# Ratio error propagation for dependent, paired variables

I've been struggling for some days with a statistical issue that is beyond my knowledge.

Context: I'm studying the volume of cells, as well as their nucleus volume. These cells are categorized in 3 different phases according to their state of development. From phases 1 to 3, the cells are expected to grow in volume, as well as their nuclei volume, but we don't know about the behavior of the ratio of the volume cell/volume nucleus. In each phase, I was able for individual cells to measure their cell volume and their nucleus volume.

Phase 1 cells:

• Cell: 1, 2, 3, ... 30
• Associated cell volume: 15, 11, 18, ... 21
• Associated nucleus volume: 2, 6, 4, ... 4
• Ratio: 7.5, 1.83, 4.5, ... 5.25

So my issues are:

1. What is the standard deviation of the ratio ? I doubt that I could just calculate the standard deviation of the ratio numbers (7.5 1.83 4.5 ... 5.25) as the values are dependent + paired. However, when I use the error propagation formula, it gives me something a bit smaller. Am I overcomplicating things, or am I missing something ?
2. The reason of this post is that I want to compare the ratio of the different phases. Right now I'm doing the method of not using the standard deviation with the error propagation formula. However, it gives me non-intuitive results: from phase 1 to phase 2, the cell volume significantly increases (P<0.001) while the nucleus volume does not change and the resulting ratio of the two seem to not change either. Am I wrong to assume it should also change significantly ?

I hope I was clear enough, thanks for any tip!

• I don't understand, how are the values dependet? You have 30 different cells and each has its own values? Paired in what way? Also for ratios you can estimate a binomial proportion CI. How are you comparing the ratios? Really you should use a linear mixed model for this. – user2974951 Oct 12 at 9:04
• Indeed, I have 30 cells, for each I have two observable, that's all basically. These two observable are dependent right ? (that's my old memory that observable variables = dependent variables, but maybe I'm totally wrong ?). But, my bad, they are not paired, I was totally wrong. I just meant they were "associated" (i.e. in "cell 1", I will measure "volumeA 1" and "volumeB 1" and retain the information for each data of VolumeA or B, from which cell it was taken). I compared the ratio with a Mann–Whitney U test as the ratio does not necessarily have a Gaussian distribution. – Caracole Oct 16 at 15:05
• I've tracked down the origin of the second issue: I had a shift from one value in my Volume cell compared to Volume nucleus... Now, it makes more sense: if the volume of one significantly change while the other remain the same, the ratio also significantly change. – Caracole Oct 16 at 15:07
• So it the issue resolved? – user2974951 Oct 17 at 7:28