# Looking for a statistical test for significance of association between a treatment and non-mutually exclusive categories

I am looking for a statistical test which will give a p value for the association between a treatment and two potentially co-occurring labels:

I have two categories of cells: state A and state B.

A cell can exist in one of the states, both of the states or neither.

A treatment has been applied to some of the cells.

I would like a test for whether the treatment significantly differentiates the two categories. I.e. whether applying the treatment would make it more likely for a cell to belong to one group or the other, and get a significance for this difference.

As far as I understand, a Fishers exact test would not work because the categories are not mutually exclusive, and using something like 'only state 1' and 'only state 2' would not work because the distributions will be unknown. Any advice would be great!

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Had the categorical variable been an independent variable, this is a somewhat standard problem of multiple regression. For a multiple choice variable with $$k$$ categories, instead of having $$k-1$$ indicator variables to represent mutually exclusive categories, you have $$k$$ indicators to represent each category, assuming that “no choices” is a valid answer. The key assumption is that of additivity / lack of interaction / lack of synergism between the choices, i.e., that certain combinations of choices don’t have specify effects in predicting Y. To get an overall frequentist assessment of association between the multiple choice variable and Y, do a chunk test with $$k$$ degrees of freedom (likelihood ratio $$\chi^2$$ or $$F$$ test depending on model, or possibly a Wald test).
But for your situation you have the categories as $$k=2$$ dependent variables and need to use a multivariate logistic regression model as used in econometrics. This will provide a similar chunk test of the treatment effect as mentioned above.
• You say "pairwise". It does not seem to me that you have "pairs"; while it may be the same "cell type", the treated cell is physically different from the untreated cell; you do not have paired data, but independent data. As far as the result of a 4x2 Fisher (or $\chi^2$ test for that matter), in a 4x2 (or 3x2 if you remove A&B data), a significant result will tell you that the proportions of your 4 (or 3) states is significantly different between treated and untreated cells. Since there are more than 2 states it will not tell you which state occurs more/less often. ...cont Commented Jul 12 at 16:05