p-values down to 10^(-287) are discussed here:
https://statmodeling.stat.columbia.edu/2018/12/02/p-value-4-76x10%E2%88%92264/
from
Ritchie, Stuart J, Simon R Cox, Xueyi Shen, et al. 2018. “Sex
Differences in the Adult Human Brain: Evidence from 5216 UK Biobank
Participants.” Cerebral Cortex 28 (8): 2959–75.
https://doi.org/10.1093/cercor/bhy109.
People regularly ask on Stack Overflow about what to do about
p-values that are too small to be represented in double-precision
floating-point (i.e. less than about 1e-308):
https://stackoverflow.com/a/46416222/190277
https://stackoverflow.com/q/40144267/190277
On 2025-10-28 9:26 a.m., Michael Dewey wrote:
On 28/10/2025 10:13, Peter Dalgaard wrote:
I suspect this is more like a relic from times when people would do
(say) 1 - pchisq(x,f) instead of pchisq(x, f, lower=FALSE) and
intended to avoid the embarrassment of printing 0 for things that
weren’t actually impossible.
People have been known to have unexpected uses for the tiny
probabilities (one case came from theoretical physics - I think it got
recorded as a fortune() entry) but rarely as low as 10^-16 in actual
significance testing. Things like whole genome scans may suggest some
hefty Bonferroni multipliers, but the numer of tests are not (yet?) in
the trillions (US).
A paper in Nature https://www.nature.com/articles/nature14151 by
stensola and colleagues (sorry there does not seem to be a DOI but it is
also vol 518, pages 207–212 (2015)) reports a p-value of Z = 405, P =
2.2 * 10^{−226} which is believed to be the current record. To give
credit where it is due this was posted in a comment by user amoeba on
CrossValidated.
Michael
- pd
On 26 Oct 2025, at 23.34, Ben Bolker <[email protected]> wrote:
One possible source of confusion is that the `print.Coefmat`
function uses .Machine$double.eps as its threshold for printing "<
[minimum value]" rather than the precise computed p-value (presumably
on the grounds that a number smaller than this is likely to be
unrealistic as an accurate statement of the unlikeliness of an
outcome in the real world).
On 10/26/25 10:41, Richard O'Keefe wrote:
No, 0 and 5-19 are not "equalled". THey are quite distinct.
As for pt() returning something smaller than double.eps, why
wouldn't it?
If I calculate 10^-30, I get 1e-30, which is much smaller than
double.eps,
but is still correct. It would be a serious error to return 0 for
10^-30.
Welcome to the wonderful world of floating-point arithmetic.
This really has nothing to do with R.
On Sun, 26 Oct 2025 at 09:38, Christophe Dutang <[email protected]>
wrote:
Thanks for your answers.
I was not aware of the R function expm1().
I’m completely aware that 1 == 1 - 5e-19. But I was wondering why
pt() returns something smaller than double.eps.
For students who will use this exercise, it is disturbing to find 0
or 5e-19 : yet it will be a good exercise to find that these
quantities are equalled.
Regards, Christophe
Le 25 oct. 2025 à 12:14, Ivan Krylov <[email protected]> a écrit :
В Sat, 25 Oct 2025 11:45:42 +0200
Christophe Dutang <[email protected]> пишет:
Indeed, the p-value is lower than the epsilon machine
pt(t_score, df = n-2, lower=FALSE) < .Machine$double.eps
[1] TRUE
Which means that for lower=TRUE, there will not be enough digits
in R's
numeric() type to represent the 5*10^-19 subtracted from 1 and
approximately 16 zeroes.
Instead, you can verify your answer by asking for the logarithm of
the
number that is too close to 1, thus retaining more significant
digits:
print(
-expm1(pt(t_score, df = n-2, lower=TRUE, log.p = TRUE)),
digits=16
)
# [1] 2.539746620181249e-19
print(pt(t_score, df = n-2, lower=FALSE), digits=16)
# [1] 2.539746620181248e-19
expm1(.) computes exp(.)-1 while retaining precision for numbers that
are too close to 0, for which exp() would otherwise return 1.
See the links in
https://cran.r-project.org/doc/FAQ/R-FAQ.html#Why-doesn_0027t-R-
think-these-numbers-are-equal_003f
for a more detailed explanation.
--
Best regards,
Ivan
(flipping the "days since referring to R FAQ 7.31" sign back to 0)
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______________________________________________
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PLEASE do read the posting guide https://www.R-project.org/posting-
guide.html
and provide commented, minimal, self-contained, reproducible code.
--
Dr. Benjamin Bolker
Professor, Mathematics & Statistics and Biology, McMaster University
Associate chair (graduate), Mathematics & Statistics
Director, School of Computational Science and Engineering
* E-mail is sent at my convenience; I don't expect replies outside of
working hours.
______________________________________________
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and provide commented, minimal, self-contained, reproducible code.
______________________________________________
[email protected] mailing list -- To UNSUBSCRIBE and more, see
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide https://www.R-project.org/posting-
guide.html
and provide commented, minimal, self-contained, reproducible code.
--
Dr. Benjamin Bolker
Professor, Mathematics & Statistics and Biology, McMaster University
Director, School of Computational Science and Engineering
> E-mail is sent at my convenience; I don't expect replies outside of
working hours.
______________________________________________
[email protected] mailing list -- To UNSUBSCRIBE and more, see
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide https://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.
