EDITED BY JOHN HEALY
As with other point estimates, we often want to calculate the confidence interval about the estimate. The intent is to determine the range of reasonable values for the true and unknown population parameter. For MTBF, this no different.
Keep in mind that when calculating MTBF, we are using the time to failure data that is often censored in some manner. For example, if we have 100 systems operating and have experienced only 12 failures, the bulk of the systems have never failed.
The discussion here is for either an exponential distribution based point estimate of the $- \theta-$ parameter or for a homogeneous Poisson process (HPP). If the system is best described by a non-homogeneous Poisson process (NHPP), then the confidence intervals described below are not appropriate as the intervals well depend on the specific NHPP model.
When conducting a test to estimate MTBF, we may run the systems in the test for a specific amount of time, or until we experience some number of failures. When the data ends at a point in time that does not correspond to a time of failure, the data is said to be time censored. If the ending time corresponds with a failure, then we have failure censoring.
I bring up the nature of the censoring as it changes the formula for the confidence interval for the MTBF estimate.
χ2 Distribution
MTBF is commonly associated with the exponential distribution, so when either assuming or deliberately using the exponential distribution and the statistic MTBF the confidence intervals are described in part by the χ2 distribution.
Keep in mind that the chi-squared distribution is not symmetrical, like the normal or t distributions, thus we need to find the appropriate lower and/or upper χ2 value to complete the calculation.
It also means the confidence intervals are likely not symmetrical either.
Lower Confidence Limit for Type I Censoring
Type I censoring is time terminated. For example, when the data collection period ends, say at 2,000 hours, there was not a failure at 2,000 hours. Lower confidence is often of interest as it indicates the lower range of the MTBF value, or how bad might the true result actually be.
The formula for the Type I lower confidence interval is
$$ \large\displaystyle \theta \ge \frac{2T}{\chi _{\left( \alpha ,2r+2 \right)}^{2}}$$
Where,
- θ is the calculated mean life (MTBF)
- T is the total time the samples operated before failing (or the test was ended)
- χ2 is the Chi-squared distribution
- α is the level of risk (1 – confidence)
- r is the number of failures, 2r+2 is then the degrees of freedom for the χ2 distribution table
Lower Confidence Limit for Type II Censoring
In this case, the censoring is failure based, meaning the test ended with a failure. The lower value is
$$ \large\displaystyle \theta \ge \frac{2T}{\chi _{\left( \alpha ,2r \right)}^{2}}$$
Note the small change in the calculation of the degrees of freedom.
Two Sided Confidence Interval Formulas
Again the censoring matters. For time censored data or Type I the equations are
$$ \large\displaystyle \frac{2T}{\chi _{\left( \frac{\alpha }{2},2r+2 \right)}^{2}}\le \theta \le \frac{2T}{\chi _{\left( 1-\frac{\alpha }{2},2r \right)}^{2}}$$
The 2r+2 degrees of freedom is only for the lower bound.
For the Type II or failure censored data the equations are
$$ \large\displaystyle \frac{2T}{\chi _{\left( \frac{\alpha }{2},2r \right)}^{2}}\le \theta \le \frac{2T}{\chi _{\left( 1-\frac{\alpha }{2},2r \right)}^{2}}$$
Again the lower bound does not have the extra two degrees of freedom. The key when calculating these confidence intervals is to know if the data is time or failure censored, then use the correct formula for degrees of freedom.
You may also be interested in how setting the confidence level alters the result. You may see data sheets reporting MTBF with a 60% confidence, well what does the mean? See the article Lower Confidence for more information.
Related:
Reliability with confidence (article)
Laplace’s Trend Test (article)
Confidence Limits (article)
Hi sir, thanks for sharing.
When components in a system have constant Failure Rates, Mean Time Between Failure (MTBF) of the system can be used to (represent) calculate Reliability at any time within the Useful Life period
The mean life function (often denoted as “MTBF”) is not a good measurement when used as the sole reliability metric. Instead, the use of a reliability value with an associated time, along with an associated confidence level, is a more versatile and powerful metric for describing a product’s reliability
For people who are unable to establish a Failure/Time distribution to calculate reliability of their product, the easiest way to track Reliability is to use MTTF(MTBF) periodically. “Single Point” calculations are not suitable for warranty, spares allocation, etc. Should calculate the MTTF(MTBF) number at a Confidence Level.
Confidence limits for the mean are an interval estimate for the mean. Interval estimates are often desirable because the estimate of the mean varies from sample to sample. Instead of a single estimate for the mean, a confidence interval generates a lower and upper limit for the mean. The interval estimate gives an indication of how much uncertainty there is in our estimate of the true mean. The narrower the interval, the more precise is our estimate.
Incredible MTBF Prediction Confidence Intervals
Use Credible Reliability Predictions
Has anyone ever asked you for a confidence interval on an MTBF prediction? You won’t find anything in MIL-HDBK-217 or Telcordia SR332. How confident can one be about a prediction? It’s not a legitimate statistical question, that uses a random sample of life data to estimate the mean (MTBF) and quantify the sample uncertainty around the mean based on the sample distribution of the mean.
“Credible Reliability Prediction” (former ASQ RD monograph) shows how to make credible reliability predictions based on observed field reliability of older, related products, by adjusting failure rate functions assuming proportional hazards scaled by ratio(s) of MTBF(old)/MTBF(new). This is based on the observations that the failure rates of generations of products resemble each other because some reliability don’t change: e.g., processes, shipment, installation, environments, training, and customers.
What if you don’t have life data: times to failures perhaps censored? Oscarsson and Halberg used least squares to estimate nonparametric reliability functions for Ericsson electronics using ships and returns counts. Harris and Rattner used least squares to estimate nonparametric survival functions for Virginia HIV+ and AIDS case counts. I use maximum likelihood and least squares to do both, currently of corona-virus using case, death, and recovery counts. Life data is sufficient but not necessary, and it is required by GAAP.
Last week I programmed bootstrap confidence bands on corona-virus survival function estimates, for the sake of forecasting using transient Markov SIR model. It occurred to me that the bootstrap could also be used to quantify uncertainty in MTBF predictions; I call them Incredible MTBF Prediction Confidence Intervals.
The incredible MTBF prediction is the integral of R(t)dt, where the integration extends from t = 0 to t-value for which R(t) = 0.0. (This can be proved by integration by parts. I’ve done it.) The incredible MTBF prediction is the sum of (R(t) for t=0,1,2,… where R(t) is the credible reliability prediction according to the monograph.
Lest you think statistical confidence intervals on MTBF predictions are impossible, because there is no sample life data, I admit some decree of skepticism is warranted. Typically, field reliability functions don’t extend from 1.0 to 0.0; they should extend past warranty, perhaps to end of useful or support life, hopefully to old-age reliability somewhat greater than 0.0. So I extrapolate them, using the FORECAST.ETS() excel function and exponential smoothing extrapolation.
When you want an Incredible MTBF Prediction Confidence Interval for a new product, let me know. I will need field reliability data on older, related product(s) and MTBF predictions for the old and new product(s). I can provide MTBF prediction workbooks or read Credible Reliability Prediction, https://sites.google.com/site/fieldreliability/credible-reliability-prediction.
It wasn’t obvious to me that the formula for LCL should be CHISQ.INV(alpha,2r+2) in denominator instead of CHISQ.DIST(t, alpha,2r+2)! If you want a demo spreadsheet, let me know.
Technically, the chi-square confidence limits on MTBF are approximations to gamma confidence limits. This is because, for TTT = total time on test, Poisson probability P{N(TTT) TTT] where T1, T2,…,Tn are iid exponentially distributed failure times.
If you want a spreadsheet to compare GAMMA.INV() and CHISQ.INV alternatives, let me know. The difference is surprising. For large numbers of failures, confidence intervals may not even overlap!
What is the source for the 2T/X2 equation. I find many resources describing the X2 distribution and confidence intervals, but I’ve never seen it get boiled down to this form. I see it in other pages (TI, Micronote, etc) but again there is no stated reference as to where it comes from. Any pointers to a reference text would be appreciated. Thanks.
Hi Mitch,
thanks for the question – John wrote this article so I am not sure which reference he used. The equations are fairly common in older reliability texts. I found the equations and discussion in my copy of Handbook of Reliability Engineering and Management 2nd edition on page 25.28. I’m sure you can find other references by searching for Failure or Time censored life testing.
cheers,
Fred
thanks
Hello Fred,
Thanks for sharing. Could you advise me how the formula for the Type I lower confidence interval is derived?
Thanks
John Man
Hi John, From what I could tell, Benjamin Epstein in a paper in 1960 first suggested the lower bound formula. The paper describes what appears to be the derivation, yet I didn’t look at it very closely.
Epstein, B. (1960). Estimation from Life Test Data. Technometrics, 2(4), 447–454. https://doi.org/10.1080/00401706.1960.10489911
It might help with where the lower bound formula came from.
Apparently there are multiple ways to estimate the intervals, not are all that great. Best to just avoid using MTBF and use a meaningful metric, such as R(t) reliability instead.
cheers,
Fred