Why can't I get a p-value smaller than 2.2e-16?

Question

I\'ve found this issue with t-tests and chi-squared in R but I assume this issue applies generally to other tests. If I do:

a <- 1:10
b <- 100:110
t.t         


        

Answer 1

I'm puzzled by several things in the exchange of answers and comments here.

First of all, when I try the OP's original example I don't get a p value as small as the ones that are being debated here (several different 2.13.x versions and R-devel):

a <- 1:10
b <- 10:20
t.test(a,b)
## data:  a and b 
## t = -6.862, df = 18.998, p-value = 1.513e-06

Second, when I make the difference between groups much bigger, I do in fact get the results suggested by @eWizardII:

a <- 1:10
b <- 110:120
(t1 <- t.test(a,b))
# data:  a and b 
# t = -79.0935, df = 18.998, p-value < 2.2e-16
#
> t1$p.value
[1] 2.138461e-25

The behavior of the printed output in t.test is driven by its call to stats:::print.htest (which is also called by other statistical testing functions such as chisq.test, as noted by the OP), which in turn calls format.pval, which presents p values less than its value of eps (which is .Machine$double.eps by default) as < eps. I'm surprised to find myself disagreeing with such generally astute commenters ...

Finally, although it seems silly to worry about the precise value of a very small p value, the OP is correct that these values are often used as indices of strength of evidence in the bioinformatics literature -- for example, one might test 100,000 candidate genes and look at the distribution of resulting p values (search for "volcano plot" for one example of this sort of procedure).