Back in 2010, researcher Daniel Daryl Bem at Cornell University supposedly proved that psychic precognition exists. He did a variety of tests on college students in which he reversed some common psychological tests and found that the students were able to predict something that had not yet happened. For example, he put pornographic photos behind one of two curtains. The students had to guess which curtain the photo was behind before seeing the answer and they were correct a statistically significant amount of the time. When statisticians use the term “statistically significant” they are typically referring to the p-value. If the study had a p-value cut-off of 5%, that would mean that if students were truly guessing randomly and did not have any psychic powers, they would still “pass” the test 5% of the time just by coincidence. So, supposing no bias, intentional or otherwise, answer the following question:

If Bem’s study on psychic powers had a p-value of 5%, what is the probability that the students were able to “pass” the test merely by coincidence rather than a true psychic ability?

My guess is that you probably answered this question with “Uh…Jamie, you already told us. There would be a 5% probability that the students passed the psychic test by coincidence.” Now, let me ask you a slightly different version of the same question:

Do you think it is more likely that the student’s in Bem’s study are truly psychic or that they just got really lucky?

Since you are currently reading Skepchick, you’re probably a skeptic and probably do not believe in psychic powers, so you likely answered this question by saying that you believe that the students in Bem’s study are not actually psychic. However, this means that you think it’s more likely that the students fell into the 1 in 20 tests that would pass by coincidence rather than that they are actually psychic. In other words, you believe there is an over 50% chance that the students are not psychic even though you previously said that according to the p-value there was only a 5% chance of the study being wrong. Regardless of what the p-value says, you know that the chance of there being real psychic powers is extremely unlikely based on the fact that there is no previous evidence of psychic powers. The problem with p-value is that it doesn’t take any past evidence into consideration.

So, if using a basic p-value for statistical significance of such an unlikely phenomena as psychic powers is flawed, then what can we use? Well, this is where Bayes’ Theorem comes in. You see, Bayes’ Theorem is a way of considering our prior knowledge in our calculation of probability. Using Bayes’ Theorem we can calculate the probability that the students in Bem’s study are really psychic or just got lucky in their guesses, while considering prior evidence as well as the new evidence. Lucky for us Bayes’ Theorem has a simple formula: A formula is not very intuitive though, so let’s just ignore that for now because an equation is not needed for actually understanding Bayes’ Theorem. Instead of looking at a formula, let’s forget about Bem and his psychic students for a second and instead talk through the following scenario. One of our newest contributors here at Skepchick is Courtney Caldwell. The posts she’s done so far are pretty good, but what if she’s not who she says she is? What if she’s actually a spy! I mean, she says she a skeptic and a feminist and all that, but can we really know? She could have joined Skepchick just to spy on us for …. ummm….. whatever, look the point is that we just don’t know that she’s not a spy.

You see, it turns out that in this data that is totally 100% true and not something I’m just making up for purposes of this scenario, 15% of new writers at Skepchick turn out to be spies. To determine if Courtney is one of them, we’ll give her a polygraph test. Polygraphs are correct in determining lying versus truth-telling 80% of the time. Courtney says she is not a spy during the polygraph, but the polygraph tells us she is lying! So, what is the probability that Courtney is a real spy?

If polygraphs are right 80% of the time and the polygraph says Courtney is a spy, does that mean there’s an 80% chance she is an actual spy? How does the 15% probability of new Skepchicks being spies factor in? To answer this, let’s step back for a second and consider a theoretical scenario where we have 100 new Skepchicks. The following box represents 100 new Skepchicks. The blue skepchicks are the ones telling the truth while the purple ones are the secret spies (15% — but not to scale in the image just to make it clearer).

We don’t know which people fall in the purple box or which in the blue, so we give them all a polygraph. They all claim they are not spies. The polygraph is correct 80% of the time, so out of our spies, 80% of them will be correctly identified as spies, while 20% will be misidentified as telling the truth even when they are not. In the following box, you can see that the 80% of the true spies that the polygraph labels as spies are represented in a red box and the 20% of the true spies that the polygraph has mislabeled as not spies are in the green box.

However, the spies aren’t the only ones that took the polygraph. All the new Skepchicks that were telling the truth about not being a spy also took the test. Because the polygraph is correct 80% of the time, 80% of the true Skepchicks were correctly labeled as not being spies, whereas 20% of them were misidentified as spies. The following box adds the polygraph results for the truth-tellers. Those in the green boxes were identified as telling the truth by the polygraph whereas those in the red boxes were identified as being lying liars.

However, we don’t know which Skepchicks are actual spies versus those that are telling the truth. All we know are their polygraph results. When considering the above box, all we can see is whether a person falls in the green box (pass polygraph) or red box (fail polygraph). For the sake of clarity, let’s label each of those boxes with a letter to make it easier to identify which box I am talking about at any given moment.

Let’s consider Courtney again. Courtney failed the polygraph, so what is the probability she is a real spy?

Considering the above boxes, you can see that Courtney falls only in one of the bottom two red boxes, boxes C or D, with her failed polygraph. We know that she is either a spy correctly identified by the polygraph (box C) or she is telling the truth but received a false positive on the test (box D). We don’t need to consider box A or B at all because we already know she failed the polygraph. Another way to phrase our question then would be: Given that we know Courtney failed the polygraph, what is the probability that she is in box C versus box D? Or, you can think of it as: C/(C+D) = probability Courtney is a spy!

In fact, we already have enough information to calculate the size of each box.

Box C is the probability that Courtney is a spy (15%) multiplied by the probability the polygraph is correct (80%): 15% * 80% = 12%

Box D is the probability that Courtney is not a spy (85%) multiplied by the probability that polygraph is wrong (20%): 85% * 20% = 17%

We can now easily get C/(C+D) –> 12%/(12%+17%) = 41.4% and…HOLY SHIT YOU GUYS WE JUST DERIVED THE FORMULA FOR BAYES’ THEOREM! Seriously, this complicated thing, which is the formula from before modified to incorporate our spy scenario:

Just means that the probability that Courtney is a true spy given that she failed the polygraph is equal to the probability that the polygraph says she is a spy and she is a spy divided by the probability that the polygraph says Courtney is a spy regardless of whether she is or not. It’s just C/(C+D) in our box figure. In this version of the formula, I labeled each section to show how it relates to our box: So, what does this mean exactly and how does it relate to Bem’s psychic research? The probability of Courtney being a spy pre-polygraph was 15%, but since we know she failed the polygraph the probability she was a spy went up to 41.4%. That means that even though the polygraph says that Courtney is a spy and the polygraph is 80% accurate, it is still more likely that Courtney is not a spy. In fact, there is a 58.6% chance that Courtney is not a spy! Yay!

You can apply Bayes’ Theorem to any type of “test” where a true positive result is quite rare. For example, the fact that breast cancer is so rare is why if you get a positive result from a mammogram you still have a pretty big chance of not having breast cancer even if the test if fairly accurate. It’s also why the World Anti-Doping Association’s protocol requires athletes taking tests for doping to take a second different test if they get a positive result on the first one, rather than just trusting one test. Psychic abilities being real are almost infinitely more unlikely than breast cancer or doping so even though Bem’s research passed a 5% p-value threshold, it’s more likely that the test was wrong than not.

In fact, now that we know the formula for Bayes’ Theorem we can calculate it. Let’s say the chance of psychic powers being real was 1%, which is frankly a lot higher than the lack of evidence for such a phenomena would suggest. Let’s further say that Bem’s test has a 5% p-value, which is close to what he claims to have had on some of his tests. The p-value may suggest that there is a 95% chance that Bem’s students passed the test because they are truly psychic, but we can use Bayes’ Theorem to consider the previous evidence regarding psychic powers.

Going back to our boxes, we want to calculate the values of C and D.

Box C = The probability psychic powers are real (1%) multiplied by the probability Bem’s test is correct (95%) = 95% * 1% = 0.95%

Box D = The probability psychic powers are not real (99%) multiplied by the probability Bem’s test is wrong (5%) = 5% * 99% = 4.95%

Baye’s Theorem = C / (C+D) = 0.95%/(0.95%+4.95%) = 16.1%

In other words, even in extremely favorable conditions where the chance of psychic powers existing is 1% and Bem had a 5% p-value, the probability that psychic powers are real is still only a hair above 16%. The probability that psychic powers exist is much, much lower than 1% and none of this rules out the rather high probability of there being some sort of bias in the test protocols. It’s extremely likely that the positive result he got on his research is not a real one and in fact future researchers have been unable to replicate his results. Bem is certainly not the only researcher to “prove” a phenomena that does not actually exist. This is why many science-based researchers and skeptics have argued that Bayes’ Theorem should be used when considering whether results are statistically significant in research involving paranormal phenomena that are extremely unlikely to exist. Next time a study like Bem’s is released and you read articles asking why p-value was used instead of Bayes’ Theorem, you’ll know exactly what they are talking about.