Comparison of multiple proportions between two groups of treatment My situation is the following:
I'm working on mice. I apply them a treatment or a sham of treatment (two groups). Then I do immunohistochemistry to analyse the intensity of the pathology in brains. For that, I divide the brain of each mice in 10 regions avec I divide each regions in a lot (> 1000) of pixels. Each one of these pixel are given an intensity of signal : 0, 1, 2 or 3. I would like, for each region, analyse if the distributions of these signals (1, 2, 3) is different between my two groups of treatment.
For example, let's take the region X. I have :

*

*Treatment A (10 mice) :

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*Proportions of signal 0 : 2, 16, 55, 46, 2, 12, 0.4, 11, 65, 0.6

*Proportions of signal 1 : 34, 55, 30, 37, 30, 42, 8, 78, 31, 27

*Proportions of signal 2 : 45, 27, 10, 18, 62, 41, 57, 10, 4, 69

*Proportions of signal 3 : 18, 1, 4, 0.1, 6, 5, 35, 0.3, 0.1, 3



*Treatment B (14 mice) :

*

*Proportions of signal 0 : 18, 2, 2, 3, 0.6, 2, 0.8, 4, 1, 6, 3, 2, 0.5, 31

*Proportions of signal 1 : 47, 24, 16, 29, 15, 22, 31, 34, 23, 52, 54, 44, 24, 60

*Proportions of signal 2 : 27, 63, 60, 55, 62, 69, 66, 56, 73, 42, 42, 53, 74, 8

*Proportions of signal 3 : 7, 10, 22, 13, 22, 7, 2, 7, 2, 0.2, 2, 1, 1.3, 0.1



I first calculated means of proportions for each group of treatments and then I thought comparing them with a chi-square (or Fisher) but these numbers are means of proportions and I'm not sure that it's the right thing to do in this case ...
I also thought about going back to the number of pixels and then sum them for each group so that I have a kind if effective for each type of signals... but I'm not sure it is the right way to do it (especially because I have different number of mice in each of the two groups and different number of pixels per regions per mice).
 A: Chi-square and Fisher tests on contingency tables are based on count values, so your hesitancy to use them is well founded. The actual counts per category are important because the reliability of an observed proportion depends on the total number of observations. Also, those tests only tell you whether there is some lack of independence among the category counts; they don't specify the source of the lack of independence. For that, a regression model can be a good choice.
As there is a natural ordering to your 4 intensity categories, ordinal logistic regression could be a useful choice for your model. With multiple measurements within individual mice, you would need to set this up as a mixed model with mice treated as involved in random effects. This page has links to implementations of mixed-model ordinal regression in R.
The way to format your data might depend on the implementation. The most general format would be with one row per pixel, annotated with intensity, treatment, region, and mouse. Some implementations might allow for rows having summed counts of each intensity level by treatment, region, and mouse, or for rows having counts of a single level of intensity by treatment, region, and mouse, along with the total number of pixels.
The model would include treatment, region, and their interactions as fixed-effect predictors. You would include a random intercept for mouse to allow for systematic overall intensity differences among mice, and perhaps a random slope for region among mice to allow for baseline intensity differences among mice for each region. The regression coefficients can be interpreted in terms of the log-odds of a pixel having a progressively higher intensity based on its region and treatment. See this UCLA web page.
This might be overkill in your situation. With each of your proportions based on >1000 pixels, the differences in reliability among proportions might not be such a big issue. Nevertheless, the "proportions" (presumably percentages) that you show don't all seem to add up to exactly 100 for each mouse, so the closer you get to the raw data the more reliable your calculations will be.
