Hi Robert, and thanks for reading!
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From my Black Belt training...
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A destructive gauge is one that possesses an inability to repeatedly assess the true value.
-Class I: The specimen is destroyed, but we are capable of sampling from homogeneous lots or subgroups
-Class II: The specimen is destroyed, and only heterogeneous lots or batches are available
-Class III: Are capable of repeated measuring, but the true value of the specimen is changing
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Each can be handled for MSA.
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Some apparently NDT are actually destructive. My first MSA was using a micrometer to measure 1-2" thick aluminum plate. Gotta be non-destructive, right? Wrong - the right angle on the carbide faces on the mic were shaving off little curls of aluminum, which when I remeasured the area got smashed, but the measurement was detectably thicker. So a mic on a huge slab of metal was a destructive measure!
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I don't know if a classification system for truly ND tests in regards to MSA. In such cases you have a (theoretically) infinite ability to remeasure the same thing, so you should be able to quantify measurement error completely.
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"Calibration" is not what we are talking here, and for gauges used for process variables, calibration is frequently irrelevant as long as you have gauge control and acceptability (see my article here: http://www.qualitydigest.com/inside/six-sigma-column/don-t-judge-gauge-… about that - in the section on Long-Term Studies). But, having said that, what you are asking about will be covered in the article on Long-Term Studies, which ought to be the one after next.

Thanks for reading, Fred!

What you have is a Class II destructive test. As it is now, you are confounding product and measurement variation. Worst case you can quantify that total variation and use a control chart to alert you when "something" goes wrong, but you won't initially be able to tell if it is process or measurement and you will need to investigate. But there are some tricks we might be able to do, so here are a couple of ideas. Keep in mind it is the measurement system we want to test, and to do so we want to test the exact same thing over and over if we can. But we do have options if we can't. I am going to have to guess at what makes sense for your process, but hopefully either what I am going to say will make sense, or you can make sense of what I say for you process! :-) Ask me if you have questions.
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Let's assume you are going to do a potential study.
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First, if you can, get a big sample of 10 runs of coated paper (to whatever different nominal moisture losses you make) and put each in a moisture-neutral environment. Not sure how this would look since I don't know what your process or product looks like, but let's say it is a high-humidity box for sake of discussion, maybe each sample has a different humidity setting on each box.
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Next, dice up each sample into test sheets. Mix them all together within sample, but keep the different samples separate in their little boxes.
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Maintain these in the controlled atmosphere. You now have 10 homogeneous samples from which to draw to perform your MSA - each operator/gauge draws a sample at random from each box (in different random orders!) and measures it. We know we are confounding product with measurement, but by randomizing the test sheets within sample, we hope to have homogenized across that, so anything that stands out on top of that inherent variation (like operator-to-operator or gauge to gauge) is probably gauge related. Basically, we are trying to take it from Class II to closer to a Class I destructive, and we can do the standard MSA on Class I.
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Now this is so much trouble that, unless you have doubts about the system you have been using for years (and you might!), it probably makes more sense to do this with 8 bigger samples that you use as during the course of your long-term MSA---which you will get in the article following the next one!
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The other option is instead to test the non-destructive components of the system. We start running into concerns about external validity, though, so be cautious. So by that I mean, maybe a mass balance and a dehumidifier are parts of the process. Well, we can test to see that the balance is in control over time by performing an MSA on that, same with the dehumidifier. If both of those exhibit control and acceptability through time, and we see a problem on a measurement, it is a good first assumption that the problem is in the product.
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Hmm, maybe I should do an article on destructive gauges?

Steven,  I am attempting to measure several properties from a continuous web based coating process where most all measurements are destructive: coating weight, peel force, release force, and shear force.  We also know that our process can inherently produce material whose characteristics are dependent on location vs. independent and random.  I would be very interested in learning about how to perform appropriate measurement system analyses with destructive testing.  Can you offer suggestions on how I can learn about MSA with destructive testing?