[R] A very small p-value

[Michael Dewey]

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 <bbolker using gmail.com> wrote:
>>
>>   One possible source of confusion is that the `print.Coefmat` function uses .Machine$double.eps as its threshold for printing "< [minimum value]" rather than the precise computed p-value (presumably on the grounds that a number smaller than this is likely to be unrealistic as an accurate statement of the unlikeliness of an outcome in the real world).
>>
>> On 10/26/25 10:41, Richard O'Keefe wrote:
>>> No, 0 and 5-19 are not "equalled".  THey are quite distinct.
>>> As for pt() returning something smaller than double.eps, why wouldn't it?
>>> If I calculate 10^-30, I get 1e-30, which is much smaller than  double.eps,
>>> but is still correct.  It would be a serious error to return 0 for 10^-30.
>>> Welcome to the wonderful world of floating-point arithmetic.
>>> This really has nothing to do with R.
>>> On Sun, 26 Oct 2025 at 09:38, Christophe Dutang <dutangc using gmail.com> wrote:
>>>>
>>>> Thanks for your answers.
>>>>
>>>> I was not aware of the R function expm1().
>>>>
>>>> I’m completely aware that 1 == 1 - 5e-19. But I was wondering why pt() returns something smaller than double.eps.
>>>>
>>>> For students who will use this exercise, it is disturbing to find 0 or 5e-19 : yet it will be a good exercise to find that these quantities are equalled.
>>>>
>>>> Regards, Christophe
>>>>
>>>>> Le 25 oct. 2025 à 12:14, Ivan Krylov <ikrylov using disroot.org> a écrit :
>>>>>
>>>>> В Sat, 25 Oct 2025 11:45:42 +0200
>>>>> Christophe Dutang <dutangc using gmail.com> пишет:
>>>>>
>>>>>> Indeed, the p-value is lower than the epsilon machine
>>>>>>
>>>>>>> pt(t_score, df = n-2, lower=FALSE) < .Machine$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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>>>> and provide commented, minimal, self-contained, reproducible code.
>>> ______________________________________________
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>>
>> -- 
>> 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.
>>
>> ______________________________________________
>> R-help using r-project.org 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.
> 
> ______________________________________________
> R-help using r-project.org 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.

-- 
Michael Dewey