Mean of ratio, ratio of mean and statistic test

Question

First off, I'm sorry if this has been answered or if the answer seem obvious, but I'm currently stuck and can't really wrap my head around the problem.

So here are an example table of my situation (random number but the idea is here)

Status	Positive	Total	Area
A	123456	2345678	6.31
A	45678912	3456789123	7.3
B	12009023	29016811	4.61
B	10106243	1816308	5.23

I basically have 2 independent groups (A and B), each containing roughly 13 independent slides of tissue. I marked said tissues with an antibody, then selected roughly the same area for each of the 26 samples and scanned them.

I ended up with a number of total count, a number of positive count and the actual size of the area (in mm²) for each tissues.

Now I want to check if there's a difference between group A and group B.

Easy enough I thought.... but since the area isn't quite the same (I had to do it manually and because of the variation in tissues and other factors the size of the area can differ a lot), I want to normalize my result to have the ratio (Positive/Total)per mm² so that at equal surface I can say whether or not there's a difference and if so, how much of it.

And that's where I'm stuck. So here are my thoughts :

1. Create a new column, with the Positivity rate (positive/total), then calculate the ratio Positivity/Area for each sample, then compare the mean of the ratio for group A versus the mean of the ratio for group B with a Mann-Whitney-Wilcoxon test
2. Calculate the mean for Positive, Total and Area then calculate the Positive/Total/Area ratio and compare both group with a Mann-Whitney-Wilcoxon test
3. Use a prop.test() to compare the ratio Positive/Total of both group, but then I can't take account for the difference in area size

I noticed that the ratio is slightly different depending on whether I calculate the mean of all sample then the ratios (which I think is called ratio of the means), or if I calculate the ratio for each sample and then calculate the mean (mean of ratios).

If anyone has any idea, or if I'm just complicating some really simple or anything..... Thanks in advance

Answer 1

First, if you calculate the density of total antibody (total/area) and the density of positive antibody (positive/area), the ratio of densities (positive/area)/(total/area) is the same as the ratio positive/total. Normalizing by area does not help. If you suspect the area has any impacts on your measurements (i.e., positive, total, or positive/total), you can check it by linear regression.

Secondly, assume you have two samples with measurements p1, t1 and p2, t2 for positive and total, respectively. Of course (p1+p2)*0.5/[(t1+t2)*0.5] is not the same with (p1/t1 + p2/t2)*0.5. Anyway, if you just sum p values together (and t values together), why do not you take a very large area and measure a single p and t in the first place? Since you take 13 samples (areas) each group, you already have 13 ratios of p/t. That is when statistics comes to play.