For each dataset, there is a limit to what we can use that dataset to test. Using the standard p-value based methods of science, the more hypotheses we check against the data, the more likely it will be that some of these checks give inaccurate conclusions. And this presents a big problem for the way science is practiced.
Let’s take an example to illustrate the principle. Suppose that you have information about 1000 people selected at random from the U.S. adult population. Your dataset includes these people’s heights, weights, ages, shoe sizes, and so forth. Now, if your goal is to know the mean height of all people in America, you can produce an estimate of this quantity by averaging the heights of the 1000 people you have information about. Despite the fact that your sample contains just 1000 people, rather than the full set of 230,000,000 or so American adults of interest, your estimate will, with high probability, be within a couple of inches of the total population mean height. This is due to the fact that the 1000 people were sampled at random (so we shouldn’t expect our sample to differ from the entire population in a systematic way) and because the standard deviation of heights is not very large (if there were tremendous outliers in the data, such as 500 foot tall giants, we would need more samples to get an accurate estimate). This idea is made precise by the central limit theorem. It tells us how likely the true entire population mean is to fall different distances from our sample estimate, and says that the error of our estimate decreases like one over the square root of the size of our sample.
The same technique could work to approximate the mean weight of adult Americans, or age, or shoe size, or number of children. And in each case, the estimate would, with high probability, be quite accurate. We could even, if we liked, estimate all these quantities simultaneously if we collected all of this information about each of our 1000 people. But the more quantities we estimate, the greater the chance that at least one estimate is quite inaccurate. Since each estimate has some chance of being bad, if we make a sufficiently large number of estimates we should expect to get unlucky at some point and end up with one or more bad ones. So, if we aren’t just estimating mean height, but rather the mean of 50 different traits, we cannot claim that all 50 of these estimate are likely to be good. We should expect that some of them will be inaccurate, though we don’t know which ones.
This is where problems arise. Suppose that you are a researcher who is trying to find interesting differences between, say, southerners and northerners in the United States. Your dataset of 1000 adults contains 500 people from each group. What do you do? Well, it might seem reasonable to go ahead and compute the mean value of many different traits, and look at how these means differ between the two groups, to see if you can find any large differences that seem interesting. For instance, you may compute the average salary of each group, and see if they deviate from each other by a large enough amount to be deemed statistically significant. If they don’t, you can try another trait like IQ, or number of children, and repeat the process. If you try enough different traits, hopefully you’ll eventually find an intriguingly large difference between the groups.
The trouble is, we know that if you estimate a large number of quantities, some of them will be inaccurate, and so some of the apparent differences between your two groups may just be due to these inaccuracies. If you test enough traits, you will eventually find differences between the populations that look significant, even though it is just the result of chance.
In fact, even if northerners and southerners had no systematic differences between them, there would still be apparent differences that arose just from the particular sample of 1000 people you happened to have data on. For example, in your dataset, it just might happen that the northerners have lower numbers of children than southerners, even if this isn’t true for the underlying populations of all northerners and southerners. If you were to publish this finding, without making mention of the number of hypotheses you tested before finding it, it may seem that you had produced a meaningful result. In fact, the assessment of this result should take into account the number of hypotheses (e.g. northerners have smaller shoe sizes than southerners, northerners have greater salaries than southerners, etc.) that you tested before you discovered this one (and the p-values can be modified to include this information). The most significant seeming deviation between the groups found after testing 100 different hypotheses is very likely greatly inflated by chance. Whereas if you had only tested a small number of hypotheses against your data, and found a strong result, this would likely be a meaningful finding.
As a general rule, the greater the number of data points you have, the larger the number of quantities you can accurately estimate from your dataset. On a set of just 10 points, you may not even be able to get an accurate estimate of the mean value of a single trait (unless the trait had very slow standard deviation). Whereas on a dataset of a billion points, you probably could estimate dozens of quantities accurately.
Unfortunately, when you’re reading a paper, there is no way to tell how many hypotheses the researcher tested on his dataset unless he chooses to publish it. And there is a strong incentive to obscure this information. If a researcher releases the fact that he tested 20 hypotheses before finding 1 which was statistically significant, readers may discredit the result, or reviewers may reject it for publication. And if the researcher spent a lot of time and money collecting his dataset, it would feel like a waste to give up on the data just because his first five hypotheses tested on it don’t pan out. It might take a lot of restraint to not just keep testing hypothesis after hypothesis until he finds something publishable.
But even if researchers were excessively careful, that wouldn’t fully resolve the problem. When a hypothesis is confirmed by a dataset, we must consider whether it is truly a confirmation of the hypothesis being tested, or a result of the fact that 20 researchers tested 20 false hypotheses, and this one of the 20 happened to seem true by chance. That is, if enough hypotheses are tested over all, we may find a large number of false hypotheses among them that just happen to seem true.
What makes this problem more pernicious is that when a hypothesis fails to pan out, the result is often not published. This is due to the fact that hypothesis disconfirmations (e.g. “no association was found between cabbage eating and longevity”) are generally less interesting and harder to publish than confirmations (e.g. “an association was found between cabbage eating and longevity”). But since most new hypotheses in science turn out to be false, we should expect the number of negative results to be very large (except in situations where previously well validated results are being confirmed). Hence, the number of published test results will be much less than the number of total tests conducted, with test failures substantially underreported. So there is no good way to tell how many times a hypotheses failed to be confirmed by tests before one researcher finally ran one that seemed to confirm it. And if a very large number of false hypotheses are tested, but mostly just the ones that turn out to look true are published, you could end up with a field’s journals being flooded with false but seemingly verified hypotheses. In exploratory fields where almost all hypotheses are false, and where disconfirmations of a hypothesis are almost never published, you might even get into a situation where most published research findings are false.