Process Performance and Statistical Process Control for Lower Detection Limits

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    William A. Levinson
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    May 20, 2021
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    13:00 PM EST
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    60 Min
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Some quality characteristics, such as trace contaminants and trace impurities, cannot be detected at less than the instrument's lower detection limit (LDL). Zero does not mean zero in these situations; it means only that the amount is below the LDL. Such measurements are also called "nondetects."


This means the process performance index (Ppk) cannot be calculated with traditional textbook methods, and traditional statistical process control (SPC) charts cannot be used either. If however the underlying statistical distribution is known (and it is unlikely to be the normal or bell curve distribution), its parameters can be determined with the maximum likelihood estimation (MLE) method that can account for nondetects. Once the parameters are available, the nonconforming fraction above the upper specification limit can be calculated, which yields Ppk = PPU (there is probably no PPL as there is no lower specification limit). The median of the distribution is then the SPC chart's center line, and its upper 99.865% percentile is its upper control limit. StatGraphics uses the MLE method, which can also be used with Excel's Solver feature.



What Will You Learn?


1.   When the critical to quality (CTQ) characteristic is something undesirable like a trace contaminant, pollutant, or trace impurity, the instrument or gage may be unable to detect less than a specified amount (the lower detection limit, or LDL).

2.  Reliability statistical methods can however assess censored data sets. "Censoring" means certain measurements are not available because not all units fail by the end of the test; this is known as right-censoring. This application involves left-censored data, which means measurements below the LDL (nondetects) are not available; we know only that they are somewhere between 0 and the LDL. Maximum likelihood estimation (MLE) is the commonly used method to estimate the distribution's parameters.

·     The webinar will provide two examples, one of the traditional normal distribution and one of the gamma distribution, as handled by StatGraphics. The appendix of the handout uses Excel's Solver feature to get the same results.

·   The deliverables are the normal distribution's mean and standard deviation, or the gamma distribution's shape and scale parameters. The method can also be used on other distributions such as lognormal (the natural log of the quality characteristic follows the normal distribution), Weibull, and exponential.

·     Tests for goodness of fit to the selected distribution should always be performed, and StatGraphics includes quantitative tests (chi square test for goodness of fit) and subjective ones (quantile-quantile plot).

3.     The EPA's free ProUCL software offers regression on order statistics (ROS) as an alternative way to estimate the distribution's parameters.

4.     The process performance index Ppk = PPU measures the process' ability to meet the upper specification limit.

·      As the characteristic is usually something undesirable, there is unlikely to be a lower specification limit for it.

·      PPU can be determined for any distribution by (1) using the distribution's parameters to calculate the nonconforming fraction and (2) converting this into the corresponding PPU for a normal distribution with the same nonconforming fraction.

5.     The SPC chart is unlikely to have a lower control limit (which can easily be below the lower detection limit).

·        The center line is the median as determined from the distribution parameters.

·        The upper control limit is the 99.865 percentile of the distribution, noting that the upper 3 sigma control limit for a traditional Shewhart chart also is the 99.865 percentile of the normal distribution.


Benefits for Attending:


Attendees will learn how to estimate the parameters of statistical distributions for quality characteristics that are not measurable below an instrument's lower detection limit (LDL), and use this information to (1) calculate the process performance index (Ppk) and (2) set up a control chart to track the characteristic in question.


Attendees will receive a pdf copy of the slides and accompanying notes, and the data set used in the presentation.



Who Should Attend?

·       C-level Executive

·       Quality Professionals

·       Engineers

·       All people with responsibility for application of statistical methods

William A. Levinson, P.E., FASQ, CQE is the principal of Levinson Productivity Systems, P.C, which specializes in lean manufacturing, quality management, and industrial statistics. He is also the author of several books on quality, productivity, and management, of which the most recent is The Expanded and Annotated My Life and Work: Henry Ford's Universal Code for World-Class Success.


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