# multiple testing adjustment for fisher exact test

I have 4 groups (G1, G2, G3, G4) consisting of samples of biological material, and we test them in order to identify are there any bacteria present. We also know that all of the samples contain virus X. So the idea is to see is there at least one bacteria present (coinfection) in a sample, and then to test whether the coinfection rate/proportions differ among groups. The data can be represented as:

Factor Group 1 Group 2 Group 3 Group 4
Bacteria A (number of samples for which pCR test was positive) 10 0 3 2
Bacteria B (number of samples for which pCR test was positive) 4 2 2 4
Bacteria C (number of samples for which pCR test was positive) 4 2 2 4
Bacteria D (number of samples for which pCR test was positive) 1 8 0 0
Coinfection (total number of samples with at least one bacteria present) 15 9 6 3
No Coinfection (only virus X present) 5 15-9 45-6 24-3
Total Sample 20 15 45 24

So I want to test the difference in proportions for each bacteria among G1 and G3, G1 and G4, G2 and G3, G2 and G4, G1 and G2. I did Fisher exact test by creating a 2*2 table for each Bacteria and two respective groups, for instance:

Factor Group 1 Group 2
Bacteria A 10 0
No Bacteria A 15-10 9-0

The question is: Is this an appropriate solution? how do i adjust for multiple comparisons? the vector of p-values which i get should include all the p-values for each combination i tested overall (G1 and G3, G1 and G4, G2 and G3, G2 and G4, G1 and G2), or just pairwisely (5 vectors of p-values to be adjusted)?

(Since bacteria can co-occur, the sum of elements does not add to total sample size, so i can not use pairwise_fisher_test, which includes adjustment)

Additional information: Subjects were tested for co-infection, so all of them have virus X+ either none (then subject is classified in 'no coinfection' group) or one or several of bacteria. Indeed, there are individuals who have 3 bacteria present at the same time, and that individual is added to the total count sum of each respective row. The idea is to test whether the proportions of bacterial infections differ among groups.

I am not sure what kind of test is possible in this regard, as there are many possible combinations of viruses that are exhibited in subjects, if we want to make the table disjoint. Theoretically, virus X can coexit with all the bacteria, so the idea of the proportion testing is whether certain kind of bacteria is more likely to be observed together in a particular group.

• Welcome to Cross Validated! (1) What does "bacteria can co-occur" mean? Is it just that there is 1 bacterium in Group 1 that is neither A, B, C, nor D types, & so on for the other groups? (2) Are you sure it's reasonable to assume counts are independent? - that the 10 Bacteria A in Group 1 aren't progeny of a single spore in a Petri dish, for example. (3) Overall it's hard to give advice without knowing the experimental set-up - not just the data but how they were obtained is relevant to the statistical analysis Mar 20 at 8:45
• @Scortchi-ReinstateMonica thank you for the clarification, indeed there is overlap in counts. I am not sure what would be the other method applicable in this situation: the sample size is relatively small for proportion testing using z-statistic. i thought about a stratification of different combinations of bacteria to make table disjoint, but as there are 11 bacteria in total (i wrote 4 as an example), listing all possible combinations would be a tedious task, or is this a way to go? many thanks in advance! Mar 20 at 9:22
• So for each virus-infected subject you have the group to which he (uniquely) belongs, plus the observed presence or absence of four different types of bacteria? And do the groups distinguish different treatment regimes to which the subjects are randomly allocated? Or reflect other attributes the subjects are observed to have? Or ... ? Mar 20 at 10:24
• @Scortchi-ReinstateMonica not exactly, individual can be assigned to different categories. So say i have subject with ID '1', he is tested for the presence of 11 different bacteria, it is found that he has bacterea A, C,D . So he is added to the total count of each group . All samples are tested in such a way, so the numbers reflect total count of subjects infected with this particular bacteria type. So they are not disjoint. Mar 20 at 11:52
• @Scortchi-ReinstateMonica the samples contain biological material collected gtom four groups, which is tested, so there are no treatment or anything of that sort. Just counting number a particular bacteria present or not, and if yes in how many samples. And compare the proportion of each virus among groups, whether they differ. Mar 20 at 11:56