If you have a larger table than a two-by-two, then you can still use the χ² test for association. The problem in this case is presenting the results and interpreting them.

The overall test, if statistically significant, simply shows that there is some association other than “chance” in the table. But it does not tell you what the association is.

Many times, it is fairly obvious what the association is because there are clear patterns in the data. In those cases, you can usually get by with a bit of slightly vague verbiage.

This association is evidently due to the generally increasing incidence of darker feather color with the larger breeds of parrot.

This type of verbiage, along with some graphical displays, may be more than enough to make the point.

What are your choices if you want more quantitative statements? Not many, unfortunately. You will likely have to use a log-linear model to make comparisons that are analogous to those available in the linear model situation.

Sometimes it is reasonable to collapse the data in the table to fewer categories in order to summarize the relationships.