model for a failure rate function that captures infant mortality, useful life reliability, and MTBF is a piecewise linear failure rate function resembling the left-hand portion of the bathtub curve. This model allows MTBF estimation even if no failures occur at an age anywhere near MTBF. It also allows estimation of the percentage of failures in infant mortality and the age at which infant mortality ends.

Accelerated tests help demonstrate MTBF and reliability with limited sample size and time. Unfortunately, system acceleration models are not as simple as those of their parts, so traditionally used acceleration models don’t apply to systems. Any and perhaps all of the parameters of the piecewise linear model can be accelerated by use of a power function of stress. The greatest acceleration is achieved by accelerating the rate of random failures as testing goes on, the continuous version of step-stress testing.

Part 2 of this article will analyze alternative piecewise linear failure rate functions and recommend designs for power law accelerated tests, estimate reliability parameters including MTBF, provide confidence intervals, and test hypotheses about the parameters. It will show how to test at only one acceleration level and estimate MTBF and reliability at working stress, using predicted or hypothesized MTBF and accelerated test results.

If you have data, would like to see whether it fits the piecewise linear model, and  would like estimates of the acceleration, MTBF, and reliability, send your data. I will send back the analysis, free of charge.

References

Bagdonavicius, Vilijandas and Mikhail Nikulin, Accelerated Life Models, Modeling and Statistical Analysis, Chapman and Hall, New York, 2002

George, L. L., “Design of Ongoing Reliability Tests (DORT),” ASQ Reliability Review, Vol. 22, No. 4, pp 5-13, 28, Dec. 2002

Kalbfleisch, John D. and Ross L. Prentice, The Statistical Analysis of Failure Time Data, Second Edition, Wiley, New York, 2002

Meeker, William Q. and Gerald J. Hahn, How to Plan an Accelerated Life, Test: Some Practical Guidelines, Vol. 10, ASQ, 1985

Nelson, Wayne, Accelerated Testing, Wiley, New York, 1990

NIST, Engineering Statistics Handbook, Ch. 8.3.1.4, “Accelerated Life Tests,” http://www.itl.nist.gov/div898/handbook/apr/section3/apr314.htm

Shaked, Moshe, “Accelerated life testing for a class of linear hazard rate type distributions,” Technometrics, Vol. 20, No. 4, pp 457-466, November 1978

Viertl, Reinhard, Statistical Methods in Accelerated Life Testing, Vandenhoeck & Ruprecht, Göttingen, 1988

Acknowledgements

Professor Emeritus Richard E. Barlow gave the interview at the beginning of this article in the newsletter of the Operations Research Society of New Zealand, http://www.esc.auckland.ac.nz/Organisations/ORSNZ/Newsletters/dec99.pdf. I recommend reading the entire interview. Professor Barlow was the chairman of my thesis committee, Department of Industrial Engineering and Operations Research of the University of California, Berkeley.

Mark Felthauser, Activant (formerly CCI/Triad), reviewed the manuscript and suggested the Neyman design for stratified sampling.Harold Williams, Reliability Review Editor, suggested this article.

Design of Accelerated Reliability Tests, Part 2:

Design and Analysis of the Piecewise Linear Model

This part describes how to design and analyze accelerated reliability tests, assuming a piecewise linear failure rate function and power law acceleration. It shows how to obtain credible results, with limited sample size and test time, at one accelerated stress level. It provides estimators for parameters, reliability, MTBF, confidence intervals, and it shows how to test model assumptions and verify MTBF. It is inspired by the observation that failure rates are not constant, often because of infant mortality.

Figure 1 plots Intel IC FITs (Failures in Thousands (of millions) of hours). Intel used to post the data behind this plot on http://www.intel.com/services. I converted units from DPM (defects per million) in the first 50 hours to FITs, for comparison with FITs thereafter. Perhaps Intel reported DPM in the first 50 hours and FITs thereafter, so you wouldn’t notice the infant mortality.

Figure 1. Intel IC FITs

Process defects cause infant mortality, and estimating infant mortality can help decide whether to do anything about process defects. You’ll need to estimate infant mortality to decide whether its cost exceeds the cost of fixing the defects. Why not estimate infant mortality while verifying MTBF and reliability.

Statistical design of experiments (DoE) minimizes the variance of some function of estimated parameters, usually assuming that all samples fail. However, if you don’t test long enough, you get few failures and high variance. Accelerated tests without failures at ages after accelerated MTBF make MTBF estimation impossible, without strong assumptions. Jensen, Meeker and Hahn, Nelson, the National Institute of Standards and Technology (NIST), Viertl, and many others recommend testing at at least two acceleration levels, so that the acceleration model can be verified and reliability at working stress (normal operating conditions) can be extrapolated. Fortunately, two or more stress levels are unnecessary!

Management wants test results ASAP, so we have test at the greatest stress level we think won’t alter failure modes and corrupt extrapolation to working stress. Recently, a test of three power supplies at 105°C resulted in three failures at age 98 days due to the same electrolytic capacitor. This result, due to apparent electronic fatigue failure, prevents verifying any acceleration model and extrapolating reliability to working stress, unless some information or field data is available under working stress.

Usually MTBF or reliability demonstration tests are for new products, so there is probably no field data, even about infant mortality. However, an MTBF prediction¾often the MTBF to be verified in the demonstration test¾should be available. If not, make one by using http://www.fieldreliability.com (MIL-HDBK-217F, Telcordia (Bellcore), or Credible Reliability Prediction, chapter 3, [George, 2003]. Although an MTBF prediction isn’t as credible as an observed MTBF, it still provides enough information about reliability under working stress to estimate the parameters of an accelerated piecewise linear failure rate function, extrapolate MTBF to working stress, and verify whether the predicted MTBF and actual MTBF differ statistically significantly.

The piecewise linear failure rate function model (Part 1) provides means for modeling reliability and acceleration, testing the model, and estimating MTBF credibly, even with only one accelerated stress level. It also suggests ways to allocate samples to different acceleration levels, if you have that opportunity, and to make product and process improvement decisions, using test results.

The piecewise linear failure rate model that captures infant mortality for MTBF and reliability demonstration testing is

a(t) = a+b(to-t) for t < to and

a(t) = a for t ³ to,

where to is the age at which infant mortality ends. The power law acceleration model multiplies the unaccelerated failure rate function by stress factor, x, to power p, xp. A more general acceleration model motivated by step-stress testing and Miner’s rule rescales the failure rate function and the age variable, t, by the same multiplicative stress factor q(x) [Xiong and Ji]; i.e.,

aAcc(t) = aUnAcc(t/q(x))/q(x).

The stress factor is lnq(x) = a + bx incorporates Arrhenius and Eyring models. Additional terms and parameters may represent increasing failure rate, other stress factors, or other phenomena, for estimating parameters and testing hypotheses about them.

Put All Your Eggs in One Basket

According to NIST, “Recent work on designing accelerated life tests has shown it is possible, for a given choice of models and assumed values of the unknown parameters, to construct an optimal design (one which will have the best chance of providing good sample estimates of the model parameters). These optimal designs typically select stress levels as far apart as possible and heavily weight the allocation of sample units to the lower stress cells.” However, failures may not occur at lower stresses during test time.

The DoE objective is to minimize the variance of parameter estimators or functions of them, by the choices of independent, controllable variables such as sample sizes, test times, and stress levels. Under pressure to demonstrate reliability or MTBF ASAP, I test enough samples, under high acceleration, for a long enough time to obtain credible reliability and MTBF estimates and to verify the accelerated piecewise linear model.

Under reasonable conditions, maximum likelihood estimates (mles) of the piecewise linear model parameters are asymptotically normally distributed and unbiased, even from censored samples. Their asymptotic variances and covariances are given by the inverse of the “Fisher information” matrix, the expected values of the second derivatives (with respect to model parameters) of the log-likelihood function [Kalbfleisch and Prentice]. Reliability metrics such as infant mortality proportion, warranty failure probability, useful life reliability, and MTBF are functions of the piecewise linear model parameters, so confidence intervals on reliability metrics can be estimated, by simulation, if necessary.

|D|-optimal designs are popular, because they subsume all metrics that might be constructed from parameters of the experimental model. In other words, if you use a |D|-optimal design, you are reasonably assured of having an efficient design, whether you need to estimate infant mortality, warranty failure probability, useful life reliability, MTBF, or another reliability metric. |D|-optimal designs minimize the determinant of the Fisher information matrix. Unfortunately, the |D|-optimal design for piecewise linear model recommends equal observations at zero, the knot point to, and anywhere thereafter. It is pointless to test for zero time, because that will probably result in zero failures and little information about infant mortality, accelerated or not, so alternatives to statistical DoE must be used.

Neyman Design

The optimum design allocates strata samples proportional to the standard deviations within strata and inversely proportional to the square roots of the cost per sample within the strata [George 2002]. The optimum stratified sample sizes satisfy

(ni/n ) = ((Nisi)/ Ö(ci))/S((Nisi)/ Ö(ci)),

where ni is the stratum sample size out of a total sample of n, each stratum is assumed to have Ni members, ci = co +c1ti is the cost per unit tested, and the sum runs over all strata [Neyman]. Call these “Neyman proportions.” If you observe only the proportion failing, pi, then the standard deviation si is Ö[nipi(1–pi)] from the binomial distribution. That design provides the sizes of strata relative to the total sample but doesn’t help choose the total sample size, n, and total test time.

Minimally Credible Design

Suppose the objective is to test enough samples long enough to provide at least two failures, one within infant mortality and one thereafter, with probability of at least 50%. (Two failures are sufficient to provide estimates of the piecewise linear model parameters.) The probability of failure in infant mortality is

1–exp[–ato–bto2/2].

The probability of failure after infant mortality but before test termination at time t is

–exp[–at–bto2/2]+exp[–ato–bto2/2].

Given a sample of size n, the probability of one failure before  to and one after to but before t is

Bin(j,n,1–exp[–ato–bto2/2])Bin(k,n–j, –exp[–at–bto2/2]+exp[–ato–bto2/2]),

for j and k of at least 1, where Bin (k,n,p) represents the probability of exactly k out of n events with probability p. (The probability of at least one failure in either interval is what I need, but I wanted to keep the formula simple.) Guess values of a, b, and to, and pick the sample size n and test time t so that the probability of at least one observation before to and one after to but before t is at least 50%.

Table 1 shows a spreadsheet for approximating the probability of at least one failure before and after to as a function of assumed parameter values, sample size n, and test time t. Cases 1 to 3 show alternative designs, all with probability of approximately 50%, in the last row. The probability of at least one failure after to is in the next-to-the-last row. The probability of at least one failure before to is in the row above the next-to-the-last row. Case 1 shows what happens when the sample size is too small and the test time is too big; the probability of failure before to is less than the probability after. Case 2 shows the opposite. Case 3 shows design balanced by choice of n and t.

Table 1. The probability of at least one failure before and after to. (Rows aren’t containing probabilities of failure combinations before and after to aren’t shown.)

Minimum Variance Design

Suppose the objective is to test enough samples long enough to have adequately small variance of estimates of the piecewise linear model parameters, acceleration parameter, reliability, and MTBF. There are so many possible choices of parameters to estimate and such complicated formulas that there are only two ways to proceed:

1.      With one parameter to estimate, choose sample size and test time to minimize the asymptotic variance of the mle of the parameter.

2.      Choose some sample size and test time. Simulate values from the asymptotic normal distribution of parameter estimates and their asymptotic variance-covariance matrix. Plug simulated values into the formula for the reliability, MTBF, or percentiles in terms of the piecewise linear parameters. If the resulting variability of the formula values is small enough, accept the sample size and test time.

For an example of alternative 1, suppose you want the variance of the parameter b to be less than 25% of its estimated value. As you might expect, the variance of b does not depend on test time t, because t has no effect on b as long as t > to. If the only failure before to is at age x1, the variance of b is

(–1/a2–1/(a+bto–bx1)2)/(to2/(a2(a+bto–bx1)2)–(2tox1)/(a2(a+bto–bx1)2)+x12/(a2(a+bto–bx1)2))/Sqrt(n).

As before, you’ll have to guess values of a, b, and to to obtain a value of n so that var(b)/b = .25. When x1 = to, the sample size solution is

 n = [(–2/a2)/(to2/a4)–(2to2/a4)+ to2/a4]/[b2/16].

For accelerated life tests, the same kind of designs as above can be created, based on reasonable guesses. They are complicated but computable.

Analysis of Data

Maximum likelihood and least squares are statistical methods for estimating parameters and testing hypotheses about them. I prefer maximum likelihood estimators (mle) for reliability analysis with censored data, because the properties of mles are well known so it is easy to design the tests and construct confidence intervals likelihood ratio tests about the parameters and models.

Mles maximize the likelihood or equivalently the log likelihood of the observations. If you’re lucky, you can find explicit mle estimator formulas by solving for the parameter values that satisfy the system of simultaneous equations obtained by setting the first derivatives of the log likelihood with respect to the parameters equal to zero. The piecewise linear failure rate function yields a likelihood function that does not have continuous derivatives, so the maximum likelihood method sometimes requires numerical methods. 

Table 2 shows the data used in the analyses. The ages at failures have been deliberately chosen to fit the piecewise linear model and not lie entirely within infant mortality, for testing the infant mortality hypothesis.

Table 2. Data with infant mortality followed by an approximately constant failure rate. (Not all rows are shown. There were 20 samples, and 15 survived 45 time units.) 

Table 3 shows the mles of the a, b, c, and to parameters of the piecewise linear failure rate models:

a(t) = a+b(to-t) for t < to and

a(t) = a for t ³ to

and the model with linearly increasing failure rate,

a(t) = b(to-t)+ct for t < to and

a(t) = ct for t ³ to.

The first row of table 3 lists the infant mortality portion (if any) of the model. The second through fifth rows contain parameter estimates corresponding to the models. The MTBF estimates were approximated from the parameter estimates. Entries in the LR (likelihood ratio) test statistic row were computed from the model in its own column and the model in the column to its left. The significance level values were chosen arbitrarily, and the chi-square values result from the significance level values and the numbers of parameters that were estimated, three for the a+b(to-t) and b(to-t)+ct models and four for the a+b(to-t)+ct model.  The likelihood ratio test statistic and the chi-square value are for the asymptotic distribution of the likelihood ratio test statistic under the null hypothesis that there is no infant mortality.

I used ExcelÔ Solver to maximize the log likelihood as a function of the parameters a, b, c, and to. Under the null hypothesis of constant failure rate, b = c = 0, the mle of a is the total time on test divided by the number of failures. Under the null hypothesis of linearly increasing failure rate, c > 0, and no infant mortality, b = 0, the mle of c is the total time on test divided by twice the number of failures.

The mle of parameter a for the a+b(to-t)+ct  model is 0. That means that model coincides with the b(to-t)+ct  model. (The MTBFs for the two models differ slightly, because different approximations were used. The LR test statistics differ, because they come from different degrees of freedom, the numbers of parameters estimated.)

The piecewise linear models are statistically significant at around the 25% level, when compared with the monotonically linear or constant models: a, ct, and a+ct. The mles of to agree that infant mortality ends around age 3.3.

The winning model is the b(to-t)+ct model, because it has the maximum likelihood (closest negative value to zero). It is statistically significantly different from the ct model at the 10% significance level. (The LR test statistic is 14.389, compared to the 10% chi-square value of 6.251.)

Table 3. Maximum likelihood estimates for a piecewise linear failure rate function

If there are too few failures, explicit formulas for parameter estimators for the piecewise linear model may not exist. When there is only one failure, before age to, there is no solution to the maximum likelihood equations. When there is only one failure, after age to, then the maximum likelihood estimator of the constant failure rate a is 1/[age at failure], the estimator one would expect. When there are two failures, one before age to and one after, the mles of parameters a and b can be obtained symbolically. The asymptotic variance-covariance matrix also exists. When there are two failures, both after age to, the mles of coefficients a and b can be found, and the mle of coefficient a is the usual one, 2/[total time on test]. However, the Fisher information matrix is not full rank, so its inverse does not exist. If there are more than two failures, with at least one prior to age to, the estimators can be found symbolically or numerically and the Fisher information matrix is full rank so you can estimate the variances of the mles.

Accelerated

Suppose the acceleration of the failure rate follows a power law. This requires estimation of one more parameter, the power p. It is convenient to model the acceleration by using a power model xp, where known parameter x characterizes the stress as follows. Let stress level j be represented by

xj = (Sj–So)/(Smax–So),

where So is the working stress, Sj is the stress at level j, and Smax is the stress at the highest level being tested. This power model includes Arrhenius’ law when Sj = 1/Tj [Shaked].

If there is only one level of acceleration, additional information is necessary to estimate the power parameter so that reliability and MTBF can be estimated at working stress. When a reliability demonstration test takes place, little is known about product reliability, except an MTBF prediction, so maximize the likelihood function subject to the constraint that the MTBF under working stress equals the predicted MTBF, a function of the parameters a and b. That constraint gives enough information to provide reasonable estimators of the power parameter too. Formulas for the mles of parameters a, b, and p can be derived using MathematicaÔ.

Table 4 shows data for an accelerated test, and table 5 shows the mles of the parameters, including the power p for the piecewise linear, accelerated linear model,

a(t) = xp(a+b(to-t)+ct) for t < to and

a(t) = xp(a+ct) for t ³ to.

The parameter estimates are functions of an MTBF prediction equal to 125, the theoretical MTBF from the a+ct model using the unaccelerated parameter estimates in the a+ct column of table 3. The acceleration model xp(a+ct) is preferred to the fancier model in table 5, because its log likelihood is greater, closer to zero, and the likelihood ratio is not significantly different from zero. 

Table 4. Data from accelerated test. (Not all rows are shown. There were 20 samples, and 9 survived 45 time units. )

Table 5. Analyses of accelerated test data for x = 1.5

If there are fewer than two failures, explicit formulas for the maximum likelihood estimators of the parameters a, b, and p do not exist. If there’s only one observed failure, assume that another failure would have occurred at the age at which testing is terminated, the censoring time. That will provide conservative estimates.

Example

A customer asked us to demonstrate that the product’s MTBF was at least 39,500 hours, half of the MTBF prediction. Each product contains 16 transceivers. We planned an accelerated MTBF demonstration test so that, if two failures occurred, the test would demonstrate that MTBF/2 was greater than 39,500 hours with 75% confidence. We used the chi-square confidence limit for a constant failure rate. That is the assumption that the test statistic, [number of failures]/[total time on test], has the chi-square distribution with four degrees of freedom, two times the number of failures.

We planned the MTBF demonstration using the assumption of constant failure rate. We decided to test seven products for six weeks (1008 hours) at 60º C, about as hot as the product could stand. (The working temperature is 25º C. That test would yield 7056 hours and a chi-square lower confidence limit (LCL) of ~39,000 hours, with a system MTBF acceleration factor of 14.6 computed by the parts-count MTBF prediction method and Arrhenius’ law for parts.

Two different products had single transceiver failures, at ages 486 and 660 hours, well after infant mortality. What LCL on transceiver and product MTBF can be estimated, given transceivers are wearing out and have increasing failure rate?

Assume that transciever failure rate function is linearly increasing with age, a+ct. Table 6 shows the parameter estimates for the transceiver failure rate and MTBF. The estimated a-value was negative, an inadmissible value, so the failure rate model became simply ct. The likelihood ratio to test the hypothesis of constant vs. linearly increasing failure rate was huge, indicating that the failure rate really was increasing.

People agree that transceiver MTBF follows Arrhenius’ law with an activation energy of between 0.8 and 1.0 electron volts. I used 0.87 eV activation energy, so the acceleration factor from 25º to 60º Celsius is 35. The unaccelerated MTBFs are accelerated MTBFs multiplied by the acceleration factor.

Table 6 also shows the MTBF of the 16-transceiver product and its 25th percentile, excluding the other product parts. The 25th percentiles were simulated by use of the failure rate function ct and the inverse Ö[-2ln(Rand)/c] of the corresponding reliability function exp[-ct2/2]. (The reason for the ~1000 hour percentile is that the simulation was only 100 units with 16 transceivers. I repeated the simulation many times, and the eyeball average 25th percentile was ~1000.)

The 16-transceiver product has demonstrated a 25th percentile MTBF of ~35,000 hours, and, at the time of this writing, no other product failures had occurred, so the client’s product MTBF has been almost demonstrated, without assuming a constant failure rate and without the magic of chi-square confidence limits on product MTBF with no failures.

The result that this product’s failure rate was linearly increasing is one reason I developed the design and analysis methods for piecewise linear failure rate functions.  

Table 6. Example transceiver failure rate parameter estimates

Conclusions and Free Offer

If you’re forced to defend assumptions, make the simplest assumptions that capture important features, such as infant mortality and constant failure rate, and test those assumptions. Take advantage of the piecewise linear failure rate models resembling the left-hand portion of the bathtub curve, because of their simplicity and usefulness for short tests of small samples and to quantify infant mortality. Statisticians start with linear models, because they’re simple and can be tested. The piecewise linear model allows credible MTBF and infant mortality estimation, with a simple, testable failure rate assumption.

If management wants to demonstrate MTBF with few samples and little time, test at an accelerated level of stress and extrapolate to working stress. Consider accelerating the rate of random failures continuously, to achieve the greatest reliability acceleration. Don’t use Weibull reliability models when there is infant mortality; their MTBFs are ridiculous.

If you can, test at more than one stress level to estimate the acceleration model and its effects on reliability at working stress. If not, use the predicted or specified MTBF to anchor to working stress an accelerated test at one high stress level. Multiplying accelerated MTBF by an acceleration factor is valid only if the standard deviation of life is linearly changed by acceleration.

Design of tests, analysis of test data, and estimation of piecewise linear model parameters, reliability, and MTBF are available, from the author. I'll try to upload them an publish links here.

References (Please refer to the references in part 1 for references not listed here.)

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

Neyman, J., “On the Two Different Aspects of the Representative Method: The Method of Stratified Sampling and the Method of Purposive Selection,” J. of the Roy. Statist. Soc., Vol. 97, pp 558-606, 1934

Xiong, Chengjie, and Ming Ji, “Analysis of Grouped and Censored Data from Step-Stress Life Test,” IEEE Trans. on Rel., Vol. 53, No. 1, pp. 22-28, March 2004