I have a dataset of two genetically different cell types, "A" and "B", and each of them can have four different morphologies, let's say "spiky cell", "elongated cell", "round cell", "triangular cell". This yields a 2x4 contingency table:
Spiky Long Round Triangular Type "A": 1 10 13 44 Type "B": 10 3 3 20
How do I test if the differences in morphologies are different? As in "Cell group B has a significantly higher proportion of spiky cells than cell group A".
Normally, I would use a Fisher's exact test, and I have thought about just doing four Fisher's tests. However, since there are four different morphologies, I feel like I should be using a statistical model that models all four proportions. Like in the above example, it seems that there are more spiky "B" cells and more elongated "A" cells, and they are somehow not independent, since a cell can only be one of the four. Should I be using some kind of multilevel model?
I have tried searching for answers in previous questions, but I didn't find a solution. I hope someone can suggest a good test or maybe direct me to a previous discussion.