I have N 2x2 observed contingency tables. Each one was sampled from the same distribution. The problem is the column labels are obfuscated so that I do not know the true label name of each column, and in some contingency table the first column refers to the same label that in a different contingency table is the second column.
If I had one contingency table, I would not care that the column labels are obfuscated in order to check independence between the variables, I can still calculate the chi-squared statistic and run the chi-squared test.
But since I have N contingency tables with the columns maybe permuted and no way to discern the column labels, I cannot build one big contingency table.
Would it make sense here to calculate the chi-squared statistic for each table separately then take the sum of all of these and calculate a p-value on that?
It seems to make sense but I want to make sure I am not missing something obvious