The Fallacy of the Base Rate

Posted: October 6, 2011 in analysis

This isn’t an analysis up to my usual standards, but I have been struggling for content these last few weeks.  Nothing out there seems to fit the somewhat particular requirements I have for a subject to write about; that it be a factual statement which requires aggressive or unorthodox analysis to get to the bottom of, but that there is a path of analysis to enlighten.  So, today I have a much simpler error, from wnd.com.

Writing about restrictions on unpasteurized milk, Bob Unruh observes:

The reason cannot be safety, the report said, since a report from the Weston A. Price Foundation revealed that from 1980 to 2005 there were 10 times more illnesses from pasteurized milk than from raw milk.

Unfortunately, this is citing a report (without giving details or linking to it) that cites another report that makes this claim.  I eventually found the original report here, only to discover that it has almost no further details about what it means.  From that report, we learn that there were 41 outbreaks and 19,531 illnesses attributed to pasteurized milk, and this is 10.7 times the illnesses for raw milk.  If you use the raw milk numbers later in the report, it comes out to 8.4 times, but they may be using different numbers.   It sounds damning, until you realize that this does not mean that an illness is ten times more likely from pasteurized milk.  The problem is that more people drink pasteurized milk, so even if illness is less likely, there will be more in total.

In fact, according to the CDC in 2002, about 80% of Americans drink milk, and about 3.5% drink raw milk.  Other numbers (cited in the report linked about) have the consumption of raw milk much lower, at around 0.5%, but we’ll stick with the higher number.  To understand what this means, consider a notional 1000 person town.  35 people drink raw milk, and 765 drink regular milk.  1 raw milk drinker gets ill, and 10 regular milk drinkers do.

So, the danger to a raw milk drinker is 1/35, or a 2.8% chance of illness.  The danger to a regular milk drinker is 10/765, or a 1.3% chance of illness.  Remember these specific chances are made up (in reality both numbers are probably lower), but the key fact is, even though ten times more regular milk drinkers got sick, there are so many more of them to choose from that the rate of sickness among regular milk drinkers is less than half that among raw milk drinkers.

This is a version of the fallacy of the base rate, where one can get sidetracked by what numbers really mean if one forgets the base probabilities underlying the situation (in this case, the chance that a random sick person drank raw milk or regular milk).  However, in the usual case of the base rate there are two intersecting probabilities, usually a population and a test, and without doing the math carefully, the fallacy is easy to fall into.  Here, there is no probability, just a number of illnesses, and the math to show how the base rate affects the probabilities is trivial.  So in this case, the fallacy is less forgivable.

I have opinions on the larger issues about the safety of raw milk, and whether regulating a farm as described in the article is an unlawful violation of personal property rights or properly applying the law to ignore legal trickery.  But this article is simply about the real meaning of the “ten times as many” claim.  Using that as evidence that pasteurized milk is more dangerous than raw milk is like saying that driving is more dangerous than hang-gliding because there are more car than hang-glider accidents.  Both claims are preposterous if given any real thought.

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