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.test(a,b) 

I get: t = -64.6472, df = 18.998, p-value < 2.2e-16. I know from the comments that 2.2e-16 is the value of .Machine$double.eps – the smallest floating point number such that 1 + x != 1, but of course R can represent numbers much smaller than that. I know also from the R FAQ that R has to round floats to 53 binary digits accuracy: R FAQ.

A few questions: (1) am I correct in reading that as 53 binary digits of precision or are values in R < .Machine$double.eps not calculated accurately? (2) Why, when doing such calculations does R not provide a means to display a smaller value for the p-value, even with some loss of precision? (3) Is there a way to display a smaller p-value, even if I lose some precision? For a single test 2 decimal significant figures would be fine, for values I am going to Bonferroni correct I’ll need more. When I say “lose some precision” I think < 53 binary digits, but (4) am I completely mistaken and any p-value < .Machine$double.eps is wildly inaccurate? (5) Is R just being honest and other stats packages are not?

In my field very small p-values are the norm, some examples: http://www.ncbi.nlm.nih.gov/pubmed/20154341, http://www.plosgenetics.org/article/info%3Adoi%2F10.1371%2Fjournal.pgen.1002215 and this is why I want to represent such small p-values.

Thanks for your help, sorry for such a tortuous question.

Answer 1

Try something like this t.test(a,b)$p.value see if that gives you the accuracy you need. I believe it has more to do with the printing of the result than it does the actual stored computer value which should have the necessary precision.