## Why most published findings are false: revisiting the Ioannidis argument

It has been more than a decade that Ioannidis published his paper entiled Why most published research findings are false. Forstmeier et al. (2016) recently revisited the Ioannidis argument, and I thought it worthwhile to prepare a blog post on the topic to cement my understanding.

## Looking for a novel effect

Let’s consider 1000 hypotheses we might want to test. Given that novelty is a key criteria to publish results, it would not be unreasonable that only 10% of these hypotheses were in fact true.

Of the 1000 hypotheses tested, 10% are in fact true.

### alpha and false-positive findings

By setting $\alpha = 0.05$, we accept that 5% of the hypotheses that we test will be false positives. In statistics, these are called type I errors, with false positives being worse errors to make than false negatives.

There are 100 hypotheses that are true and 900 that are not. With$\alpha = 0.05$, 5% of the 900 hypotheses that are not true will have p-values below 0.05. This means we will have 45 false positives.

With alpha set to 0.05, 5% of the 900 hypotheses that are not true will have p-values below 0.05. Thus, 45 tested hypotheses will be statistically significant when they are in fact not true (i.e., false-positives)

### beta and false-negative findings

In statistics, beta represents the chance of making a type II error; i.e., false negative findings.

beta is equal to the statistical power of a study, with 80% power being a common target for studies. As previously discussed, statistical power depends on the sample size, the variability in the data and the size of the studied effect.

With $\beta = 20%$, 80% of the hypotheses that are in fact true will be statistically significant. Conversely, 20% of the hypotheses that are true will not be statistically significant (i.e., false negatives).

With beta = 20%, 80% of the hypotheses that are true will have p-values below 0.05. These are true positives. This also means that 20% of the hypotheses that are true will have p-values above 0.05; these are false-negatives. With 100 true hypotheses, 80 of them will be statistically significant.

### The crux of the Ioannidis argument

In our current example, we saw that 45 significant results were produced when we tested the 900 hypotheses that were not true (because $\alpha$ was set to 0.05). We also saw that 80 significant results were produced when we tested the 100 hypotheses that were true (because $\beta$ was equal to 20%).

What is the percentage of statistically significant hypotheses that are false? This is known as the false-positive report probability (FPRP).

$FPRP = 100 * (falsePositive / (falsePositive + truePositive))$

$FPRP = 100 * (45 / (45 + 80))$

$FPRP = 36\%$

This means that with $\alpha = 0.05$ and $\beta = 20%$, values considered respectable in most fields, 36% of the significant results are falsely associated with hypotheses that are not true. While this number may alarm some readers, things get even scarier when studies are underpowered.

## The consequence of low statistical power

As we have previously discussed, studies in neuroscience are often underpowered. That is to say, they have beta values much greater than 20%. In fact, studies can have $\beta$ values as high as 80%. These studies would therefore have 20% statistical power (i.e., if there is a true effect to be found, a study would have a 20% chance of finding it).

If we continue with our previous example, but now have $\beta = 80%$, we see that the majority of hypotheses associated with statistically significant effects are in fact false positives.

$FPRP = 100 * (falsePositive / (falsePositive + truePositive))$

$FPRP = 100 * (45 / (45 + 20))$

$FPRP = 69\%$

With beta = 80%, 20% of the hypotheses that are true will have p-values below 0.05. These are true positives. This also means that 80% (!) of the hypotheses that are true will have p-values above 0.05; these are false negatives. With 100 true hypotheses, only 20 of them will be found to be statistically significant. When we also consider that we will also discover 45 false positive effects (5% of 900), the false-positive report probability will be 69%

The effect of low statistical power on the the false positive report probability is obviously less for intermediate values. For example, a study with with a statistical power of 40% (i.e., $\beta = 60%$) is associated with a false-positive report probability of 53%.

## How the likelihood of a hypothesis effects the chance of false-positive findings

### Many true effects to be discovered

The previous example of 1000 hypotheses considered a typical situation where true effects are unlikely, only 10% were true. If we work in a field where half the studied hypotheses are in fact true, the chance of reporting a false positive decreases greatly.

The example below considers a field of research that is underpowered, with $\beta = 60%$. Despite being underpowered, the fact that 50% of the studied hypotheses are true greatly reduces the false positive report probability.

$FPRP = 100 * (falsePositive / (falsePositive + truePositive))$

$FPRP = 100 * (25 / (25 + 200))$

$FPRP = 11\%$

With beta = 60%, 40% of the hypotheses that are true will have p-values below 0.05. These are true positives. With only 10 true hypotheses, 4 of them will be found to be statistically significant. Given that alpha = 0.05, 5% of the hypotheses that are false will be statistically significant. This means there will be 50 false positives. Thus, the false-positive report probability will be 93%.

With beta = 60%, 40% of the hypotheses that are true will have p-values below 0.05. These are true positives. With 500 true hypotheses, 200 of them will be found to be statistically significant. Given that alpha = 0.05, 5% of the hypotheses that are false will be statistically significant. This means there will be 25 false-positives. Thus, the false-positive report probability will be 11%

### Few true effects to be discovered

In a field where the vast majority of hypotheses are false, the chance that a discovery is in fact a false positive is extremely high. Such fields are in search of extremely novel, and thus improbable results.

If we repeat the above example ($\beta = 60%$ $\alpha = 0.05$), but this time with only 1% of hypotheses being true, the false positive discovery rate is 93%.

$FPRP = 100 * (falsePositive / (falsePositive + truePositive))$

$FPRP = 100 * (50 / (50 + 4))$

$FPRP = 93\%$

## Summary

If science followed common statistical guidelines, the threshold for statistical significance would be set to $\alpha = 0.05$ and $\beta = 20%$ (i.e, 80% power). Based on the Ioannidis argument, these guidelines will result in 36% of results reported as statistically significant to be false positives when conducting research in a field that values novelty (i.e., 10% of studied hypotheses are true).

The Ioannidis argument also highlights the influence of underpowered studies and the search for novel but unlikely discoveries on the likelihood that a statistically significant result is in fact true.

Since it was initially published, numerous researchers have pursued the line of thinking presented by Ioannidis (2005). An important concept that we will visit in our next post is the influence of p-hacking and research flexibility on the chance that a statistically significant result is in fact false.

### Reference

Forstmeier W, Wagenmakers EJ, Parker TH (2016. Detecting and avoiding likely false-positive findings – a practical guide. Biol Rev Camb Philos Soc. doi:10.1111/brv.12315 [Epub ahead of print].

Ioannidis JP (2005). Why most published research findings are false.. PLoS Med 2:e124.