Stanley's ICA, Intensity Correlation Analysis; ICQ, Intensity Correlation Quotient

[Daniel White]

Hi All,

>Date:    Sun, 16 Mar 2014 21:56:50 +0100
>From:    Johannes Schindelin <[hidden email]>
>Subject: Re: Stanley's ICA, Intensity Correlation Analysis; ICQ, Intensity
>Correlation Quotient
>
>Hi Jeremy,
>
>On Sun, 16 Mar 2014, Jeremy Adler wrote:
>
>> So I suggest sticking with the Pearson and rank Spearman for correlation
>> analysis, they have a long history. If you really want to go binary, try
>> Kendall's tau.

>For your interest: I am happy to report that Coloc 2 recently learnt to
>calculate Kendall's Tau.

>Ciao,
>Johannes

I think I'm worried that most of our statistical assumptions are actually
false in the case of using Pearsons etc. in coloc analysis of fluorescence
images of biological systems.... why?
Because the intensities distribution of the fluorescence signal are almost
never anything close to being normally distributed: a Gaussian,  bell
 curve.

The histograms of whole images tend to be dominated by low values, which
make the mean much lower than the mean of the interesting stuff, if that
mean value is actually meaningful in anyway, .... and worse, even if we
choose a region of interest where the biology really is, the histogram in
that ROI is also seldom a normal distribution either....

Do we need a totally different statistical approach? What methods don't
assume a normal distribution?

Best

Dan (of coloc_2)

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[dscho]

Hi Dan,

On Mon, 17 Mar 2014, Daniel White wrote:

> >From:    Johannes Schindelin <[hidden email]>
>
> >On Sun, 16 Mar 2014, Jeremy Adler wrote:
> >
> >> So I suggest sticking with the Pearson and rank Spearman for
> >> correlation analysis, they have a long history. If you really want to
> >> go binary, try Kendall's tau.
>
> >For your interest: I am happy to report that Coloc 2 recently learnt to
> >calculate Kendall's Tau.
>
> I think I'm worried that most of our statistical assumptions are actually
> false in the case of using Pearsons etc. in coloc analysis of fluorescence
> images of biological systems.... why?
>
> Because the intensities distribution of the fluorescence signal are almost
> never anything close to being normally distributed: a Gaussian,  bell
> curve.

It is true that already a Poisson distribution is a much better model for
fluorescence (because there is no negative fluorescence).

However, we cannot in general assume a Poisson distribution. There are
many challenges in typical images, such as offsets, noise, sometimes
signals which are non-linearly related to the underlying value we want to
measure.

That is why I implemented Kendall's Tau. It only assumes ranked values.
The only remaining problem with that is noise: Kendall's Tau is a
frequentist method that assumes that your data are exact, without flaw. I
guess it would not be too difficult to add a "grace difference" in which
values are considered equal as far as the Tau calculation is concerned
when the numerical values acutally differ. But that looks too ad-hoc to
me.

Ciao,
Johannes

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[Leonardo Guizzetti]

Hi all,

I thought I would offer my two cents on the matter of correlation as
co-localization. In short, I second the idea to use Pearson's correlation
or Spearman's rank correlation coefficients over ICA and ICQ.

There is a widely held belief that the Pearson's test can only operate
under bivariate normal distributions. While this was the inital idea when
conceived, there is a large body of literature examining the robustness of
PCC on bivariate non-normal distributions as well as skewed normal
variables. Under a variety of continuous, non-normal bivariate data, the
Pearson's correlation coefficient reflects the normal distribution value
reasonably well (or better, even), even when the distributions are horribly
skewed [1]. The second paper discusses further cases in which the PCC is
robust to deviations from normal distributions, and that the usual
hypothesis testing applies [2]. Generally, with large sample size (pixels
in this case), and reasonably acquired images (high signal-to-noise ratio),
the PCC can well reflect the correlation it is intended to measure.
Situations where the PCC value can become unstable can arise when the PCC
value is very close to its minimal or maximal value (-1 or 1) [1].

With respect to the "true" distribution of fluorescence signals, there are
really two distributions to consider. The first is the noise of the system
(usually shot noise is the big concern), and this is a Poisson
distribution. The distribution of "signal" fluorescence is probably better
characterized as a log-normal distribution and is ideally much greater than
the shot noise. Therefore, we can consider the Pearson's assumption to be
less violated. As to the biology, I would hazard that Pearson's is best
suited for linearly correlated variables, and since colocalization is
usually employed in the context of identifying protein complexes, the
Pearson's test is probably only valid in the case of fixed stoichiometry of
the complex subunits. When multiple stoichiometries (or higher order
complexes) are possible, then a more sensitive test to use would be the
Spearman's rank correlation because its rank nature is well suited to
detect non-linear correlations. The Spearman's is really just the rank
transform of the Pearson's, and has the nice property that the fluorescence
intensity is not completely discarded by transforming to a binary scale,
where the intensity usually gives very valuable information.

1) Kowalski, CJ. (1977) On the Effects of Non-Normality on the Distribution
of the Sample Product-Moment Correlation Coefficient. (
http://www.jstor.org.proxy1.lib.uwo.ca/stable/2346598)
2) Fowler, Robert. (1987). Power and Robustness in Product-Moment
Correlation. (http://dx.doi.org.proxy1.lib.uwo.ca/10.1177/014662168701100407
)



Leonardo (Lenny) Guizzetti, BSc

Ph.D. Candidate (Medical Biophysics and Collaborate Graduate Program in
Molecular Imaging)
Lawson Health Research Institute
268 Grosvenor St., Room #E5-114
London, Ontario, N6A 4V2
Lab: 519.646.6100 ext 64461
Personal: 226.688.7669


On Tue, Mar 18, 2014 at 3:19 PM, Johannes Schindelin <
[hidden email]> wrote:

> Hi Dan,
>
> On Mon, 17 Mar 2014, Daniel White wrote:
>
> > >From:    Johannes Schindelin <[hidden email]>
> >
> > >On Sun, 16 Mar 2014, Jeremy Adler wrote:
> > >
> > >> So I suggest sticking with the Pearson and rank Spearman for
> > >> correlation analysis, they have a long history. If you really want to
> > >> go binary, try Kendall's tau.
> >
> > >For your interest: I am happy to report that Coloc 2 recently learnt to
> > >calculate Kendall's Tau.
> >
> > I think I'm worried that most of our statistical assumptions are actually
> > false in the case of using Pearsons etc. in coloc analysis of
> fluorescence
> > images of biological systems.... why?
> >
> > Because the intensities distribution of the fluorescence signal are
> almost
> > never anything close to being normally distributed: a Gaussian,  bell
> > curve.
>
> It is true that already a Poisson distribution is a much better model for
> fluorescence (because there is no negative fluorescence).
>
> However, we cannot in general assume a Poisson distribution. There are
> many challenges in typical images, such as offsets, noise, sometimes
> signals which are non-linearly related to the underlying value we want to
> measure.
>
> That is why I implemented Kendall's Tau. It only assumes ranked values.
> The only remaining problem with that is noise: Kendall's Tau is a
> frequentist method that assumes that your data are exact, without flaw. I
> guess it would not be too difficult to add a "grace difference" in which
> values are considered equal as far as the Tau calculation is concerned
> when the numerical values acutally differ. But that looks too ad-hoc to
> me.
>
> Ciao,
> Johannes
>
> --
> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>

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