Of all the statistical tricks or treats I like to think about, the base rate (and it’s associated fallacy) are probably the most interesting to me. It’s a common fallacy, in large part because it requires two steps of math to work out what’s going on. I’ve referenced it before, but I wanted a definitive post where I walked through what a base rate is and why you should remember it exists. Ready? Let’s go.

First, let’s find an example.
Like most math problems, this one will be a little easier to follow if we use an example.In my

In my Intro to Internet Science series, I mentioned the troubling case of a couple of former CIA analysts whose house was raided by a SWAT team after they were spotted shopping at the wrong garden store. After spotting the couple purchasing what they thought was marijuana growing equipment, the police had tested their trashcans for the presence of drugs. Twice the police got a positive test result, and thus felt perfectly comfortable raiding the house and holding the parents and kids at gunpoint for two hours while they searched for the major marijuana growing operation they believed they were running. In the end it was determined the couple was actually totally innocent. There’s a lot going on with this story legally, but what was up with those positive drug tests?

Let’s make a contingency table!
In last week’s post, I discussed the fact that there is almost always more than one way to be wrong. A contingency table helps us visualize the various possibilities that can arise from the two different types of test results and the two different realities:

So here we have four options, two good and two bad:

1. True positive (yes/yes): we have evidence of actual wrongdoing¹
2. False negative (no/yes): someone with drugs appears innocent
3. False positive (yes/no): someone without drugs appears guilty
4. True negative (no/no): an innocent person’s innocence is confirmed

In this case, we ended up with a false positive, but how often does that really happen? Is this just an aberration or something we should be concerned about?

Picking between the lesser of two evils.
Before we go on, let’s take a step back for a minute and consider why the police department may have had to consider when they selected a drug screening test to use. It’s important to recognize that in this situation (as in most of life), you actually do have some discretion over which way you chose to be wrong.  In a perfect world we’d have unlimited resources to buy a test that gets the right answer every time, but in the real world we often have to go the cheap route and consider the consequences of either type of error and make trade-offs.

For example, in medicine false positives are almost always preferable to false negatives. Most doctors (and patients!) would prefer that a screening test told them they might have a disease that they did not have (false positive) than to have a screening test miss a disease they did have (false negative).

In criminal justice, there is a similar preference. Police would rather have evidence of activity that didn’t happen (false positive) then not get evidence when a crime was committed (false negative).

So what kind of trade-offs are we talking about?
Well, in the article I linked to above, it mentioned that one of the downfalls of the drug tests many police departments use is a very high false positive rate…..as high as 70%. This means that if you tested 100 trashcans that were completely free of drugs, you’d get a positive test for 70 of them.

Well that sounds pretty bad….so is that the base rate you were talking about?
No, but it is an important rate to keep in mind because it influences the math in ways that aren’t particularly intuitive for most people. For example, if we test 1000 trash cans, half with drugs and half without, here’s what we get:

When the police are out in the field, they get exactly one piece of information: whether or not the trash can tested positive for drugs.  In order to use this information, we actually have to calculate what that means. In the above example, we have 495 true positive trash cans with drugs in them. We also have 350 false positive trash cans with no drugs in them, but with a positive test. So overall, we have 845 trash cans with a positive test. 495/845 is about 59%…..so under these circumstances, a positive test only means drugs are present about 60% of the time.

Now about that base rate……
Okay, so none of that is great, but this actually can get worse. You see, the rate of those who do drugs and those who don’t do drugs isn’t actually equal. The rate of those who don’t do drugs is actually much much higher, and this is the base rate I was talking about before.

According to many reports, about 10% of the US adult population used illegal drugs in the past month (mostly marijuana, FYI….not controlled for states that have legalized it). Presumably this means that about 10% of trash cans might contain drugs at any given time. That makes our numbers look like this:

Using the same math as above, we get 99/(630+99) = 14%. Now we realize that for every positive test, there’s actually only about a 14% chance there are drugs in that trash can. I’m somewhat curious how much worse that is than just having a trained police officer take a look.  In fact, because the base rates are so different, you actually would need a test with an 11% false positive rate (as compared to the 70% we currently have) to make the chances 50/50 that your test is telling you what you think it’s telling you. Yikes.

Now of course these numbers only holds if you’re testing trash cans randomly….but if you’re testing the garbage of everyone who goes to a garden store on a Saturday morning, that may be a little closer to the truth than you want to admit.

So what’s the takeaway?
The crux of the base rate fallacy is that a small percentage of a large number can easily be larger than a large percentage of a small number. This is basic math, but it becomes hard to remember when you’re in the moment and the information is not being presented in a straightforward way. If you got a math test that said “Which value is larger….11% of 900 or 99% of 100?” You’d probably get it right pretty quickly. However, when it’s up to you to remember what the base rate is, people get much much worse at this problem. In fact, the vast majority of medical doctors don’t get this type of problem correct when it’s presented to them and they’re specifically given the base rate….so my guess is the general population success rate is quite low.

No matter how accurate a test is, if the total number of entries in one of the rows (or columns) is much larger than the total of the other, you should watch out for this.

Base rate matters.
^{1. Note for the libertarians: It is beyond the scope of this post to discuss current drug policy and whether or not this should actually constitute wrongdoing. Just roll with it.↩}