# Test of significance of multiple proportions in two groups

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.

Thanks!

## 1 Answer

If you use a chi-square you can partition it into 2x2 contrasts - possibly with the aid of collapsing some groups for some of the contrasts. (Your lowest expected values are 3.81 and 4.50, which for most purposes is plenty.)

Alternatively, you could use a chi-square and perform the various 2x2 post hoc comparisons. If you want them, you can even do the usual kinds of significance level adjustments for those.

If you don't have to have formal tests, you can just look at the Pearson residuals or (the signed square root of) the contribution to chi-square from the independence model to see which cells contributed substantively to significance.

Similar things can be done with the G-test (the likelihood-ratio-test form of chi-square).

You might instead fit a multinomial model via GLMs and test whichever contrasts or post hoc comparisons matter for you.

• Thanks, but I was hoping an idea of a general model taking all variables into account. Here is an example of a 3x2 table, mine is a 2x4 table: stats.stackexchange.com/a/9721/29337 – Chrelli Apr 10 '14 at 14:38
• Thanks, but I was hoping an idea of a general model taking all variables into account. Perhaps comparing all possibilities of things being equal or different. Here is an example of a 3x2 table, mine is a 2x4 table: stats.stackexchange.com/a/9721/29337 The example is good, and the soultion is elegant given that every condition can be described by one $\theta_{Dead}$ (they can be dead or not-dead). In my case, there are 4 outcomes (Spiky,Long,Round,Triangular). This means, that for each group (A or B) I should somehow model the hypothesis using three variables \$\theta_{Spiky}, \theta_{Lon – Chrelli Apr 10 '14 at 14:45