# [BioC] RMA-bimodality:

Peter G. Warren peter.warren at verizon.net
Wed Jun 7 01:28:17 CEST 2006

```Hi, Wolfgang,

Just a short follow-up. The example you provided to Noel has embedded within
it two distributions. Your example is as follows:

set.seed(123)
n = 100000
z = 20 + exp(c(rnorm(n), 3+rnorm(n)))
par(mfrow=c(1,2))
plot(density(log2(z)))
plot(density(log2(z-20)))

Continuing, if we separate the two distributions in z and overlay them, the
plot illustrates the point I've been attempting to make:

z1=20+exp(rnorm(n))
z2=20+exp(3+rnorm(n))
# Note that z = c(z1+z2)
# The left plot shows your bimodal combined density, before the "background
correction", as before.
plot(density(log2(z)))
# The right plot shows the two component distributions separately.
plot(density(log2(z1)))
lines(density(log2(z2)))

Your example in fact models quite well what I see with real, uncorrected
data. If the two distributions are not "absent" and "present" intensity
distributions, I'm open to other suggestions.

Regards,
- Peter

> -----Original Message-----
> From: Wolfgang Huber [mailto:huber at ebi.ac.uk]
> Sent: Tuesday, June 06, 2006 12:15 PM
> To: Peter G. Warren
> Cc: noel0925 at sbcglobal.net; bioconductor at stat.math.ethz.ch
> Subject: Re: [BioC] RMA-bimodality:
>
> Hi Peter,
>
> - doesn't the distribution of mRNA abundances (i.e. physical
> concentrations measured e.g. in average no. of molecules per
> cell) span the whole range from just undetectably above zero
> to very large? I am not sure what mechanism would then result
> in two distinct peaks of fluorescences, one for "absent" and
> and one for "present" mRNAs.
>
> - I tried find a definition of "truely unimodal
> distributions" (and I suppose, "falsely unimodal
> distributions"), but couldn't find one, can you advise?
>
> Cheers
>  Wolfgang
>
> Peter G. Warren wrote:
> > Hi, Wolfgang, Noel,
> >
> > It is true that a non-linear transformation can change the
> number of
> > nodes of the data, and that that transformation can be
> sufficient to
> > explain the bimodality we see in background-corrected data.
> However,
> > in my experience, the raw probe-level data is itself bimodal. When
> > there is some real signal present, the probe-level intensities are
> > actually from two different distributions. The first ("absent") is
> > where there is no positive transcript binding, only cross-hyb,
> > non-specific binding, and background. The second
> > ("present") is all that, plus true target transcript binding. This
> > bimodality is more evident with log-transformed values. (In
> contrast,
> > a log-transformation of a truly unimodal distribution, such as
> > density(rnorm(...), is still unimodal.) In every case I've
> looked at,
> > the "absent" distribution dwarfs the "present" one, so it
> often looks
> > like one mode, before log transformation. After log
> transformation, I
> > have been unable to model the data successfully with a single
> > distribution; it always takes two.
> >
> > Regards,
> > - Peter Warren
> >
> >> Hi Noel,
> >>
> >>> Just so that I am clear- the point is that the bimodality
> is not an
> >>> artifact of the convolution, but simply the fact that the
> number of
> >>> modes of a distribution is not conserved under monotonous
> >>> transformations.
> >> No, I did not say that, and I do not know how to understand this
> >> sentence, since "the convolution" is directly related to "the
> >> monotonous transformation" that we are talking about
> >>
> >>> This is why the paper points to the
> >>> fact that the histograms of log2 (PMs/MMs) stratified by
> log2(PMs)
> >>> is bimodal
> >> I leave the exegesis of the paper to its authors.
> >>
> >>> -so bimodality is a more
> >>> general property of the probe level data.
> >> As you have just said yourself, the number of modes is not
> a property
> >> of the data, but of the data plus the particular (non-linear)
> >> transformation that you choose to apply to them.
> >>
> >>
> >> Best wishes
> >>  Wolfgang.
> >
>
>
> --
> ------------------------------------------------------------------
> Wolfgang Huber  EBI/EMBL  Cambridge UK  http://www.ebi.ac.uk/huber
>
>

```