 Joint conditional independence test for stratified contingency tables

For $K$ independent strata, each containing an $r\times c$ contingency table, test independence of the two table variables within every stratum. Under adequate expected counts and the usual multinomial distribution sampling assumptions, each stratum's Pearson chi-squared statistic has asymptotic chi-squared distribution with $(r-1)(c-1)$ statistical degrees of freedom. Their sum has asymptotic chi-squared distribution with $K(r-1)(c-1)$ statistical degrees of freedom, by the additivity of independent chi-squared distributions. This is a test of conditional independence, not of equality of a common effect across strata; the latter needs an interaction contrast or another specified homogeneity test.