Robust Rule of Thumb: Quasi-Nonparametric UTL

Davis & Wambach (2015). “Quasi Nonparametric” Upper Tolerance Limits for Occupational Exposure Evaluations. Journal of Occupational and Environmental Hygiene, 12: 342-349.

Rules of thumb can simplify complex statistical analysis of exposure monitoring data with a 1-to-1 comparison. They also allow for fewer samples to be collected compared to conventional approaches. The EN689 preliminary test or the AIHA 6-10 samples below 10% OEL are prime examples.

Unsurprisingly these examples are typically provided with no mathematical explaination given that their intent is to remove complexity. The user is left to assume the rule of thumb is reasonable reflection of the likelihood to comply with the corresponding “full” statistical test.

A skeptical hygienist may reasonably question:

Are 3 or so samples ever sufficiently representative to prove compliance?
Is the rule of thumb based on the same compliance metric I intended for? (70% vs 90% vs 95% confidence etc)
Are the (unspecified) assumptions of the rule of thumb met for my current data set?

This is not to dismiss existing rules of thumb. I, and others, have shown that the preliminary test is a fairly reliable test (with caveats). But those skeptics reading this, I present the “Quasi- Nonparametric Upper Tolerence Limit” test. An awful and complex name for a terrific and simple (but robust!) rule of thumb.

What is the rule?

The QNP UTL basically compares the largest value of a data set to the OEL. If the largest value is lower than a ratio of the OEL, then you can be 95% confident that at least 95% of exposures are below the OEL. As the ratio values are predetermined based on sample size, there is no statistics or mathematics needed for the user. Just “Is this number bigger or smaller than that number?”.

To make your life easier, I have created a tool for you to enter infomation and have a decision provided to you. You can even change the confidence and acceptable exceedance fraction to whatever you want!

How it works

The full explanation of the mathematics is available in the paper. I encourage you to read it as it’s not overly complicated, but more than is necessary for a general audience here.

Instead we can understand the fundamentals by breaking down the name:

Parametric - makes assumptions that the data follows a certain distribution (e.g. that it’s log-normally distributed

Non-parametric - makes no assumption about the distribution

“Quasi non-parametric” - a made up term by the authors. There are very loose assumptions about the data, but the test still works if the assumptions aren’t met.

Upper tolerance limit (UTL) - the value at which you are x% confidence that y percentile is below. Usually 95% confidence of the 95th percentile.

Other tests may be parametric and make assumptions that aren’t necessarily met, making the decision void. The beauty of this test is that it works in essentially every OH scenario.

What are the loose assumptions of the quasi-nonparametric test? First, that the data is lognormal (which in OH it usually is). Second, that the GSD is less than 7.39 which is also a safe bet. Again, even if the assumptions aren’t met, the worst thing that happens is that the test is too conservative.

Extra notes

Because the QNP UTL only relays on the largest value of the dataset, censored data is no problem. No need to process them in any way.

If you had a guess at the largest exposure you expected to measure, you could use this tool to calculate how many samples you need to prove compliance (assuming your estimation is correct).

The QNP UTL lines up perfectly with the AIHA rule! If you have 8 samples less than 10% of the exposure standard, both the AIHA rule and the QNP UTL find the data compliant.

Conclusion

Please, have a play with the tool available on the website. I would love to know if this is something you might use in future exposure assessment.

Reference