Fisher’s Bias – focusing on a low FPR without regard to TPR – is the mirror image of the Confirmation Bias – focusing on a high TPR without regard to FPR. They both neglect the fact that what matters is the ratio of the two – the Likelihood Ratio. As a result, they both give rise to major inferential pitfalls.
The Confirmation Bias explains weird beliefs – the ancient Greeks’ reliance on divination and the Aztecs’ gruesome propitiation rites, as well as present-day lunacies, like psychics and other fake experts, superstitions, conspiracy theories and suicide bombers, alternative medicine and why people drink liquor made by soaking a dried tiger penis, with testicles attached, into a bottle of French cognac.
Fisher’s Bias has no less deleterious consequences. FPR<5% hence PP>95%: ‘We have tested our theory and found it significant at the 5% level. Therefore, there is only a 5% probability that we are wrong.’ This is the source of a deep and far-reaching misunderstanding of the role, scope and goals of what we call science.
‘Science says that…’, ‘Scientific evidence shows that…’, ‘It has been scientifically proven that…’: the view behind these common expressions is of science as a repository of established certainties. Science is seen as the means for the discovery of conclusive evidence or, equivalently, the accumulation of overwhelmingly confirmative evidence that leaves ‘no room for doubt or opposition‘. This is a treacherous misconception. While truth is its ultimate goal, science is not the preserve of certainty but quite the opposite: it is the realm of uncertainty, and its ethos is to be entirely comfortable with it.
Fisher’s Bias sparks and propagates the misconception. Evidence can lead to certainty, but it often doesn’t: the tug of war between confirmative and disconfirmative evidence does not always have a winner. By equating ‘significance’ with ‘certainty beyond reasonable doubt’, Fisher’s Bias encourages a naïve trust in the power of science and a credulous attitude towards any claim that manages to be portrayed as ‘scientific’. In addition, once deflated by the reality of scientific controversy, such trust can turn into its opposite: a sceptical view of science as a confusing and unreliable enterprise, propounding similarly ‘significant’ but contrasting claims, all portrayed as highly probable, but in fact – as John Ioannidis crudely puts it – mostly false.
Was Ronald Fisher subject to Fisher’s Bias? Apparently not: he stressed that ‘the null hypothesis is never proved or established, but is possibly disproved, in the course of experimentation. Every experiment may be said to exist only in order to give the facts a chance of disproving the null hypothesis’, immediately adding that ‘if an experiment can disprove the hypothesis’ it does not mean that it is ‘able to prove the opposite hypothesis.’ (The Design of Experiments, p. 16). However, the reasoning behind such conclusion is typically awkward. The opposite hypothesis (in our words, the hypothesis of interest) cannot be tested because it is ‘inexact’ – remember in the tea-tasting experiment the hypothesis is that the lady has some unspecified level of discerning ability. But – says Fisher – even if we were to make it exact, e.g. by testing perfect ability, ‘it is easy to see that this hypothesis could be disproved by a single failure, but could never be proved by any finite amount of experimentation’ (ibid.). Notice the confusion: saying that FPR<5% disproves the null hypothesis but FPR>5% does not prove it, Fisher is using the word ‘prove’ in two different ways. By ‘disproving’ the null he means considering it unlikely enough, but not certainly false. By ‘proving’ it, however, he does not mean considering it likely enough – which would be the correct symmetrical meaning – but he means considering it certainly true. That’s why he says that the null hypothesis as well as the opposite hypothesis are never proved. But this is plainly wrong and misleading. Prove/disprove is the same as accept/reject: it is a binary decision – doing one means not doing the other. So disproving the null hypothesis does mean proving the opposite hypothesis – not in the sense that it is certainly true, but in the correct sense that it is likely enough.
Here then is Fisher’s mistake. If H is the hypothesis of interest and not H the null hypothesis, FPR=P(E|not H) – the probability of the evidence (e.g. a perfect choice in the tea-tasting experiment) given that the hypothesis of interest is false (i.e. the lady has no ability and her perfect choice is a chance event). Then saying that a low FPR disproves the null hypothesis is the same as saying that a low P(E|not H) means a low P(not H|E). But since P(not H|E)=1–P(H|E)=1–PP, then a low FPR means a high PP, as in: FPR<5% hence PP>95%.
Hence yes: Ronald Fisher was subject to Fisher’s Bias. Despite his guarded and ambiguous wording, he did implicitly believe that 5% significance means accepting the hypothesis of interest. We have seen why: prior indifference. Fisher would not contemplate any value of BR other than 50%, i.e. BO=1, hence PO=LR=TPR/FPR. Starting with prior indifference, all is needed for PP=1-FPR is error symmetry.
Fisher’s Bias gives rise to invalid inferences, misplaced expectations and wrong attitudes. By setting FPR in its proper context, our Power surface brings much needed clarity on the subject, including, as we have seen, Ioannidis’s brash claim. Let’s now take a closer look at it.
Remember Ioannidis’s main point: published research findings are skewed towards acceptance of the hypothesis of interest based on the 5% significance criterion. Fisher’s bias favours the publication of ‘significant’ yet unlikely research findings, while ‘insignificant’ results remain unpublished. As we have seen, however, this happens for a good reason: it is unrealistic to expect a balance, as neither researchers nor editors are interested in publishing rejections of unlikely hypotheses. What makes a research finding interesting is not whether it is true or false, but whether it confirms an unlikely hypothesis or disconfirms a likely one.
Take for instance Table 4 in Ioannidis’s paper (p. 0700), which shows nine examples of research claims as combinations of TPR, BO and PP, given FPR=5%. Remember the match between our and Ioannidis’s notation: FPR=α, TPR=1-β (FNR=β), BO=R and PP=PPV. For the moment, let’s just take the first two columns and leave the rest aside:
So for example the first claim has TPR=80%, hence LR=16 and, under prior indifference (BO=1, BR=50%), PO=16 and therefore PP=94.1%. In the second, we have TPR=95%, hence LR=19, BO=2 and BR=2/3, hence PO=38 and therefore PP=97.4%. And so on. As we can see, four claims have PP>50%: there is at least a preponderance of evidence that they are true. Indeed the first three claims are true even under a higher standard, with claim 2 in particular reaching beyond reasonable doubt, as it starts from an already high prior, which gets further increased by powerful confirmative evidence. In 3, powerful evidence manages to update a sceptical 25% prior to an 84% posterior, and in 6 to update an even more strongly sceptical prior to a posterior above 50%. The other five claims, on the other hand, have PP<50%: they are false even under the lowest standard of proof, with 8 and 9 in particular standing out as extremely unlikely. Notice however that in all nine cases we have LR>1: evidence is, in various degrees, confirmative, i.e. it increases prior odds to a higher level. Even in the last two cases, where evidence is not very powerful and BR is a tiny 1/1000 – just like in our child footballer story – LR=4 quadruples it to 1/250. The posterior is still very small – the claims remain very unlikely – but this is the crucial point: they are a bit less unlikely than before. That’s what makes a research finding interesting: not a high PP but a LR significantly different from 1. All nine claims in the table – true and false – are interesting and, as such, worth publication. This includes claim 2, where further confirmative evidence brings virtual certainty to an already strong consensus. But notice that in this case disconfirmative evidence, reducing prior odds and casting doubt on such consensus, would have attracted even more interest. Just as we should expect to see a preponderance of studies confirming unlikely hypotheses, we should expect to see the same imbalance in favour of studies disconfirming likely hypotheses. It is the scientific enterprise at work.
Let’s now look at Ioannidis’s auxiliary point: the preponderance of ‘significant’ findings is reinforced by a portion of studies where significance is obtained through data manipulation. He defines bias u as ‘the proportion of probed analyses that would not have been “research findings”, but nevertheless end up presented and reported as such, because of bias’ (p. 0700).
How does Ioannidis’s bias modify his main point? This is shown in the following table, where PP* coincides with PPV in his Table 4:
Priors are the same, but now bias u causes a substantial reduction in LR and therefore in PP. For instance, in the first case u=0.10 means that 10% of research findings supporting the claim have been doctored into 5% significance through some form of data tampering. As a result, LR is lowered from 16 to 5.7 and PP from 94.1% to 85%. So in this case the effect of bias is noticeable but not determinant. The same is true in the second case, where a stronger bias causes a big reduction in LR from 19 to 2.9, but again not enough to meaningfully alter the resulting PP. In the third case, however, an even stronger bias does the trick: it reduces LR from 16 to 2 and PP from 84.2% all the way down to 40.6%. While the real PP is below 50%, a 40% bias makes it appear well above: the claim looks true but is in fact false. Same for 6, while the other five claims, which would be false even without bias, are even more so with bias – their LR reduced to near 1 and their PP consequently remaining close to their low BR.
This sounds a bit confusing so let’s restate it, taking case 3 as an example. The claim starts with a 25% prior – it is not a well established claim and would therefore do well with some confirmative evidence. The appearing evidence is quite strong: FPR=5% and TPR=80%, giving LR=16, which elevates PP to 84.1%. But in reality the evidence is not as strong: 40% of the findings accepting the claim have been squeezed into 5% significance through data fiddling. Therefore the real LR – the one that would have emerged without data alterations – is much lower, and so is the real PP resulting from it: the claim appears true but is false. So is claim 6, thus bringing the total of false claims from five to seven – indeed most of them.
How does bias u alter LR? In Ioannidis’s model, it does so mainly by turning FPR into FPR*=FPR+(1-FPR)u – see Table 2 in the paper (p. 0697). FPR* is a positive linear function of u, with intercept FPR and slope 1-FPR, which, since FPR=5%, is a very steep 0.95. In case 3, for example, u produces a large increase of FPR from 5% to 43%. In addition, u turns TPR into TPR*=TPR+(1-TPR)u, which is also a positive linear function of u, with intercept TPR and slope 1-TPR which, since the TPR of confirmative evidence is higher than FPR, is flatter. In case 3 the slope is 0.2, so u increases TPR from 80% to 88%. The combined effect, as we have seen, is a much lower LR*=TPR*/FPR*, going down from 16 to 2.
I will post a separate note about this model, but the point here is that, while Ioannidis’s bias increases the proportion of false claims, it is not the main reason why most of them are false. Five of the nine claims in his Table 4 would be false even without bias.
In summary, by confusing significance with virtual certainty, Fisher’s Bias encourages Ioannidis’s bias (I write it with a small b because it has no cognitive value: it is just more or less intentional cheating). But Ioannidis’s bias does not explain ‘Why Most Research Findings Are False’. The main reason is that many of them test unlikely hypotheses, and therefore, unless they manage to present extraordinary or conclusive evidence, their PP turns out to be lower and often much lower than 50%. But this doesn’t make them worthless or unreliable – as the paper’s title obliquely suggests. As long as they are not cheating, researchers are doing their job: trying to confirm unlikely hypotheses. At the same time, however, they have another important responsibility: to warn the reader against Fisher’s Bias, by explicitly clarifying that, no matter how ‘significant’ and impressive their results may appear, they are not ‘scientific revelations’ but tentative discoveries in need of further evidence.
