On the hazards of significance testing.
Part 2: the false discovery rate, or how not to make a fool of yourself with P values

This post is now a bit out of date: there is a summary of my more recent efforts (papers, videos and pop stuff) can be found on Prof Sivilotti’s OneMol pages.

What follows is a simplified version of part of a paper that appeared as a preprint on arXiv in July. It appeared as a peer-reviewed paper on 19th November 2014, in the new Royal Society Open Science journal. If you find anything wrong, or obscure, please email me. Be vicious.

There is also a simplified version, given as a talk on Youtube..

It’s a follow-up to my very first paper, which was written in 1959 – 60, while I was a fourth year undergraduate(the history is at a recent blog). I hope this one is better.

‘”. . . before anything was known of Lydgate’s skill, the judgements on it had naturally been divided, depending on a sense of likelihood, situated perhaps in the pit of the stomach, or in the pineal gland, and differing in its verdicts, but not less valuable as a guide in the total deficit of evidence” ‘George Eliot (Middlemarch, Chap. 45)

“The standard approach in teaching, of stressing the formal definition of a p-value while warning against its misinterpretation, has simply been an abysmal failure”  Sellke et al. (2001) `The American Statistician’ (55), 62–71

The last post was about screening. It showed that most screening tests are useless, in the sense that a large proportion of people who test positive do not have the condition. This proportion can be called the false discovery rate. You think you’ve discovered the condition, but you were wrong.

Very similar ideas can be applied to tests of significance. If you read almost any scientific paper you’ll find statements like “this result was statistically significant (P = 0.047)”. Tests of significance were designed to prevent you from making a fool of yourself by claiming to have discovered something, when in fact all you are seeing is the effect of random chance. In this case we define the false discovery rate as the probability that, when a test comes out as ‘statistically significant’, there is actually no real effect.

You can also make a fool of yourself by failing to detect a real effect, but this is less harmful to your reputation.

It’s very common for people to claim that an effect is real, not just chance, whenever the test produces a P value of less than 0.05, and when asked, it’s common for people to think that this procedure gives them a chance of 1 in 20 of making a fool of themselves. Leaving aside that this seems rather too often to make a fool of yourself, this interpretation is simply wrong.

The purpose of this post is to justify the following proposition.

This makes slightly less startling the assertion in John Ioannidis’ (2005) article, Why Most Published Research Findings Are False. That paper caused quite a stir. It’s a serious allegation. In fairness, the title was a bit misleading. Ioannidis wasn’t talking about all science. But it has become apparent that an alarming number of published works in some fields can’t be reproduced by others. The worst offenders seem to be clinical trials, experimental psychology and neuroscience, some parts of cancer research and some attempts to associate genes with disease (genome-wide association studies). Of course the self-correcting nature of science means that the false discoveries get revealed as such in the end, but it would obviously be a lot better if false results weren’t published in the first place.

How can tests of significance be so misleading?

Tests of statistical significance have been around for well over 100 years now. One of the most widely used is Student’s t test. It was published in 1908. ‘Student’ was the pseudonym for William Sealy Gosset, who worked at the Guinness brewery in Dublin. He visited Karl Pearson’s statistics department at UCL because he wanted statistical methods that were valid for testing small samples. The example that he used in his paper was based on data from Arthur Cushny, the first holder of the chair of pharmacology at UCL (subsequently named the A.J. Clark chair, after its second holder)

The outcome of a significance test is a probability, referred to as a P value. First, let’s be clear what the P value means. It will be simpler to do that in the context of a particular example. Suppose we wish to know whether treatment A is better (or worse) than treatment B (A might be a new drug, and B a placebo). We’d take a group of people and allocate each person to take either A or B and the choice would be random. Each person would have an equal chance of getting A or B. We’d observe the responses and then take the average (mean) response for those who had received A and the average for those who had received B. If the treatment (A) was no better than placebo (B), the difference between means should be zero on average. But the variability of the responses means that the observed difference will never be exactly zero. So how big does it have to be before you discount the possibility that random chance is all you were seeing. You do the test and get a P value. Given the ubiquity of P values in scientific papers, it’s surprisingly rare for people to be able to give an accurate definition. Here it is.

If this probability is low enough, the conclusion would be that it’s unlikely that the observed difference (or a still bigger one) would have occurred if A and B were identical, so we conclude that they are not identical, i.e. that there is a genuine difference between treatment and placebo.

This is the classical way to avoid making a fool of yourself by claiming to have made a discovery when you haven’t. It was developed and popularised by the greatest statistician of the 20th century, Ronald Fisher, during the 1920s and 1930s. It does exactly what it says on the tin. It sounds entirely plausible.

What could possibly go wrong?

Another way to look at significance tests

One way to look at the problem is to notice that the classical approach considers only what would happen if there were no real effect or, as a statistician would put it, what would happen if the null hypothesis were true. But there isn’t much point in knowing that an event is unlikely when the null hypothesis is true unless you know how likely it is when there is a real effect.

We can look at the problem a bit more realistically by means of a tree diagram, very like that used to analyse screening tests, in the previous post.

In order to do this, we need to specify a couple more things.

First we need to specify the power of the significance test. This is the probability that we’ll detect a difference when there really is one. By ‘detect a difference’ we mean that the test comes out with P < 0.05 (or whatever level we set). So it’s analogous with the sensitivity of a screening test. In order to calculate sample sizes, it’s common to set the power to 0.8 (obviously 0.99 would be better, but that would often require impracticably large samples).

The second thing that we need to specify is a bit trickier, the proportion of tests that we do in which there is a real difference. This is analogous to the prevalence of the disease in the population being tested in the screening example. There is nothing mysterious about it. It’s an ordinary probability that can be thought of as a long-term frequency. But it is a probability that’s much harder to get a value for than the prevalence of a disease.

If we were testing a series of 30C homeopathic pills, all of the pills, regardless of what it says on the label, would be identical with the placebo controls so the prevalence of genuine effects, call it P(real), would be zero. So every positive test would be a false positive: the false discovery rate would be 100%. But in real science we want to predict the false discovery rate in less extreme cases.

Suppose, for example, that we test a large number of candidate drugs. Life being what it is, most of them will be inactive, but some will have a genuine effect. In this example we’d be lucky if 10% had a real effect, i.e. were really more effective than the inactive controls. So in this case we’d set the prevalence to P(real) = 0.1.

We can now construct a tree diagram exactly as we did for screening tests.

Suppose that we do 1000 tests. In 90% of them (900 tests) there is no real effect: the null hypothesis is true. If we use P = 0.05 as a criterion for significance then, according to the classical theory, 5% of them (45 tests) will give false positives, as shown in the lower limb of the tree diagram. If the power of the test was 0.8 then we’ll detect 80% of the real differences so there will be 80 correct positive tests.

The total number of positive tests is 45 + 80 = 125, and the proportion of these that are false positives is 45/125 = 36 percent. Our false discovery rate is far bigger than the 5% that many people still believe they are attaining.

In contrast, 98% of negative tests are right (though this is less surprising because 90% of experiments really have no effect).

The equation

You can skip this section without losing much.

As in the case of screening tests, this result can be calculated from an equation. The same equation works if we substitute power for sensitivity, P(real) for prevalence, and siglev for (1 – specificity) where siglev is the cut off value for “significance”, 0.05 in our examples.

The false discovery rate (the probability that, if a “signifcant” result is found, there is actually no real effect) is given by

\[FDR = \frac{siglev\left(1-P(real)\right)}{power.P(real) + siglev\left(1-P(real)\right) }\; \]

In the example above, power = 0.8, siglev = 0.05 and P(real) = 0.1, so the false discovery rate is

\[\frac{0.05 (1-0.1)}{0.8 \times 0.1 + 0.05 (1-0.1) }\; = 0.36 \]

So 36% of “significant” results are wrong, as found in the tree diagram.

Some subtleties

The argument just presented should be quite enough to convince you that significance testing, as commonly practised, will lead to disastrous numbers of false positives. But the basis of how to make inferences is still a matter that’s the subject of intense controversy among statisticians, so what is an experimenter to do?

It is difficult to give a consensus of informed opinion because, although there is much informed opinion, there is rather little consensus. A personal view follows.  Colquhoun (1970), Lectures on Biostatistics, pp 94-95.

This is almost as true now as it was when I wrote it in the late 1960s, but there are some areas of broad agreement.

There are two subtleties that cause the approach outlined above to be a bit contentious. The first lies in the problem of deciding the prevalence, P(real). You may have noticed that if the frequency of real effects were 50% rather than 10%, the approach shown in the diagram would give a false discovery rate of only 6%, little different from the 5% that’s embedded in the consciousness of most experimentalists.

But this doesn’t get us off the hook, for two reasons. For a start, there is no reason at all to think that there will be a real effect there in half of the tests that we do. Of course if P(real) were even bigger than 0.5, the false discovery rate would fall to zero, because when P(real) = 1, all effects are real and therefore all positive tests are correct.

There is also a more subtle point. If we are trying to interpret the result of a single test that comes out with a P value of, say, P = 0.047, then we should not be looking at all significant results (those with P < 0.05), but only at those tests that come out with P = 0.047. This can be done quite easily by simulating a long series of t tests, and then restricting attention to those that come out with P values between, say, 0.045 and 0.05. When this is done we find that the false discovery rate is at least 26%. That’s for the best possible case where the sample size is good (power of the test is 0.8) and the prevalence of real effects is 0.5. When, as in the tree diagram, the prevalence of real effects is 0.1, the false discovery rate is 76%. That’s enough to justify Ioannidis’ statement that most published results are wrong.

One problem with all of the approaches mentioned above was the need to guess at the prevalence of real effects (that’s what a Bayesian would call the prior probability). James Berger and colleagues (Sellke et al., 2001) have proposed a way round this problem by looking at all possible prior distributions and so coming up with a minimum false discovery rate that holds universally. The conclusions are much the same as before. If you claim to have found an effects whenever you observe a P value just less than 0.05, you will come to the wrong conclusion in at least 29% of the tests that you do. If, on the other hand, you use P = 0.001, you’ll be wrong in only 1.8% of cases. Valen Johnson (2013) has reached similar conclusions by a related argument.

A three-sigma rule

As an alternative to insisting on P < 0.001 before claiming you’ve discovered something, you could use a 3-sigma rule. In other words, insist that an effect is at least three standard deviations away from the control value (as opposed to the two standard deviations that correspond to P = 0.05).

The three sigma rule means using P= 0.0027 as your cut off. This, according to Berger’s rule, implies a false discovery rate of (at least) 4.5%, not far from the value that many people mistakenly think is achieved by using P = 0.05 as a criterion.

Particle physicists go a lot further than this. They use a 5-sigma rule before announcing a new discovery. That corresponds to a P value of less than one in a million (0.57 x 10⁻⁶). According to Berger’s rule this corresponds to a false discovery rate of (at least) around 20 per million. Of course their experiments can’t be randomised usually, so it’s as well to be on the safe side.

Underpowered experiments

All of the problems discussed so far concern the near-ideal case. They assume that your sample size is big enough (power about 0.8 say) and that all of the assumptions made in the test are true, that there is no bias or cheating and that no negative results are suppressed. The real-life problems can only be worse. One way in which it is often worse is that sample sizes are too small, so the statistical power of the tests is low.

The problem of underpowered experiments has been known since 1962, but it has been ignored. Recently it has come back into prominence, thanks in large part to John Ioannidis and the crisis of reproducibility in some areas of science. Button et al. (2013) said

“We optimistically estimate the median statistical power of studies in the neuroscience field to be between about 8% and about 31%”

This is disastrously low. Running simulated t tests shows that with a power of 0.2, not only do you have only a 20% chance of detecting a real effect, but that when you do manage to get a “significant” result there is a 76% chance that it’s a false discovery.

And furthermore, when you do find a “significant” result, the size of the effect will be over-estimated by a factor of nearly 2. This “inflation effect” happens because only those experiments that happen, by chance, to have a larger-than-average effect size will be deemed to be “significant”.

What should you do to prevent making a fool of yourself?

The simulated t test results, and some other subtleties, will be described in a paper, and/or in a future post. But I hope that enough has been said here to convince you that there are real problems in the sort of statistical tests that are universal in the literature.

The blame for the crisis in reproducibility has several sources.

One of them is the self-imposed publish-or-perish culture, which values quantity over quality, and which has done enormous harm to science.

The mis-assessment of individuals by silly bibliometric methods has contributed to this harm. Of all the proposed methods, altmetrics is demonstrably the most idiotic. Yet some vice-chancellors have failed to understand that.

Another is scientists’ own vanity, which leads to the PR department issuing disgracefully hyped up press releases.

In some cases, the abstract of a paper states that a discovery has been made when the data say the opposite. This sort of spin is common in the quack world. Yet referees and editors get taken in by the ruse (e.g see this study of acupuncture).

The reluctance of many journals (and many authors) to publish negative results biases the whole literature in favour of positive results. This is so disastrous in clinical work that a pressure group has been started; altrials.net “All Trials Registered | All Results Reported”.

Yet another problem is that it has become very hard to get grants without putting your name on publications to which you have made little contribution. This leads to exploitation of young scientists by older ones (who fail to set a good example). Peter Lawrence has set out the problems.

And, most pertinent to this post, a widespread failure to understand properly what a significance test means must contribute to the problem. Young scientists are under such intense pressure to publish, they have no time to learn about statistics.

Here are some things that can be done.

• Notice that all statistical tests of significance assume that the treatments have been allocated at random. This means that application of significance tests to observational data, e.g. epidemiological surveys of diet and health, is not valid. You can’t expect to get the right answer. The easiest way to understand this assumption is to think about randomisation tests (which should have replaced t tests decades ago, but which are still rare). There is a simple introduction in Lectures on Biostatistics (chapters 8 and 9). There are other assumptions too, about the distribution of observations, independence of measurements), but randomisation is the most important.
• Never, ever, use the word “significant” in a paper. It is arbitrary, and, as we have seen, deeply misleading. Still less should you use “almost significant”, “tendency to significant” or any of the hundreds of similar circumlocutions listed by Matthew Hankins on his Still not Significant blog.
• If you do a significance test, just state the P value and give the effect size and confidence intervals (but be aware that this is just another way of expressing the P value approach: it tells you nothing whatsoever about the false discovery rate).
• Observation of a P value close to 0.05 means nothing more than ‘worth another look’. In practice, one’s attitude will depend on weighing the losses that ensue if you miss a real effect against the loss to your reputation if you claim falsely to have made a discovery.
• If you want to avoid making a fool of yourself most of the time, don’t regard anything bigger than P < 0.001 as a demonstration that you’ve discovered something. Or, slightly less stringently, use a three-sigma rule.

Despite the gigantic contributions that Ronald Fisher made to statistics, his work has been widely misinterpreted. We must, however reluctantly, concede that there is some truth in the comment made by an astute journalist:

“The plain fact is that 70 years ago Ronald Fisher gave scientists a mathematical machine for turning baloney into breakthroughs, and °flukes into funding. It is time to pull the plug“. Robert Matthews Sunday Telegraph, 13 September 1998.

There is now a video on YouTube that attempts to explain explain simply the essential ideas.  The video  has now been updated. The new version has better volume and it used  term ‘false positive risk’, rather than the earlier term ‘false discovery rate’, to avoid confusion with the use of the latter term in the context of multiple comparisons.

The false positive risk: a proposal concerning what to do about p-values (version 2)

Follow-up

31 March 2014 I liked Stephen Senn’s first comment on twitter (the twitter stream is storified here). He said ” I may have to write a paper ‘You may believe you are NOT a Bayesian but you’re wrong'”. I maintain that the analysis here is merely an exercise in conditional probabilities. It bears a formal similarity to a Bayesian argument, but is free of more contentious parts of the Bayesian approach. This is amplified in a comment, below.

4 April 2014

I just noticed that my first boss, Heinz Otto Schild.in his 1942 paper about the statistical analysis of 2+2 dose biological assays (written while he was interned at the beginning of the war) chose to use 99% confidence limits, rather than the now universal 95% limits. The later are more flattering to your results, but Schild was more concerned with precision than self-promotion.