[R] A very small p-value

[Bert Gunter]

In my own experience, teeny tiny p-values were either markers of already
well known stuff or of errors somewhere -- often my own typos, etc. That
is, they never indicated anything of real scientific interest. Of course,
YMMV.

😟

-- Bert

On Sun, Oct 26, 2025 at 3:34 PM 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)
> >>
> >> ______________________________________________
> >> 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.
>
> --
> 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.
>

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