sum of chi-squared statistics for hypothesis testing

Question

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

Answer 1

In fact that is a common technique in meta-analysis of significance values. The original approach by Fisher transforms the $p$-values to a $\chi^2_2$ because there is an easy formula for that but Lancaster suggested doing it for any number of degrees of freedom. So, yes, just add them up and you have a $\chi^2_k$ where you had $k$ tables.