# Generalized Cochran's Q Test

I have two groups of experimental data samples that are categorical, and my goal is similar to Pearson's chi-squared test ($${ \chi ^{2}}$$), to evaluate how likely it is that any observed difference between the sets arose by chance.

A sample from the first experiment is $$D_1 = \{x_{1,1}: n_{1,1}, x_{1,2}: n_{1,2}, ... x_{1,t}: n_{1,t}\}$$ and a sample from the second experiment is $$D_2 =\{x_{2,1}: n_{2,1}, x_{2,2}: n_{2,2}, ... x_{2,t}: n_{2,t}\}$$. ( An example for data samples: $$\{CAT: 10, DOG: 22, ... FROG: 44\}$$)

Now, lets say I want to compare these data sets using $$\chi^2$$ test, but I want to stratify the data into different blocks (because observations are taken across time and I want to compare observations done at the same time directly to each other).

More concretely, let's say for each group, I add a block index: $$D^b_i = \{x^b_{i,1}: n^b_{i,1}, x^b_{i,2}: n^b_{i,2}, ... x^b_{i,t}: n^b_{i,t}\}$$ where $$i \in {1, 2}$$ and $$b$$ is the block number. If the data was dichotomous, i.e., $$n_i^j$$'s were only 0 or 1, I could use Cochran's Q test to compare the two groups.

Is there such a test that generalizes Cochran's Q test to all categorical data and not just dichotomous data?