Some industrial and laboratory activities involve measurements for which there is a lower detection limit (LDL) or "nondetects," which means the instrument cannot measure a usually undesirable characteristic, such as impurities or pollution levels, below the limit in question. The good news is however that, if the statistical distribution is known, its parameters can be fitted with the same maximum likelihood estimate (MLE) methods that are used in reliability statistics. Once the parameters are known, it is possible to estimate the process performance index and also define control limits for statistical process control purposes.
Why Should You Attend
Manufacturers often must calculate process performance indices, and use statistical process control (SPC) for processes. Some quality characteristics such as impurities, pollutants, particulates, and trace contaminants have upper specifications only because fewer are better, and zero is ideal. Complications for using statistical methods arise, however, when the instrument used to measure these characteristics has a lower detection limit, which means essentially that zero isn't zero. Zero can mean anything, for example, between zero and 0.2 parts per million inclusive.
The good news is however that reliability statistics include methods for dealing with censoring, which means some specimens' characteristics are outside the measurement capability of the experiment. While the usual context involves specimens that survive beyond the testing period, i.e. the data set is upper censored, the same methods apply to the bottom censored as well.
Objectives of the Presentation
» Know that some quality characteristics have lower detection limits, which means a measurement of "zero" can mean anything from 0 to the LDL inclusive.
» Maximum likelihood estimation methods can however be used to optimize the distribution parameters if the underlying distribution (exponential, Weibull, or gamma) is known. StatGraphics Centurion is able to handle these distributions.
» Once the parameters have been estimated, the process performance index (Ppk = PPU, there is unlikely to be a PPL because there is no lower specification limit) can be calculated along with a control limit and centerline for an SPC chart.

A key takeaway is that we can indeed estimate process performance indices, and set up practical control charts, for non-normal distributions regardless of whether data are censored; it is in fact a lot easier to do it for non-censored data.
Areas Covered in the Session
» The issue of a lower detection limit (LDL) or "nondetects" means that a measurement of 0 really means anything in the range [0, LDL].
» The usual context is an undesirable characteristic such as impurities, particles, or contaminants. This means there will be only an upper specification limit, and also that the underlying distribution is unlikely to be normal (bell curve).
» Reliability statistics addresses the issue of censored data with maximum likelihood estimates (MLEs) for the distribution parameter. The usual context is upper censoring in which some items on the test survive past the test period which means their specific failure times are unknown. The same methods can however be used on lower censoring, and StatGraphics can handle this for several distributions.
1. The webinar handout includes an appendix that shows how to deploy MLE with Excel's Solver feature; it worked for the two examples in the webinar.
2. The webinar also introduces the EPA's free ProUCL software which can perform regression on order statistics (ROS) to estimate the process parameters.
» Once the distribution parameters are known, we can estimate the nonconforming fraction (above the upper specification) and therefore the performance index Ppk. We can also determine the upper 0.99865 quantile (which corresponds to the 3 sigma control limit for a normal distribution) for a Shewhart control chart, and also set the centerline at the median (50th percentile).

Two examples will be presented, one with normally-distributed data and one with data that follow a gamma distribution.
Attendees will receive a copy of the slides and accompanying notes and simulated data with which to try the examples.
Who Will Benefit
» Quality managers
» Engineers
» Technicians
» Others with responsibility for the application of statistical methods
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