NPR’s Planet Money ran an interesting podcast in mid-January of this year. I recommend you take the time to listen to it.
The show discussed the idea that there are problems in the way that we do science — in this case that our continual reliance on hypothesis testing (or statistical significance) is leading to many scientifically spurious results. As a Bayesian, that comes as no surprise. One section of the show, however, piqued my pedagogical curiosity:
STEVE LINDSAY: OK. Let’s start now. We test 20 people and say, well, it’s not quite significant, but it’s looking promising. Let’s test another 12 people. And the notion was, of course, you’re just moving towards truth. You test more people. You’re moving towards truth. But in fact – and I just didn’t really understand this properly – if you do that, you increase the likelihood that you will get a, quote, “significant effect” by chance alone.
KESTENBAUM: There are lots of ways you can trick yourself like this, just subtle ways you change the rules in the middle of an experiment.
You can think about situations like this in terms of coin tossing. If we conduct a single experiment where there are only two possible outcomes, let us say “success” and “failure”, and if there is genuinely nothing affecting the outcomes, then any “success” we observe will be due to random chance alone. If we have a hypothetical fair coin — I say hypothetical because physical processes can make coin tossing anything but fair — we say the probability of a head coming up on a coin toss is equal to the probability of a tail coming up and therefore must be 1/2 = 0.5. The podcast describes the following experiment:
KESTENBAUM: In one experiment, he says, people were told to stare at this computer screen, and they were told that an image was going to appear on either the right site or the left side. And they were asked to guess which side. Like, look into the future. Which side do you think the image is going to appear on?
If we do not believe in the ability of people to predict the future, then we think the experimental subjects should have an equal chance of getting the right answer or the wrong answer.
The binomial distribution allows us to answer questions about multiple trials. For example, “If I toss the coin 10 times, then what is the probability I get heads more than seven times?”, or, “If the subject does the prognostication experiment described 50 times (and has no prognostic ability), what is the chance she gets the right answer more than 30 times?”
When we teach students about the binomial distribution we tell them that the number of trials (coin tosses) must be fixed before the experiment is conducted, otherwise the theory does not apply. However, if you take the example from Steve Lindsay, “..I did 20 experiments, how about I add 12 more,” then it can be hard to see what is wrong in doing so. I think the counterintuitive nature of this relates to general misunderstanding of conditional probability. When we encounter a problem like this, our response is “Well I can’t see the difference between 10 out of 20, versus 16 out of 32.” What we are missing here is that the results of the first 20 experiments are already known. That is, there is no longer any probability attached to the outcomes of these experiments. What we need to calculate is the probability of a certain number of successes, say x given that we have already observed y successes.
Let us take the numbers given by Professor Lindsay of 20 experiments followed a further 12. Further to this we are going to describe “almost significant” in 20 experiments as 12, 13, or 14 successes, and “significant” as 23 or more successes out of 32. I have chosen these numbers because (if we believe in hypothesis testing) we would observe 15 or more “heads” out of 20 tosses of a fair coin fewer than 21 times in 1,000 (on average). That is, observing 15 or more heads in 20 coin tosses is fairly unlikely if the coin is fair. Similarly, we would observe 23 or more heads out of 32 coin tosses about 10 times in 1,000 (on average).
So if we have 12 successes in the first 20 experiments, we need another 11 or 12 successes in the second set of experiments to reach or exceed our threshold of 23. This is fairly unlikely. If successes happen by random chance alone, then we will get 11 or 12 with probability 0.0032 (about 3 times in 1,000). If we have 13 successes in the first 20 experiments, then we need 10 or more successes in our second set to reach or exceed our threshold. This will happen by random chance alone with probability 0.019 (about 19 times in 1,000). Although it is an additively huge difference, 0.01 vs 0.019, the probability of exceeding our threshold has almost doubled. And it gets worse. If we had 14 successes, then the probability “jumps” to 0.073 — over seven times higher. It is tempting to think that this occurs because the second set of trials is smaller than the first. However, the phenomenon exists then as well.
The issue exists because the probability distribution for all of the results of experiments considered together is not the same as the probability distribution for results of the second set of experiments given we know the results of the first set of experiment. You might think about this as being like a horse race where you are allowed to make your bet after the horses have reached the half way mark — you already have some information (which might be totally spurious) but most people will bet differently, using the information they have, than they would at the start of the race.