If the p-value is < .05, then the probability of falsely rejecting the null hypothesis is <5%, right? That means, a maximum of 5% of all significant results is a false-positive (that’s what we control with the α rate).

Well, no. As you will see in a minute, the “false discovery rate” (aka. false-positive rate), which indicates the probability that a significant p-value actually is a false-positive, usually is much higher than 5%.

## A common misconception about p-values

Oates (1986) asked the following question to students and senior scientists:

The answer is “false” (you will learn why it’s false below). But 86% of all professors and lecturers in the sample who were teaching statistics (!) answered this question erroneously with “true”. Gigerenzer, Kraus, and Vitouch replicated this result in 2000 in a German sample (here, the “statistics lecturer” category had 73% wrong). Hence, it is a wide-spread error to confuse the p-value with the false discovery rate.

## The False Discovery Rate (FDR) and the Positive Predictive Value (PPV)

To answer the question “What’s the probability that a significant p-value indicates a true effect?”, we have to look at the positive predictive value (PPV) of a significant p-value. The PPV indicates the proportion of significant p-values which indicate a real effect amongst all significant p-values. Put in other words: Given that a p-value is significant: What is the probability (in a frequentist sense) that it stems from a real effect?

(The false discovery rate simply is 1-PPV: the probability that a significant p-value stems from a population with null effect).

That is, we are interested in a conditional probability Prob(effect is real | p-value is significant).
Inspired by Colquhoun (2014) one can visualize this conditional probability in the form of a tree-diagram (see below). Let’s assume, we carry out 1000 experiments for 1000 different research questions. We now have to make a couple of prior assumptions (which you can make differently in the app we provide below). For now, we assume that 30% of all studies have a real effect and the statistical test used has a power of 35% with an α level set to 5%. That is of the 1000 experiments, 300 investigate a real effect, and 700 a null effect. Of the 300 true effects, 0.35300 = 105 are detected, the remaining 195 effects are non-significant false-negatives. On the other branch of 700 null effects, 0.05700 = 35 p-values are significant by chance (false positives) and 665 are non-significant (true negatives).
This path is visualized here (completely inspired by Colquhoun, 2014): Now we can compute the false discovery rate (FDR): 35 of (35+105) = 140 significant p-values actually come from a null effect. That means, 35/140 = 25% of all significant p-values do not indicate a real effect! That is much more than the alleged 5% level.

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