# Mean of ratio, ratio of mean and statistic test

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

• Just speaking generally it is very seldom the case that ratios should be added or averaged. Averages of log ratios are often appropriate, hence the use of geometric means. Aug 24, 2021 at 12:12
• @FrankHarrell So you think I should calculate the ratio for each sample, then log them up and average the log ? Followed by a comparison of mean(log(ratio)) ? Aug 24, 2021 at 12:21
• I'm 0.8 certain that is the way to go. Aug 24, 2021 at 15:15

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.

• Already did the linear regression, and yeah the area definitely impact the measurement (which is to be expected since the bigger the area, the more cells you'll have).As for why not a larger area, I wasn't the one who choosed the area. If it were me, I would have scanned the whole tissue for each individual rather than a limited area, but my boss asked to only scanned a small designated area.... What I could do would be to re-do the measurement and try to get area as close to each other as I can. For curiosity sake, with area that are different, is there a way to take it into calculation ? Aug 25, 2021 at 10:50
• @YoPomdpin does the area affects the ratio positive/total? Aug 25, 2021 at 22:05
• unfortunately for me, it does =/ Aug 26, 2021 at 9:00