# Ratios of means - statistical comparison test using Fieller's theorem?

I would really appreciate any suggestions with the following data analysis issue. Please read till the end as the problem at first may appear trivial, but after much researching, I assure you it is not. The situation is a little complicated because I want to compare the ratios of means:

For example, in one experiment I have collected data (Electric current levels, a continuous variable) from 7-8 cells (replicates) which express a particular ion channel gene (Gene1). This gives me a mean current level for Gene1. I have then measured current levels from a different set of 7-8 replicate cells which also express Gene1 PLUS an activator gene (the "treatment"). Now I have a mean current value for Gene1+activator.

I repeat the two measurements for Gene2 (a different ion channel gene), again recording mean current values from 7-8 cell replicates for Gene2 alone and then for Gene2+activator.

The quantity I am interested in comparing is the %activation or fold activation caused by the activator for Gene1 versus Gene2. So, I would obtain a ratio by dividing the mean current for Gene1+activator by the mean current for Gene1 alone. This would give me the fold activation for Gene1. I would compare this to a similar ratio obtained for Gene2.

I have done some research on this and using Fieller's intervals to compute the error bars or confidence intervals seems promising. However, I don't know how to convert that to hypothesis test and get an appropriate p-value for the comparison of means being same/different. Furthermore, the best solution would also allow multiple comparisons and allow me to compare "fold-activations" for Gene1 and Gene2 and Gene3 and Gene4 at the same time.

Fieller's intervals seem like the perfect tool to compute 95% confidence intervals etc around each fold-activation but as we are submitting to a journal, they will insist on p-values for our comparisons. As of now, I am reduced to insisting the lack of CI overlap clearly signals a significant difference but I know that lack of 95%CI overlap is an overly stringent comparison test which represents an alpha<0.05. I would truly appreciate any suggestions to the appropriate comparison test (single or multiple comparison, either at this point will be helpful). Thanks in advance.

• BUMP. Still hoping for an answer. Anyone? How about using bootstrapping/permutation testing to test the null hypothesis: H0: m1/m2 = m3/m4 restated, H0: m1/m2 - m3/m4 = 0; I can see how I may bootstrap the difference in raio of means by sampling with replacement from each of m1,2,3,4 many times and recomputing the difference. But I can't see how to set this up as a permutation test to test the above null hypothesis. What groups could I randomize across to assume the null hypothesis and obtain the distribution on which to test my observed value? – Electrophys Jun 2 '13 at 23:41
• Would it make sense to use the logarithm of the currents? That way you could compare the differences of the log transformed values rather than the more awkward ratio. (Whoops, I now read EdM's answer. I agree with EdM.) – Michael Lew Oct 6 '15 at 20:26

If $I$ represents current, 1 and 2 represent the genes, and $c$ and $a$ represent control and activator cases respectively, the desired comparison is $I_{1a}/I_{1c}$ versus $I_{2a}/I_{2c}$. In the log scale the null hypothesis becomes:
$$log(I_{1a})-log(I_{1c}) = log(I_{2a})-log(I_{2c})$$