I have a problem that I will express in 2 ways: a math-y way and a biology way. Hopefully this will make it more clear.
I have N observations of a pair of binomial variables, call them R and D. Each time I observe them, I do n trials and observe k successes to get an estimated binomial probability p. For each observation, I am interested in the ratio R/D, a.k.a. the ratio of the paired binomial probabilities. I have done N observations of this ratio for group 1, and M observations of this ratio for group 2. I would like to know whether these two groups are significantly different. Some caveats are that M,N are small, i.e. 2-3, and for some of the observations, the number of trials for either R or D is low, so I'm thinking that normal approximations may not work here.
I have a community of ~30 microbes, and each one expresses the same gene (call it YFG1). I can measure the relative abundance of each microbe in the community, as well as the relative amount of each microbe's YFG1 transcription via high-throughput sequencing. So for each microbe, I have both RNA and DNA data. For each type of data, I know the number of reads that map to the microbe, and the total number of mappable reads. I'm interested in the YFG1 RNA/DNA ratio for each microbe under different conditions, and so I've collected data 2-3 times at each condition of interest. I would like to know if there are any conditions in which a particular microbe significantly up- or down-regulates YFG1 expression.
I know that the ratio of two binomial variables does not have a finite variance, since dividing by zero is a possibility, but I was wondering if it was possible to know whether two groups of binomial ratios are significantly different? I'm open to simulations, as long as I know what to simulate :). I've been running around in circles on this one and was hoping y'all could shed some light.