Rank product

Key takeaways

Rank product belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Rank product to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Rank product from memory before moving on to harder problems.

Reference excerpt

The rank product is a biologically motivated rank test for the detection of differentially expressed genes in replicated microarray experiments. It is a simple non-parametric statistical method based on 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

Given n genes and k replicates, let r g , i {\displaystyle r_{g,i}} be the rank of gene g in the i-th replicate. Compute the rank product via the geometric mean:

R P ( g ) = ( Π i = 1 k r g , i ) 1 / k {\displaystyle RP(g)=(\Pi _{i=1}^{k}r_{g,i})^{1/k}}

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 permutations 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: E R P ( g ) = c / p {\displaystyle \mathrm {E} _{\mathrm {RP} }(g)=c/p} . calculate the percentage of false positives as : p f p ( g ) = E R P ( g ) / r a n k ( g ) {\displaystyle \mathrm {pfp} (g)=\mathrm {E} _{RP}(g)/\mathrm {rank} (g)} where r a n k ( g ) {\displaystyle \mathrm {rank} (g)} is the rank of gene g in a list of all n genes sorted by increasing R P {\displaystyle \mathrm {RP} } .

Exact probability distribution and accurate approximation Permutation re-sampling requires a computationally demanding number of permutations to get reliable estimates of the p-values for the most differentially expressed genes, if n is large. Eisinga, Breitling and Heskes (2013) provide 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 thin right tail. However, exact statistical significance of large rank products may take unacceptable long amounts of time to compute. Heskes, Eisinga and Breitling (2014) provide a method to determine accurate approximate p-values of the rank product statistic in a computationally fast manner.

See also Ranking Schulze method Comparison of electoral systems Arrow's impossibility theorem

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.; Heskes, T. (2013). "The exact probability distribution of the rank product statistics for replicated experiments". FEBS Letters. 587 (6): 677–682. Bibcode:2013FEBSL.587..677E. doi:10.1016/j.febslet.2013.01.037. hdl:2066/116720. PMID 23395607. S2CID 246960. Heskes, T.; Eisinga, R.; Breitling, R. (2014). "A fast algorithm for determining bounds and accurate approximate p-values of the rank product statistic for replicate experiments". BMC Bioinformatics. 15 (1): 367. doi:10.1186/preaccept-1857144210135244. PMC 4245829. PMID 25413493.

Worked examples

Example 1 — a first encounter with Rank product

Start with the simplest possible case. Write down what Rank product claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Rank product before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Rank product ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Rank product

In research: Rank product appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Rank product in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Rank product is common in secondary-school and first-year university syllabi. It links to neighbouring topics Gene expression, Microarrays, Nonparametric statistics, so understanding it makes those chapters shorter.
In everyday life: Look for Rank product outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Rank product in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Rank product means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Rank product out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Rank product in simple terms?

The rank product is a biologically motivated rank test for the detection of differentially expressed genes in replicated microarray experiments. It is a simple non-parametric statistical method based on ranks of fold changes.

Why does Rank product matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Rank product?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Rank product.