Treatment of zero fail read points

[Tom Anderson]

Stress testing with interval read points, a typical burn in experiment. Sample size determined from chi-square - set to prove at 90% confidence that a particular upper limit of failures (parts per million) {note assuming 0 fails} indicating after some maximum time the goal is demonstrated.

The problem becomes one of sparse data and limited accelerated read points - the electronics industry is capable of making very reliable devices....and little infant mortaility.

Get the Weibull parameters from the results. I think it is viable to include read points that do not exhibit any failures - with a 90% confidence upper bound, use Kaplan-Meier product treatment on all intervals to estimate the maximum possible result. These points can then be fit by regression to obtain a slope parameter. Following this, the cumulative failures found through a particular read point can be used at 90% confidence (or any other choice) to estimate the scale parameter.

Now, the flaws in this are several - but you need to start somewhere. First, unlikely to really have an exponential failure distribution - at least for burn in and infant mortality regions. So the sampling and statistics are a little shaky. Second is the treatment of non-failing read points - but I am not too unhappy about adding points describing detectability - I do not represent these as the mean/median of the data because I now use an upper estimate based on known cumulative failures. - unless I have none at all.

My question is to the validity of this thought process and overall methodology - most Reliability texts and papers deal with non-sparse information. I am reasonably confident a Weibull model is appropriate.

[Pantelis]

Tom,

I am not sure I am following what you are presenting. Specifically in the third paragraph – can you elaborate…

[Tom Anderson]

Sorry -

Say I have set up a burn in evaluation for a product that has an initial sample size of ~2606 units. In theory (assuming both chi-square statistics and an exponential distribution) this allows a detection of no better than 500 PPM at 90% Confidence.

I then set intervals from zero to 168 hours - say I even choose to divide them into equal time intervals - based on their log(times) - maybe 5 equally spaced (in log-time) read points.

At each read point I test all units, at some of these read points I do not find any failures. Upon completing the study, I find only 4 failures and two of the read points did not have any. (I may also not have any failures for a different product)

I now wish to find a reasonable estimate and I also want to assume a Weibull distribution. At each read point I would like to assume the detection limit at some confidence and again use chi-square statistics to calculation that read point's reliability - even those without failures as I am sampling.

Now I have ln(time) values for each read point. I then take the R(estimate) for each read point and determine the cumulative R through the entire set of read points - more or less Kaplan-Meier treatment of the read point R's.

I can then take ln(-ln(R)) of each of the cumulative R values for each read point and do a regression. and determine the Weibull parameters from the regression coefficients. (Keep in mind that I am, at this point, most interested in an estimate of a slope.) The scale parameter may be set later by having a much larger sample tested at a fixed read point from burn in and the resulting reliability estimate through that time of burn in.

What are the flaws in this approach? - I propose taking this approach due to many cases of either zero failures or a very low number of failures and the economics of running a much larger sample to have a higher probability of sufficient failures for a more rigorous treatment.

I hope that this explains the proposition a little more clearly.

[Pantelis]

Tom based on my understanding of what you are attempting to do, I am going to provide you with an analysis method. I think the way you are approaching it is questionable – I am not sure how you are using chi-square to compute the reliability in paragraph 4. Now in paragraph 1 your use of the chi-square assumes an exponential distribution – with a constant failure rate --. When you jump over to a Weibull distribution the failure rate is not constant (except for the trivial case of beta=1). The shape parameter (beta) of the Weibull distribution dictates the behavior of the failure rate function, thus for you to be able to predict how the failure rate changes with time you need to observe some failures.

To do that, consider the following scenarios:

You have n units under test. The units are inspected at some predefined intervals. During each interval inspection you may observe none, one or more failures.

Case 1: Some failures are found during specific intervals. The failures are then treated as interval censored data and the non-failed are treated as right censored (if unclear on these classifications see discussion at http://www.weibull.com/LifeDataWeb/data_classification.htm). Based on this MLE methods can be used to estimate the parameters of the Weibull - or any other - distribution (see http://www.weibull.com/LifeDataWeb/likelihood_function.htm -- note that Kaplan-Meier methods may also be utilized – but I would recommend an MLE solution.)

Now to implement this method – at the absolute minimum - you will need to see at least one unit failure out of the sample size (and this may or may not converge to a solution depending on the intervals utilized). Realistically, and if you want any confidence in the results more failures would need to be observed. The number of failures observed will have a direct impact on the confidence intervals of the parameters and subsequent reliability estimates. Furthermore interval selection will also impact this.

Case 2: No failures are observed. With no failures observed – trying to compute a beta from the data is impossible – no matter how many tricks one uses. To put it simply without observing any failures one cannot make any statements as to how the failure rate would change over time. In this case you two options (a) assume a constant failure – i.e. use an exponential distribution (b) assume a value of beta using prior knowledge and use the one parameter weibull distribution (see http://www.weibull.com/LifeDataWeb/weibull_probability_density_function.htm)

Please let me know if this helped.

[Artimus the Just]

The answer to the "little data" problem is to use Bayesian methods (See Jaynes, Probability Theory, 2003).