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The '''rank product''' is a biologically motivated test for the detection of differentially [[Gene expression|expressed]] genes in replicated [[DNA microarray|microarray]] experiments.
It is a simple [[Non-parametric statistics|non-parametric]] statistical method based on [[ranking|ranks]] of fold changes. In addition to its use in [[expression profiling]], it can be used to combine ranked lists in various application domains, including [[proteomics]], [[metabolomics]], statistical [[meta-analysis]], and general [[feature selection]].

==Calculation of the rank product==

[[Image:Rank product.png|thumb|right|225px|Filled circles represent ranks of one gene in the different replicates.  The rank product for this gene would be (2&times;1&times;4&times;2)<sup>1/4</sup> ≈ 2]]

Given ''n'' genes and ''k'' replicates, let <math>e_{g,i}</math> be the fold change and <math>r_{g,i}</math> the rank of gene ''g'' in the ''i''-th replicate.

Compute the rank product via the [[geometric mean]]:
: <math>RP(g)=(\Pi_{i=1}^kr_{g,i})^{1/k}</math>

==Determination of significance levels==
Simple permutation-based estimation is used to determine how likely a given RP value or better is observed in a random experiment.
# generate ''p'' [[permutation]]s of ''k'' rank lists of length ''n''.
# calculate the rank products of the ''n'' genes in the ''p'' permutations.
# count how many times the rank products of the genes in the permutations are smaller or equal to the observed rank product. Set ''c'' to this value.
# calculate the average expected value for the rank product by: <math>\mathrm{E}_{\mathrm{RP}}(g)=c/p</math>.
# calculate the percentage of false positives as : <math>\mathrm{pfp}(g)=\mathrm{E}_{RP}(g)/\mathrm{rank}(g)</math> where <math>\mathrm{rank}(g)</math> is the rank of gene ''g'' in a list of all ''n'' genes sorted by increasing <math>\mathrm{RP}</math>.


==Exact probability distribution of the rank products statistics==
Permutation re-sampling  requires  a computationally demanding large number of permutations to get reliable estimates of the ''p''-values for the most differentially expressed genes. Eisinga, Breitling and Heskes (2013) provided the exact probability mass distribution of the rank product statistic. Calculation of the exact ''p''-values offers a substantial improvement over permutation approximation, most significantly for that part of the distribution rank product analysis is most interested in, i.e., the tin right tail.


   


==References==
* Breitling, R., Armengaud, P., Amtmann, A., and Herzyk, P. (2004) Rank Products: A simple, yet powerful, new method to detect differentially regulated genes in replicated microarray experiments, FEBS Letters, 573:83–-92
* Eisinga, R., Breitling, R., and Heskes, T. (2013). The exact probability distribution of the rank product statistics for replicated experiments. FEBS Letters, 587:677--682  http://dx.doi.org/10.1016/j.febslet.2013.01.037 [http://robeisinga.ruhosting.nl/MyOpenAccess/FEBS%202013%20preprint.pdf preprint] 

[[Category:Gene expression]]

[[Category:Meta-analysis]]
[[Category:Non-parametric statistics]]
[[Category:Microarrays]]