— What’s a p-value? —

If you run a study, then (all else equal, aside from rare edge cases) the lower the p-value, the lower the chance that your results are due to random chance or luck.

More precisely: a p-value is the probability you’d get a result at least as extreme as what you got IF there were actually no effect (or if some other pre-specified “null hypothesis” is true).

So it’s a probability calculated based on assuming that there is no effect (or assuming that a pre-specified “null hypothesis” is true). Here the phrase “no effect” would mean, in the case of a study on a new medicine, that the medicine doesn’t do anything.

To put it in terms of coin flips: suppose you’re trying to decide if a coin is fair (i.e., if it has an equal chance of landing on heads and tails – so that’s your “null hypothesis” in this context). You flip the coin 100 times and get 60 heads. You calculate the p-value (p=0.06).

This p-value tells you there’s a 6% chance you’d get 60 or more heads OR 60 or more tails out of 100 flips if the coin were actually fair.

What makes p-values useful is that when they are high, you usually can’t rule out your effect being due to random chance or luck. And, when they are very low, random chance is (in most cases) unlikely to be the explanation for your result.

— What’s the problem with p-values? —

In social science, p<0.05 is often used as the cutoff for a “successful” result (i.e., they treat the effect as real and potentially publishable). This is an arbitrary cutoff; there’s nothing special about 0.05. The phrase “statistically significant” is defined simply to mean that p<0.05.

There are many ways that p-values get commonly misused, creating lots of problems. For instance:

• p-values often get misinterpreted as the probability that an effect is not real (recall: p-values are actually the probability of getting a result this extreme if there is no effect, which is not the same thing)

• If you see one study where the main finding’s p-value is, say, 0.05, and another study where the main finding’s p-value is, say, 0.01, it’s tempting to conclude that the finding of the 2nd study is much less likely to be the result of chance (e.g., 1/5th as likely) than the 1st study’s finding. Unfortunately, we can’t draw this conclusion. The probability that a study’s finding is the result of chance is not the same as the p-value, and in fact, it can’t even be calculated just by knowing the p-value.

• Because a p-value threshold is often used for a result to be publishable (p<0.05 in social science), researchers sometimes engage in fishy methods to get their p-values below the threshold. This is known as “p-hacking.:

• A result’s p-value (or “statistical significance”) is sometimes focused on instead of focusing on other factors that are also important. For instance, a result may have a low p-value but be such a weak effect that it’s totally useless or uninteresting.

• While a low p-value helps you rule out the possibility that your effect is merely due to random chance, unfortunately, that’s all it helps you with. But researchers sometimes act as though it tells them more than that. Even an extremely low p-value doesn’t mean an effect is “real” or that the effect means what you think. Low p-values can result from a variety of causes, including mistakes in experimental design or confounds.

Here’s another way to think about what a p-value is and isn’t that some people find helpful: a p-value does not tell you the probability that your result is due to chance. It tells you how consistent your results are with being due to chance. (I’m paraphrasing from here.) So, the lower the p-value, the less consistent your results are with them being due to chance.

It’s interesting to note that, empirically, results with lower p-values are more likely to be genuine effects (i.e., not false positives). I looked at results for 325 psychology study replications, and when the original study p-value was at most 0.01, about 72% replicated. When p>0.01, only 48% did.

Ultimately, p-values are a useful (though often abused) statistical tool.

— BONUS APPENDIX: what’s the chance of a hypothesis being “true” if p<0.05? —

One annoying thing about p-values is that they don’t answer the question we are usually interested in. Usually, we want to know something like “What’s the probability that my hypothesis is true?” or “What’s the probability that the effect of this drug is bigger than X?” but p-values don’t tell us those things.

However, we can put a different spin on p-values to get them to answer questions that are closer to what we’re really interested in. Let’s think of p-values as giving us a decision procedure (in an overly simplified world where you either “believe” in an effect or you fail to believe in it). 

Suppose you test 100 totally separate, previously unexplored hypotheses about humans, and suppose that you commit to “believe” a hypothesis is true if and only if you get p<0.05 (and otherwise, you don’t believe it).

I think it’s realistic that in a social science context, most hypotheses studied will be false since discovering novel, publishable hypotheses about humans is hard. So let’s suppose that 80% of the hypotheses you test are *not* true. 

Finally, suppose that you use a large enough number of participants in your studies so that if you are testing for the presence of a real effect, there is an 80% chance you’ll be able to find it (this 80% figure is a common recommendation for “statistical power”). 

Under these assumptions, if you test 100 hypotheses, then you will end up believing in 20 hypotheses, and 80% of those you believe will be true (with the other 20% being false positives). That means that of the results you believe in, 80% will be correct! Of course, this assumes no mistakes are made in the process of designing the experiment, running the statistics, and so on.

Here’s how the math works out if you’re curious:

• Out of the 100 hypotheses, 20 will be true, and of those, you’ll believe 16 = 0.80 * 20 (these are the true positives) and fail to believe 4 (these are the false negatives).

• Out of the 100 hypotheses, 80 will be false, and of those, you’ll believe 4 = 0.05 * 80 (these are the false positives), and you’ll reject 76 (these are the true negatives).

Of course, if the numbers here had been different, the conclusions would be different as well. For instance, imagine if you started with 2000 hypotheses, and this time, imagine that only 1% of them were true. If the power was still 80%, then:

 • Out of the 2000 hypotheses, 20 of them would be true, and of those, you’d believe 16 (0.80 * 20) of them (these are true positives) and fail to believe 4 of them (these are false negatives).

• Out of the 2000 hypotheses, 1980 would be false, and of those, you’d believe 99 (0.05*1980) of them (these are false positives), and you’d reject the other 1881 of them (these are true negatives).

• So, altogether, you’d believe 115 (16 + 99) hypotheses, of which only 16 would’ve actually been true, so of the results you believe in, less than 14% would be correct! 

From analyses like these, we can see that the probability that a specific hypothesis is true, given that we’ve found p<0.05, depends on a variety of factors, including the sample size, the true effect size, the base rate probability that a new hypothesis tested by that researcher is true, the probability of errors being made in the experimental design or statistical analysis, and so on.

In real life:

(1) Studies often don’t use large enough numbers of participants (and so are underpowered).

(2) Researchers sometimes engage in p-hacking to artificially lower their p-values to help their papers get published.

(3) Researchers often don’t carefully track how many hypotheses they’ve really tested.

(4) The decision procedure described above is often not adhered to so strictly (e.g., a result of p=0.08 might be treated as suggestive evidence for the hypothesis, and hence the hypothesis is not rejected).

(5) Real hypotheses often have auxiliary assumptions beyond what the p-value accounts for (such as an assumption that there is a lack of confounders, a lack of serious errors in the experimental setup, and so on).

I personally don’t like thinking in terms of this decision procedure for p-values because I believe that modeling hypotheses as “true” or “false” is not a good approach to thinking clearly. This is because I believe it’s usually much better to think in terms of probabilities rather than a “true”/”false” dichotomy when trying to understand the answers to complex questions.

Some people have argued that we should switch to a Bayesian approach to hypothesis testing since such an approach avoids many of the issues of p-values (including avoiding the problematic “true”/”false” dichotomy). But it also introduces other challenges, such as how to come up with an appropriate “prior” (which represents one’s belief about the probability of the hypothesis having different strengths of effects prior to seeing the study results).

This piece was first written on December 31, 2022, and first appeared on this site on April 2, 2023.

If you read this line, please do us a favor and click here to answer one quick question.

You can make inferences using…

(1) Deduction:

As a consequence of the definition of X and Y, if X then Y.

X applies to this case. Therefore Y.

“Plato is a man, and all men are mortal; therefore Plato is mortal.”

“For any number that is an integer, there exists another integer greater than that number. 1,000,000 is an integer. So there exists an integer greater than 1,000,000.”

Especially popular among philosophers and mathematicians?

Flaws: to apply to the world, you need to add in assumptions about the world, or to apply other methods of inference on top.

(2) Frequencies:

In the past, 95% of the time that X occurred, Y occurred.

X occurred. Therefore Y (with high probability).

“95% of the time when we saw a transaction identical to this one, it was fraudulent. So this transaction is fraudulent.”

Especially popular among applied statisticians and data scientists?

Flaws: You need to have a moderately large number of examples like the current one to perform calculations on, and the method assumes that those past examples were drawn from a process that is (statistically) just like the one that generated this latest example. Moreover, sometimes it is unclear what it means for “X” to have occurred. What if it’s something that’s very similar to but not quite like X that occurred – should that be counted? If we broaden our class of what counts or change to another class that still encompasses all of our prior examples, we’ll potentially get a different answer. Though, fortunately, there are plenty of cases where the class to use is fairly obvious.

(3) Models:

Given our probabilistic model of this thing, when X occurs, the probability of Y is 0.95.

X occurred. Therefore Y (with high probability).

“Given our multivariate Gaussian model of loan prices, when this loan defaults, there is a 0.95 probability of this other loan defaulting.”

Especially popular among financial engineers and risk modelers?

Flaws: hinges on the appropriateness of the parameterized probabilistic model chosen, may require a moderately large amount of past data to estimate free model parameters, and may go haywire if modeling assumptions are suddenly violated.

(4) Regression:

In prior data, as X and Z increased, the likelihood of Y increased.

X and Z are at high levels. Therefore Y.

“Height for children can be approximately predicted as an (increasing/positive) linear function of age and weight. This child is older and heavier than the others, so we predict he is also taller than the others.”

Especially common among economists and data scientists?

Flaws: often is applied with simple assumptions (e.g., linearity) that may not capture the complexity of the inference, but very large amounts of data may be needed to apply much more complex models (e.g., to use neural networks).

(5) Bayesianism:

Given my prior odds on Y being true…

And given evidence X…

And given my Bayes factor, which is my estimate of how much more likely X is to occur if Y is true than if Y is not true…

I calculate that Y is far more likely to be true than to not be true (by multiplying the prior odds by the Bayes factor to get the posterior odds).

Therefore Y (with high probability).

“My prior odds that my boss is angry at me were 1 to 4, because he’s angry at me about 20% of the time. But then he came into my office shouting and flipped over my desk, which I estimate is 200 times more likely to occur if he’s angry at me compared to if he’s not. So now the odds of him being angry at me are 200 * (1/4) = 50 to 1 in favor of him being angry.”

Not as popular as it should be?

Flaws: it is sometimes hard to know what to set your prior odds at, and it can be very hard in some cases to perform the calculation. In practice, carrying out the calculation might end up relying on subjective estimates of the odds, which can be especially tricky to guess when the evidence is not binary (i.e., not of the form “happened” vs. “didn’t happened”), or if you have lots of different pieces of evidence that are partially correlated. On the other hand, if you can do the calculations in a given instance, and have a sensible way to set a prior, this is, in my opinion, the mathematically optimal framework to use for probabilistic prediction. In that sense, we can think of many of the other approaches on this list as (hopefully pragmatic) approximations of Bayesianism (sometimes good approximations, sometimes bad ones).

(6) Theories:

Given our theory, when X occurs, Y occurs.

X occurred. Therefore Y.

“One theory is that depressed people are most at risk for suicide when they are beginning to come out of a really bad depressive episode. So as depression is remitting, patients should be carefully screened for potentially increasing risk factors.”

“When inflation rises, unemployment falls. Inflation is rising, so unemployment will fall.”

Especially popular among psychologists and economists?

Flaws: it’s very challenging to come up with reliable theories, and often you will not know how accurate such a theory is. Even if it has substantial truth to it and is often right, there may be cases where the opposite of what was predicted actually happens, and for reasons that the theory can’t explain.

(7) Causes:

We know that X causes Y to occur.

X occurred. Therefore Y.

“Rusting of gears causes increased friction, leading to greater wearing down. In this case, the gears were heavily rusted, so we expect to find a lot of wearing down.”

“This gene produces this phenotype, and we see that this gene is present, so we expect to see the phenotype.”

Especially popular among engineers and biologists?

Flaws: it’s often extremely hard to figure out causality in a highly complex system, especially in “softer” subjects like nutrition.

(8) Experts:

This expert (or prediction market, or prediction algorithm) X is 90% accurate at predicting things in this general domain of prediction.

X predicts Y. Therefore Y (with high probability).

“This prediction market has been right 90% of the time when predicting recent baseball outcomes, and in this case, they predict that the Yankees will win.”

Not as popular as it should be?

Flaws: you often don’t have access to the predictions of experts (or of prediction markets or prediction algorithms), and when you do, you usually don’t have reliable measures of their past accuracy.

(9) Metaphors:

X, which is what we are dealing with now, is metaphorically a Z.

For Z, when W is true, then obviously Y.

Now W (or its metaphorical equivalent) is true for X. Therefore Y.

“Your life is but a boat, and you are riding on the waves of your experiences. When a raging storm hits, a boat can’t be under full sail. It can’t continue at its maximum speed. You are experiencing a storm now, and so you too must learn to slow down.”

Especially popular among self-help gurus and some ancient philosophers?

Flaws: Z working as a metaphor for X doesn’t mean that all (or even most) solutions that are good for situations involving Z are appropriate (or even make any sense) for X.

(10) Similarities:

X occurred, and X is very similar to Z in properties A, B, and C.

When things similar to Z in properties A, B, and C occur, Y usually occurs.

Therefore Y (with high probability).

“This conflict is similar to the Gulf war in that…and with “Gulf”-like wars, we have always seen that…”

“This data point (with unknown label) is closest in feature space to this other data point which is labeled ‘cat,’ and all the other labeled points around that point are also labeled ‘cat,’ so this unlabeled point should also likely get the label ‘cat.’”

Especially popular among historians and within some machine learning algorithms?

Flaws: in the history case, it is difficult to know which features are the appropriate ones to use to compare similarities, and often the conclusions are based on a relatively small number of examples. In the machine learning case, a very large amount of data may be needed to train the model.

(11) Cases:

In this handful of examples (or perhaps even just one example) where X occurred, Y occurred.

X occurred. Therefore Y.

“The last time we elected a [insert political group you don’t like] as president, we saw what happened. Let’s not make that mistake again.”

“The last three times I went to action movies, I didn’t like them. So I don’t want to go to one again.”

Especially popular with politicians and with nearly everyone in daily living?

Flaws: unless we are in a situation with very little noise/variability, a few examples likely will not be enough to accurately generalize from.

(12) Intuition:

X occurred. My intuition (that I may have trouble explaining) predicts that when X occurs, Y is true. Therefore Y.

“The tone of voice he used when he talked about his family gave me a bad vibe. My feeling is that anyone who talks about their family with that tone of voice probably does not really love them.”

Popular with nearly everyone in daily living?

Flaws: our intuitions can be very well-honed in situations we’ve encountered many times and that we received feedback on (i.e., where there was some sort of answer we got about how well our intuition performed), but in highly novel situations or in situations where we receive no feedback on how well our intuition is performing, our intuitions may be highly inaccurate (even though we may not FEEL any less confident about our correctness).

This essay was first written on October 7, 2018, and first appeared on this site on December 31, 2021.