# [Rd] Calculation of e^{z^2/2} for a normal deviate z

Ben Bolker bbo|ker @end|ng |rom gm@||@com
Sun Jun 23 10:39:51 CEST 2019

```  I agree with many the sentiments about the wisdom of computing very
small p-values (although the example below may win some kind of a prize:
I've seen people talking about p-values of the order of 10^(-2000), but
never 10^(-(10^8)) !).  That said, there are a several tricks for
getting more reasonable sums of very small probabilities.  The first is
to scale the p-values by dividing the *largest* of the probabilities,
then do the (p/sum(p)) computation, then multiply the result (I'm sure
this is described/documented somewhere).  More generally, there are
methods for computing sums on the log scale, e.g.

https://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.misc.logsumexp.html

I don't know where this has been implemented in the R ecosystem, but
this sort of computation is the basis of the "Brobdingnag" package for
operating on very large ("Brobdingnagian") and very small
("Lilliputian") numbers.

On 2019-06-21 6:58 p.m., jing hua zhao wrote:
> Hi Peter, Rui, Chrstophe and Gabriel,
>
> Thanks for your inputs --  the use of qnorm(., log=TRUE) is a good point in line with pnorm with which we devised log(p)  as
>
> log(2) + pnorm(-abs(z), lower.tail = TRUE, log.p = TRUE)
>
> that could do really really well for large z compared to Rmpfr. Maybe I am asking too much since
>
> z <-20000
>> Rmpfr::format(2*pnorm(mpfr(-abs(z),100),lower.tail=TRUE,log.p=FALSE))
> [1] "1.660579603192917090365313727164e-86858901"
>
> already gives a rarely seen small p value. I gather I also need a multiple precision exp() and their sum since exp(z^2/2) is also a Bayes Factor so I  get log(x_i )/sum_i log(x_i) instead. To this point, I am obliged to clarify, see https://statgen.github.io/gwas-credible-sets/method/locuszoom-credible-sets.pdf.
>
> I agree many feel geneticists go to far with small p values which I would have difficulty to argue againston the other hand it is also expected to see these in a non-genetic context. For instance the Framingham study was established in 1948 just got \$34m for six years on phenotypewide association which we would be interesting to see.
>
> Best wishes,
>
>
> Jing Hua
>
>
> ________________________________
> From: peter dalgaard <pdalgd using gmail.com>
> Sent: 21 June 2019 16:24
> To: jing hua zhao
> Cc: Rui Barradas; r-devel using r-project.org
> Subject: Re: [Rd] Calculation of e^{z^2/2} for a normal deviate z
>
> You may want to look into using the log option to qnorm
>
> e.g., in round figures:
>
>> log(1e-300)
> [1] -690.7755
>> qnorm(-691, log=TRUE)
> [1] -37.05315
>> exp(37^2/2)
> [1] 1.881797e+297
>> exp(-37^2/2)
> [1] 5.314068e-298
>
> Notice that floating point representation cuts out at 1e+/-308 or so. If you want to go outside that range, you may need explicit manipulation of the log values. qnorm() itself seems quite happy with much smaller values:
>
>> qnorm(-5000, log=TRUE)
> [1] -99.94475
>
> -pd
>
>> On 21 Jun 2019, at 17:11 , jing hua zhao <jinghuazhao using hotmail.com> wrote:
>>
>> Dear Rui,
>>
>> Thanks for your quick reply -- this allows me to see the bottom of this. I was hoping we could have a handle of those p in genmoics such as 1e-300 or smaller.
>>
>> Best wishes,
>>
>>
>> Jing Hua
>>
>> ________________________________
>> Sent: 21 June 2019 15:03
>> To: jing hua zhao; r-devel using r-project.org
>> Subject: Re: [Rd] Calculation of e^{z^2/2} for a normal deviate z
>>
>> Hello,
>>
>> Well, try it:
>>
>> p <- .Machine\$double.eps^seq(0.5, 1, by = 0.05)
>> z <- qnorm(p/2)
>>
>> pnorm(z)
>> # [1] 7.450581e-09 1.228888e-09 2.026908e-10 3.343152e-11 5.514145e-12
>> # [6] 9.094947e-13 1.500107e-13 2.474254e-14 4.080996e-15 6.731134e-16
>> #[11] 1.110223e-16
>> p/2
>> # [1] 7.450581e-09 1.228888e-09 2.026908e-10 3.343152e-11 5.514145e-12
>> # [6] 9.094947e-13 1.500107e-13 2.474254e-14 4.080996e-15 6.731134e-16
>> #[11] 1.110223e-16
>>
>> exp(z*z/2)
>> # [1] 9.184907e+06 5.301421e+07 3.073154e+08 1.787931e+09 1.043417e+10
>> # [6] 6.105491e+10 3.580873e+11 2.104460e+12 1.239008e+13 7.306423e+13
>> #[11] 4.314798e+14
>>
>>
>> p is the smallest possible such that 1 + p != 1 and I couldn't find
>>
>>
>> R version 3.6.0 (2019-04-26)
>> Platform: x86_64-pc-linux-gnu (64-bit)
>> Running under: Ubuntu 19.04
>>
>> Matrix products: default
>> BLAS:   /usr/lib/x86_64-linux-gnu/blas/libblas.so.3.8.0
>> LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.8.0
>>
>> locale:
>>  [1] LC_CTYPE=pt_PT.UTF-8       LC_NUMERIC=C
>>  [3] LC_TIME=pt_PT.UTF-8        LC_COLLATE=pt_PT.UTF-8
>>  [5] LC_MONETARY=pt_PT.UTF-8    LC_MESSAGES=pt_PT.UTF-8
>>  [7] LC_PAPER=pt_PT.UTF-8       LC_NAME=C
>> [11] LC_MEASUREMENT=pt_PT.UTF-8 LC_IDENTIFICATION=C
>>
>> attached base packages:
>> [1] stats     graphics  grDevices utils     datasets  methods
>> [7] base
>>
>> other attached packages:
>>
>>
>>
>> Hope this helps,
>>
>>
>> �s 15:24 de 21/06/19, jing hua zhao escreveu:
>>> Dear R-developers,
>>>
>>> I am keen to calculate exp(z*z/2) with z=qnorm(p/2) and p is very small. I wonder if anyone has experience with this?
>>>
>>> Thanks very much in advance,
>>>
>>>
>>> Jing Hua
>>>
>>>       [[alternative HTML version deleted]]
>>>
>>> ______________________________________________
>>> R-devel using r-project.org mailing list
>>> https://stat.ethz.ch/mailman/listinfo/r-devel
>>>
>>
>>        [[alternative HTML version deleted]]
>>
>> ______________________________________________
>> R-devel using r-project.org mailing list
>> https://stat.ethz.ch/mailman/listinfo/r-devel
>
> --
> Peter Dalgaard, Professor,
> Center for Statistics, Copenhagen Business School
> Solbjerg Plads 3, 2000 Frederiksberg, Denmark
> Phone: (+45)38153501
> Office: A 4.23
> Email: pd.mes using cbs.dk  Priv: PDalgd using gmail.com
>
>
>
>
>
>
>
>
>
>
> 	[[alternative HTML version deleted]]
>
> ______________________________________________
> R-devel using r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-devel
>

```