p-value < 2.2e-16

A claim: 2.2e-16 is the most popular p-value in research papers, even more popular than 0.05 (or if you’re being cynical 0.049).

Why?

2.2e-16 happens to be the epsilon of a double-precision float (i.e. a decimal number stored using 64 bits). Roughly, this means that if you try to calculate 1 - epsilon, with anything smaller than epsilon, the answer will be 1.

In R, you can calculate this by running the following code (+2 due to convention):

i <- 1
while (1-2^(-i) != 1) {i <- i + 1}
2^(-i + 2)

Which you’ll find is the same as the built in constant .Machine$double.eps (which is 2.2e-16).

Note that if you are trying to add 1e-17 to 1 you won’t get anything useful, but adding numbers with more similar exponents everything works just fine.

> 2.2e-16 + 1.3e-17
[1] 2.33e-16

In fact, doubles can store between 2^-1023 and 2^1023, around 1e308, so why are they being truncated at 2.2e-16?

R stores all of its decimal numbers in the double format, and can be quite stubborn about not reporting them below this upper bound on rounding error. You can happily compare, add and subtract values of this 2.2e-16 magnitude, but if you started working with values out of this range you might run into problems. So R truncates it for you.

With datasets of even modest size, it’s easy to get p-values below this enforced bound:

> chisq.test(c(80, 1))
    Chi-squared test for given probabilities 
data: c(80, 1) 
X-squared = 77.049, df = 1, p-value < 2.2e-16

So what should you do? Firstly, I’d start by asking whether the p-value at this level is actually meaningful. Is the model you’ve used to calculate it accurate enough to be this confident about the p calculation? Can we really make any meaningful interpretation of what say p < 1E-10 means (the rarest event I know of that has happened is around p < 1E-12). In this case, rather than reporting 2.2e-16, I’d just round your value and use something like p < 1E-10 (or p « 1E-10).

Sometimes this order of magnitude actually is needed. When you have a huge number of tests and apply a correction. To compare p-values from genome-wide association studies, the log(p-value) is very useful. R, being a sneaky so-and-so, actually has calculated the value without rounding, and you can usually get it by indexing into p.value in either the summary() or print() return of your test

print(chisq.test(c(80,1)))$p.value
[1] 1.667363e-18