# How do I take this ratio of two means, then normalize according to another mean while maintaining a correct measure of spread?

I have been trying to figure out how to correctly calculate the standard deviation for the ratio of two protein expressions as measured by ELISA, so that I may then run a hypothesis test, similar to this figure in this paper with the IL-1ra/TNF-alpha ratio. https://pubmed.ncbi.nlm.nih.gov/24013843/#&gid=article-figures&pid=figure-1-uid-0 I am seeing how expression of TNF-alpha and IL-1ra change according to two factors: the type of particle used, and the concentration of each particle. I have 2 particles, 3 concentrations each, therefore 6 treatment groups plus one contrl group. I have 7 data sets of n=6 technical replicates each for the IL-1ra, 7 data sets of n=4 technical replicates each for the TNF-alpha. I am worried that it is incorrect just to divide the respective average concentrations and propagate their standard deviations. I additionally want to normalize these ratios according to percentage cell viability data that I obtained at each concentration. Please let me know if this is the appropriate place to ask this question, or where else I could ask.

• Welcome to Cross Validated! This is a good place to ask, but please provide a few more details. I understand your 7 groups (control and 6 particle-type/concentration combinations), but it's not exactly clear to me what the n represents. Does n=4 for TNF-alpha and n=6 for IL1-ra mean that completely separate experimental runs were done for analyzing each of TNF-alpha (4 runs) and IL1-ra (6 runs), or do those represent the number of technical replicates in a single experimental run? Please edit the question to provide that information; comments are easy to overlook and can be deleted.
– EdM
Dec 27, 2023 at 16:55
• Hi EdM, apologies for the confusion. I have edited the question to provide that information. n represents the number of technical replicates in a single experimental run. Please let me know what other information would help. Thank you! Dec 27, 2023 at 19:15

As best as I can tell, the paper you cite seems to report only errors arising from technical replicates based on a single batch of macrophages. Inference should be based on biological replicates. See the Technical Perspective on "Empowering statistical methods for cellular and molecular biologists" in Molecular Biology of the Cell 30: 1359-1368 (2019) for how to distinguish technical from biological replicates (depending on experimental context) and the danger of pseudoreplication when you rely solely on technical replicates. I trust that you will perform this experiment more than once.

That said, sometimes the results of biological replicates are illustrated with one example as in the figure you link, with a statement that the example is representative of a larger number of biological replicates. As it's (perhaps unfortunately) customary for reviewers to insist on (potentially misleading) p-values based on technical replicates in such a display, I'll provide some suggestions.

Your primary measurements of IL-1a and TNF-alpha can (in principle) be handled by analysis of variance with appropriate post-modeling comparisons among the treatments and control. The assumptions might not hold well if your data are similar to those in your linked plot, with errors evidently increasing with the values of the analytes.

Sometimes working in a log scale instead improves the situation, although then you would be modeling the means of the logs of the values (generally different from the logs of the means). That's similar, however, to reporting qPCR cDNA values in Cq or delta-Cq scales, logarithmically related to the actual amount of target cDNA in the sample.*

Think carefully about the specific comparisons you will perform after the modeling. There are 21 two-way comparisons possible among 7 experimental conditions, so if you do them all you need to make a large correction for multiple comparisons. The Technical Perspective noted above discusses that issue briefly.

For ratios/normalizations, be warned that the formulas for propagation of uncertainty are approximations that can be highly misleading, depending on the nature of the data. In your situation the technical replicates of IL-1a and TNF-alpha are presumably uncorrelated, which simplifies the formula for the IL-1a/TNF-alpha ratio.

I suspect that you will be better off working in a log scale for the ratios, as:

$$\log \frac{\text{IL-1a}}{\text{TNF-alpha}} = \log \text{IL-1a} - \log \text{TNF-alpha},$$

and (with no correlation) the variance of the ratio in the log scale is just the sum of the variances of the two analytes in their log scales.

The above doesn't address the problem of the statistical tests to use once you have variance estimates for the ratios. The best would be to use the variance estimates along with the corresponding Welch-Satterthwaite degrees of freedom, which are based on the numbers of replicates of each type and the standard deviations.

There is probably no need to adjust the IL-1a/TNF-alpha ratio for cell viability. There is no reason to expect that wells sampled for IL-1a differ in viability from those sampled for TNF-alpha, so a pooled viability estimate for each particle-type/concentration combination makes the most sense. If you correct each of IL-1a and TNF-alpha for viability, that viability factor cancels out in the ratio. If you think that you need to correct the IL-1a and TNF-alpha values individually for viability, then apply the principles outlined above for ratios.

*An alternative is ordinal regression, which only uses the rank-ordering of outcome values and can be considered a generalization of non-parametric tests like the Kruskal-Wallis test.

• Thank you for the comprehensive answer! I wanted just some clarification on your last sentence. When correcting the IL-1ra and TNF-alpha values individually for viability using the principles outlined above for ratios, would this for example be correct? $\log \frac{\text{IL-1ra}}{\text{MTT Absorbance}} = \log \text{IL-1a} - \log \text{MTT Absorbance},$ Additionally, if I did do this, would there be an error associated with each individual data point of IL-1ra as I would be dividing by the average MTT absorbance over n=6 biological replicates of the respective concentration it is in. Dec 29, 2023 at 0:54
• @slatt the formula is certainly correct (although I worry about MTT as a "viability"measure ; see ncbi.nlm.nih.gov/pmc/articles/PMC8657538). Do incorporate the variance in MTT into the calculations. The formula for the variance of the ratio depends on whether the IL1a and MTT were done on the same wells or on different wells. If they were from the same wells, then use the formula for a weighted sum of correlated variables. If not, the covariance is 0 and it's just the sum of the variances.
– EdM
Dec 29, 2023 at 15:57