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The question: Is there an appropriate statistical test of a change in dispersion across a large number of variables as a result of experimental manipulation?

Background: A manipulation has been made to a regulatory 'error-checking' gene and RNA-sequencing has generated count values for the level of expression of thousands of other genes, where counts are generally considered to originate from a Poisson distribution. Thus, the data is m-genes by n-samples, with n = 2 groups (control, treatment) times 3 replicates for each group. The standard approach with these data is a gene- or row-wise GLM to test for a difference in the number of counts with treatment.

The problem: I suspect that my treatment might also disrupt the stability of expression across replicates for any given gene with some probability. Thus, I want to test for a group level difference in dispersion itself with treatment. I would naively consider computing the dispersion for each gene and treatment, possibly as Shannon entropy on row-wise probabilities, then testing if the expected dispersion for treatment is statistically greater than that of control using a non-parametric test, but I am not sure if this is valid or whether there are established tests for this.

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Some of your terminology differs slightly from what I am accustomed to, but given that I understand your problem correctly, you may use an F-test or the Brown-Forsythe test to determine a difference in variance between two samples. Depending on what conditions you require to be met you may select between them. Here are links to their Wikipedia pages:

https://en.wikipedia.org/wiki/F-test_of_equality_of_variances

https://en.wikipedia.org/wiki/Brown%E2%80%93Forsythe_test

If I have misinterpreted your question I apologize. I cannot spot the immediate reasoning for using a non-parametric alternative here, but if you have one then I do not see why it would be inappropriate.

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