Comment by nfw2 | Hacker News Reader

Comment by nfw2 | Hacker News Reader

In 2014 I consulted my father, a lawyer, in writing an amicus brief to a lawsuit relating to lead exposure in baby food.

Basically, lead exposure testing in food can have highly-variable results because of how the particles adhere to the food substrate, variability between food batches, and the way the exposure tests work. Because of this, lead exposure testing generally involves testing several samples and taking the mean to estimate the real level of exposure.

Despite several samples showing high levels of lead, the defense got off by claiming that the geometric mean was the correct average to use to infer the total exposure. This is obviously incorrect because the geometric mean is a multiplicative mean. With a geometric mean, all you need is one sample with lead exposure close to 0 to get a mean that's close to 0, even if all the other samples tested positive for extreme levels of lead exposure.

Just as a sanity check, we checked official literature on food toxicology, and every resource we could find stated that using arithmetic mean was the correct methodology, and not one mentioned geometric means.

The defense ended up winning the case, which is absolutely horrifying because 1.) baby food is the most dangerous product imaginable to contain lead and 2.) now there is a legal precedent to defend yourself against lead exposure claims using total BS statistics.

Edit: thankfully, it looks like the FDA is getting involved now https://www.usatoday.com/story/news/2021/10/01/baby-foods-co...

> With a geometric mean, all you need is one sample with lead exposure close to 0 to get a mean that's close to 0, even if all the other samples tested positive for extreme levels of lead exposure.

Not going to comment on whether arithmetic or geometric mean is more appropriate, but your comment is incorrect.

Yes, a number close to 0 brings everything down. But then you take the nth-root, which brings it back up again. Some examples - assume one of the numbers is 0.01 (close to 0).

    2 samples means square root - that becomes 0.1
    3 samples means cube root - that becomes 0.215
    10 samples means 10th root - that becomes 0.63
    20 samples means 20th root - that becomes 0.794

The more samples, the less impact an outlier like a number close to 0 will have.

Put another way, the arithmetic mean has the same problem. Extreme outliers can greatly influence the sum. But if you have enough samples, each extreme outlier is tempered a bit by dividing by n.

If you want to be precise, what I meant was that there is no level of exposure in the first n samples so great that the geoemetric mean of n+1 samples is certain to be over any arbitrary value X

This is not true of arithmetic mean. If you have 9 samples with value 10, then the arithmetic mean of 9 samples plus one more is certainly at least 9.

Ignoring the context, and speaking only to the math, the two examples you give are not analogous.

In a geometric mean, picking a number below 1 is the equivalent of picking a negative number in the arithmetic mean. The multiplicative identity is 1, and the additive identity is 0. Going below 1 for the geometric mean is like going below 0 in the arithmetic mean. So mathematically, you do not have a guarantee with the arithmetic mean either.

Of course, in your case, the numbers aren't allowed to be negative. Either approach you take, there is some "contextual" bias going on.

Why is it helpful to ignore the context? My examples clearly state I am making statements in the context of taking exposure samples

I think you're making potentially invalid assumptions about the about the relative scales of the numbers involved.

The issue is that if the small outlier leads to the product being less than 1, the degree of the root doesn't matter because the largest value possible of the n-th root approaches 1, which may be below whatever meaningful thresholds are relevant here.

> The issue is that if the small outlier leads to the product being less than 1, the degree of the root doesn't matter because the largest value possible of the n-th root approaches 1, which may be below whatever meaningful thresholds are relevant here.

That is true, but I don't see the relevance given the context. If the geometric mean approaches 1 (i.e. after the n-th root), then you have too much lead. If all your samples have lead levels well below tolerance, then the n-th root of the product will not approach 1, but will be significantly smaller. Example: If all 10 samples are 0.01, the geometric mean is 0.01.

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