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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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2 Answers 2

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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.

Another approach is to reverse the model, since you don’t have any adjustment covariates. Using an approach of Peter O’Brien discussed here you predict treatment from groups. If group can’t predict treatment, treatment can’t predict groups. So you have a multivariate 2 degree of freedom test for the groups by using a univariate model, where X and Y are swapped.

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But don't you have 4 mutually exclusive categories? A, B, A&B, neither. Per your problem statement, a cell has to be in one, and only 1 of these 4 states (after all, this is the case of the typical 2x2 contingency matrix, where one state is "outcome" -whatever that may be-, and the other is "no outcome"). So you have a 4x2 contingency matrix (4 states, 2 treatments -treated, not treated), and a Fisher test will let you know whether the treatment affects the likelihood to be in one of the 4 states.

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  • $\begingroup$ Perhaps I should clarify more: I am looking for a test which tells me pairwise, when treated is a cell significantly more likely to then occupy state A over State B. So if a Fisher test is used in the way you describe, an example might me: The odds ratios for occupying state A if treated are higher than those for state B if treated, and the single p value given by the test is < threshold. This means that the difference in odds ratio between State A and B is significant? Have I got that correct? $\endgroup$
    – Richard
    Commented Jul 12 at 11:48
  • $\begingroup$ Another approach which has been suggested is to remove any intersection between the two states and perform a Fisher test between cells ONLY in state A and ONLY in state B. $\endgroup$
    – Richard
    Commented Jul 12 at 11:49
  • $\begingroup$ 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 $\endgroup$
    – jginestet
    Commented Jul 12 at 16:05
  • $\begingroup$ ...cont. As far as remove the A&B state, that depends really on the ultimate goal of the experiment, on exactly what A and B are, etc. It may be ok, it may not; you would need domain expertise to decide. As a general rule, my gut says "do not ignore data", but... And all that would do is go from a 4x2 scenario to a 3x2, which does not change the nature of the test at all. So... $\endgroup$
    – jginestet
    Commented Jul 12 at 16:09

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