On the Limits of Configural Frequency Analysis: Working With Small Tables

Configural Frequency Analysis, or CFA, has long been treated as a dependable way to probe what is happening inside a contingency table. Most of the time researchers reach for it when their data spread across three to five variables, with each variable carrying roughly three categories. That is the comfortable middle ground where the method tends to behave. What we wanted to find out was different: can the same approach still tell us anything useful once the table shrinks all the way down to 2 x 2?

To answer that, we combined analytical work with simulation. The picture that came out was not encouraging for one common version of the method. First order CFA, the variant that adjusts for the main effects before looking at the cells, runs into trouble on a 2 x 2 table because the individual cell tests end up depending on one another. When that dependency is present, the results lose much of their interpretive value, and the format of the table is simply too cramped for the base model to do its job cleanly.

Zero order CFA fared better. Part of the reason comes down to counting: a zero order analysis of a 2 x 2 table has three degrees of freedom to work with, whereas the first order version leaves you with only one. That extra room matters. It gives the zero order approach something to test against, which is exactly what the first order model lacks in this setting.

The short version is that table size is not a detail you can wave away. CFA earns its reputation on tables of moderate dimension, and pushing it down to the smallest possible cross-classification exposes where its assumptions start to fray. If a 2 x 2 table is all you have and you still want to apply CFA, the zero order base model is the one worth considering.