Monday, March 18, 2013

What is your hearing threshold? - the joy of statistics

Statistical methods can be applied to a lot of different types of experiments and data that may not seem related at all. Four decades ago over weekends I ran hearing tests on people in the Air Force Reserve. A person sits in a soundproof booth while wearing calibrated headphones. We had an automatic audiometer system, but it used to be done even more simply.

For each frequency (in each ear) you could test by playing a tone at different loudness levels. The person pushed a button (or just raised his hand) to indicate when he heard the tone. If he heard it, then you turned the loudness down by a step. If he didn’t, you turned it up by a step. The sequence of “up-and-down” test steps was like a staircase, and it would converge on the threshold, which was about 30 dB in the example shown below.

During World War II at the Bruceton, Pennsylvania research station of the Bureau of Mines people determined how sensitive explosives were to detonation by mechanical shock. Someone would carefully (and remotely) drop a weight from different heights on top of a series of samples, and see whether they went boom or not. Their sequential test strategy was just like a hearing test.

Insecticides sometimes are tested by exposing groups of insects to different concentrations, and then counting the proportion of them that died. The Probit method is used to analyze data. Results are usually quoted as the median lethal dose, or LD50, where half of them were expected to die.

Three decades ago, in a research lab in Ann Arbor, we were running bent-beam stress corrosion cracking tests where a series of little steel specimens were deflected different amounts, and then dunked in acidified brine saturated with hydrogen sulfide gas for a two weeks. Each specimen either cracked or didn’t crack. We tried to determine the “critical stress” at which half the specimens would crack. Typically we ran the first set of four to six specimens with a coarse spacing between their deflections, and then ran another set or two with half or a third of that spacing. 

All four of these  situations involve the statistical analysis of pass-fail or binary data, finding the transition between nothing happening and something happening.

When I started graduate school four decades ago, another student told me to buy a cookbook called Experimental Statistics (National Bureau of Standards Handbook 91) written by Mary Gibbons Natrella and published back in 1963. She wrote the statistical equivalent to The Joy of Cooking - a comprehensive, easy-to follow collection of recipes (and discussion of how they worked) that began as a series of pamphlets on ordnance  for the U.S. Army.

A decade later I found that Chapter 10 in Experimental Statistics was all about Sensitivity Testing. That got Dave Cameron and I started on finding out more on how statistics could be applied to stress corrosion cracking data. We looked through about a hundred more recent magazine articles and books, and eventually wrote about over thirty of them. At the 1984 National Association of Corrosion Engineers meeting I presented our Corrosion 84 Paper #214, Statistical Methods for Treatment of Sulfide Stress Cracking Data.  

Later the National Bureau of Standards was renamed the National Institute of Standards and Technology (NIST). The descendant of Mary Natrella’s cookbook is the free, online NIST/Sematech Engineering Statistics Handbook. It has eight chapters on how to explore, measure, characterize, model, improve, monitor, and compare, and also discusses reliability.

The U.S. Navy image of a hearing test came from here at Wikimedia Commons.

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