Re: Kolmogorov-Smirnov

From: T. Scott Thurmond (thurmons@ehsct7.envmed.rochester.edu)
Date: Tue Nov 25 1997 - 15:47:00 EST


Kenneth Ault wrote:
> 
> As someone who made extensive use of the K-S test in a past life I would
> like to point out one important issue that is not dealt with in
> statistics text books, and I think was not explicitly mentioned in Ted
> Young's excellent paper that introduced K-S to the flow community.
> 
> The K-S test is typically used to compare two frequency distributions
> non-parametrically.  The number of degrees of freedom that one uses to
> calculate a p value from a K-S statistic is based upon the number of
> bins in the frequency distribution histogram.  For flow cytometry data
> one is tempted to use the number of channels, i.e. 256 or 1024 etc. as
> the degrees of freedom.  Doing this will result in any two histograms
> that are not identical being highly statistically significantly
> different.  In other words using channels as degrees of freedom makes
> the K-S test ridiculously too sensitive to trivial differences in the
> histograms.
>    In fact our flow histograms have far fewer degrees of freedom than
> they have channels.  The correct value is based upon the CV of your
> histogram, i.e. how many distinct distinguishable histograms can you fit
> into the number of channels that you have?  For most of our data we have
> no way (that I know of) to estimate the correct number of degrees of
> freedom.
>    For this reason I have always used the K-S statistic as a measure of
> the difference between two histograms, but have not used it to calculate
> a p value.  I am sure there are others listening to this discussion who
> are better qualified to discuss how one calculates degrees of freedom
> for a flow histogram - I for one would be interested in such a
> discussion.

	There is some argument as to whether the K-S test even fits the
conditions required for flow cytometric histogram analysis.  Chris Cox
in his paper on  flow cytometry frequency distributions makes a good
case for using the chi-square test for statistical histogram comparisons
(Cytometry 9:291-298, 1988).
Scott



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