“If you torture the data long enough, it will confess.”
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%.
Oakes (1986) asked the following question to students and senior scientists:
You have a p-value of .01. Is the following statement true, or false?
You know, if you decide to reject the null hypothesis, the probability that you are making the wrong decision.
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.
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, 0.35*300 = 105 are detected, the remaining 195 effects are non-significant false-negatives. On the other branch of 700 null effects, 0.05*700 = 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 (see also Lakens & Evers, 2014, and Ioannidis, 2005)
Together with Michael Zehetleitner I developed an interactive app that computes and visualizes these numbers. For the computations, you have to choose 4 parameters.
Let’s go through the settings!
Now, what is a good setting for the a priori proportion of true hypotheses? It’s certainly not near 100% – in this case only trivial and obvious research questions would be investigated, which is obviously not the case. On the other hand, the rate can definitely drop close to zero. For example, in pharmaceutical drug development “only one in every 5,000 compounds that makes it through lead development to the stage of pre-clinical development becomes an approved drug” (Wikipedia). Here, only 0.02% of all investigated hypotheses are true.
Furthermore, the number depends on the field – some fields are highly speculative and risky (i.e., they have a low prior probability), some fields are more cumulative and work mostly on variations of established effects (i.e., in these fields a higher prior probability can be expected).
But given that many journals in psychology exert a selection pressure towards novel, surprising, and counter-intuitive results (which a priori have a low probability of being true), I guess that the proportion is typically lower than 50%. My personal grand average gut estimate is around 25%.
That’s easy. The default α level usually is 5%, but you can play with the impact of stricter levels on the FDR!
The average power in psychology has been estimated at 35% (Bakker, van Dijk, & Wicherts, 2012). An median estimate for neuroscience is at only 21% (Button et al., 2013). Even worse, both estimates can be expected to be inflated, as they are based on the average published effect size, which almost certainly is overestimated due to the significance filter (Ioannidis, 2008). Hence, the average true power is most likely smaller. Let’s assume an estimate of 25%.
Finally, let’s add some realism to the computations. We know that researchers employ “researchers degrees of freedom”, aka. questionable research practices, to optimize their p-value, and to push a “nearly significant result” across the magic boundary. How many reported significant p-values would not have been significant without p-hacking? That is hard to tell, and probably also field dependent. Let’s assume that 15% of all studies are p-hacked, intentionally or unintentionally.
When these values are defined, the app computes the FDR and PPV and shows a visualization:
With these settings, only 39% of all significant studies are actually true!
Wait – what was the success rate of the Reproducibility Project: Psychology? 36% of replication projects found a significant effect in a direct replication attempt. Just a coincidence? Maybe. Maybe not.
The formula to compute the FDR and PPV are based on Ioannidis (2005: “Why most published research findings are false“). A related, but different approach, was proposed by David Colquhoun in his paper “An investigation of the false discovery rate and the misinterpretation of p-values” [open access]. He asks: “How should one interpret the observation of, say, p=0.047 in a single experiment?”. The Ioannidis approach implemented in the app, in contrast, asks: “What is the FDR in a set of studies with p <= .05 and a certain power, etc.?”. Both approaches make sense, but answer different questions.
Although the BF is a continuous measure of evidence, humans love verbal labels, categories, and benchmarks. Labels give interpretations of the objective index – and that is both the good and the bad about labels. The good thing is that these labels can facilitate communication (but see @richardmorey), and people just crave for verbal interpretations to guide their understanding of those “boring” raw numbers.
The bad thing about labels is that an interpretation should always be context dependent (Such as “30 min.” can be both a long time (train delay) or a short time (concert), as @CaAl said). But once a categorical system has been established, it’s no longer context dependent.
These labels can also be a dangerous tool, as they implicitly introduce cutoff values (“Hey, the BF jumped over the boundary of 3. It’s not anecdotal any more, it’s moderate evidence!”). But we do not want another sacred .05 criterion!; see also Andrew Gelman’s blog post and its critical comments. The strength of the BF is precisely its non-binary nature.
Several labels for paraphrasing the size of a BF have been suggested. The most common system seems to be the suggestion of Harold Jeffreys (1961):
|> 100||Extreme evidence for H1|
|30 – 100||Very strong evidence for H1|
|10 – 30||Strong evidence for H1|
|3 – 10||Moderate evidence for H1|
|1 – 3||Anecdotal evidence for H1|
|1/3 – 1||Anecdotal evidence for H0|
|1/3 – 1/10||Moderate evidence for H0|
|1/10 – 1/30||Strong evidence for H0|
|1/30 – 1/100||Very strong evidence for H0|
|< 1/100||Extreme evidence for H0|
Note: The original label for 3 < BF < 10 was “substantial evidence”. Lee and Wagenmakers (2013) changed it to “moderate”, as “substantial” already sounds too decisive. “Anecdotal” formerly was known as “Barely worth mentioning”.
Kass and Raftery suggested a comparable classification, only that the “strong evidence” category for them starts at BF > 20 (see also Wikipedia entry).
How much is a of 3.7? It indicates that data occured 3.7x more likely under than under , given the priors assumed in the model. Is that a lot of evidence for ? Or not?
Following Table 1, it can be labeled “moderate evidence” for an effect – whatever that means.
Some have argued that strong evidence, such as BFs > 10, are quite evident from eyeballing only:
“If your result needs a statistician then you should design a better experiment.” (attributed to Ernest Rutherford)
— Edward Tufte (@EdwardTufte) 13. Januar 2015
Is that really the case? Can we just “see” it when there is an effect?
Imagine the following scenario: When I give a present to my two boys (4 and 6 years old), it is not so important what it is. The most important thing is: “Is it fair?”. (And my boys are very sensitive detectors of unfairness).
Imagine you have bags with red and blue marbles. Obviously, the blue marbles are much better, so it is key to make sure that in each bag there is an equal number of red and blue marbles. Hence, for our familial harmony I should check whether reds and blues are distributed evenly or not. In statistical terms: : p = 0.5, : p != 0.5.
When drawing samples from the bags, the strongest evidence for an even distribution () is given when exactly the same number of red and blue marbles has been drawn. How much evidence for is it when I draw n=2, 1 red/1 blue? The answer is in Figure 1, upper table, first row: The is 0.86 in favor of , resp. a of 1.16 in favor of – i.e., anecdotal evidence for an equal distribution.
You can get these values easily with the famous BayesFactor package for R:
What if I had drawn two reds instead? Then the BF would be 1.14 in favor of (see Figure 1, lower table, row 1).
Obviously, with small sample sizes it’s not possible to generate strong evidence, neither for nor for . You need a minimal sample size to leave the region of “anecdotal evidence”. Figure 1 shows some examples how the BF gets more extreme with increasing sample size.
These visualizations indeed seem to indicate that for simple designs such as the urn model you do not really need a statistical test if your BF is > 10. You can just see it from looking at the data (although the “obviousness” is more pronounced for large BFs in small sample sizes).
The dotted lines in Figure 2 show the maximal and the minimal BF that can be obtained for a given number of drawn marbles. The minimum BF is obtained when the sample is maximally consistent with (i.e. when exactly the same number of red and blue marbles has been drawn), the maximal BF is obtained when only marbles from one color are drawn.
Figure 2 highlights two features:
Here’s a shiny widget that let’s you draw marbles from the urn. Monitor how the BF evolves as you sequentially add marbles to your sample!
When I teach sequential sampling and Bayes factors, I bring an actual bag with marbles (or candies of two colors).
In my typical setup I ask some volunteers to test whether the same amount of both colors is in the bag. (The bag of course has a cover so that they don’t see the marbles). They may sample as many marbles as they want, but each marble costs them 10 Cent (i.e., an efficiency criterium: Sample as much as necessary, but not too much!). They should think aloud, about when they have a first hunch, and when they are relatively sure about the presence or absence of an effect. I use a color mixture of 2:1 – in my experience this give a good chance to detect the difference, but it’s not too obvious (some teams stop sampling and conclude “no difference”).
This exercise typically reveals following insights (hopefully!)
The analysis so far seems to support the “interocular traumatic test”: “when the data are so compelling that conclusion hits you straight between the eyes” (attributed to Joseph Berkson; quoted from Wagenmakers, Verhagen, & Ly, 2014).
But the authors go on and quote Edwards et al. (1963, p. 217), who said: “…the enthusiast’s interocular trauma may be the skeptic’s random error. A little arithmetic to verify the extent of the trauma can yield great peace of mind for little cost.”.
In the next visualization we will see, that large Bayes factors are not always obvious.
What happens if we switch to group differences? European women have on average a self-reported height of 165.8 cm, European males of 177.9 cm – difference: 12.1 cm, pooled standard deviation is around 7 cm. (Source: European Community Household Panel; see Garcia, J., & Quintana-Domeque, C., 2007; based on ~50,000 participants born between 1970 and 1980). This translates to a Cohen’s d of 1.72.
Unfortunately, this source only contains self-reported heights, which can be subject to biases (males over-report their height on average). But it was the only source I found which also contains the standard deviations within sex. However, Meyer et al (2001) report a similar effect size of d = 1.8 for objectively measured heights.
Now look at this plot. Would you say the blue lines are obviously higher than the red ones?
I couldn’t say for sure. But the is 14.54, a “strong” evidence!
If we sort the lines by height the effect is more visible:
Again, you can play around with the interactive app:
To summarize: Whether a strong evidence “hits you between the eyes” depends on many things – the kind of test, the kind of visualization, the sample size. Sometimes a BF of 2.5 seems obvious, and sometimes it is hard to spot a BF>100 by eyeballing only. Overall, I’m glad that we have a numeric measure of strength of evidence and do not have to rely on eyeballing only.
Try it yourself – draw some marbles in the interactive app, or change the height difference between males and females, and calibrate your personal gut feeling with the resulting Bayes factor!