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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) :
    • 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).

kjetil b halvorsen
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Agrd
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1 Answers1

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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.

EdM
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  • I think the 1000 is a red herring, in a way. These pixels are not independent. So the chi-squared and Fisher exact test are not appropriate anyway? And with one pixel per row the OP will then have to figure out how to deal with the dependence between pixels which should depend on their proximity in the brain. – dipetkov Jan 12 '23 at 21:39
  • Another way to think about this is: the unit of treatment is brain, not pixel. – dipetkov Jan 12 '23 at 21:40
  • @dipetkov the OP might not need to evaluate intra-region dependences of intensity values, as interesting as that might be. In the extreme, back when I studied brain chemistry, I would dissect out and homogenize small individual brain regions and report bulk neurochemical concentrations in amount/weight units. That prevented intra-region analysis of local anatomical concentrations, but there was much to be learned that way. The approach of the OP seems to be intermediate between the extremes of grind-it-all-up and very-high-resolution cell-by-cell (or finer) analysis. – EdM Jan 13 '23 at 15:45
  • I didn't mean to suggest that the OP should look at the dependencies at all. But to point out that they shouldn't be considering a Chi-squared or a Fisher exact test either. (Because their 1000 pixels are not at all like asking 1000 participants for their preferred category.) I admit to not having studied brain chemistry though, so I might be wrong in my thinking. – dipetkov Jan 13 '23 at 15:54
  • @dipetkov you are correct in your thinking; yours is yet another reason not to use chi-square/Fisher tests here. In general with this type of study, the hope is that random effects or cluster/sandwich-type variance estimates will handle the within-mouse or within-human correlations adequately, so that analysis ends up on the individual as unit of treatment. How often or how well those methods are implemented and validated in this field of study is another question. – EdM Jan 15 '23 at 20:08