  Suppose that we are interested in the comparison of two independent
categorical variables. Suppose also that the population is divided into
subpopulations or groups. Notice that the distribution of the target variable
may vary across subpopulations, moreover, it may happen that the two
independent variables have the same distribution in the whole population, but
their distributions could differ in some groups. So, instead of testing the
homogeneity of the two categorical variables, one may be interested in
simultaneously testing the homogeneity in all groups. A novel procedure is
proposed for carrying out such a testing problem. The test statistic is shown
to be asymptotically normal, avoiding the use of complicated resampling methods
to get $p$-values. Here by asymptotic we mean when the number of groups
increases; the sample sizes of the data from each group can either stay bounded
or grow with the number of groups. The finite sample performance of the
proposal is empirically evaluated through an extensive simulation study. The
usefulness of the proposal is illustrated by three data sets coming from
diverse experimental fields such as education, the COVID-19 pandemic and
digital elevation models.
