Re: A very small p-value

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)
>>>>>
>>>>> ______________________________________________
>>>>> [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.
>>>> ______________________________________________
>>>> [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
>>> 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.
>>>
>>> ______________________________________________
>>> [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.
>>
>> ______________________________________________
>> [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.