# Odds ratio for an $I \times J$ table

I am trying to find a general expression for the odds ratio in a two-way table. I know for a $$2 \times 2$$ table odds ratio can be expressed as: $${\rm OR} = \frac{\pi_{11}\pi_{22}}{\pi_{12}\pi_{21}}.$$

I am trying to find a general expression of the odds ratio for an $$I \times J$$ table. Is there any way to write the OR in terms of $$\pi_{ij}$$?

Is there any way to write the OR in terms of $$\pi_{ij}$$? (Emphasis added.)

There is no single odds ratio for an $$I \times J$$ contingency table. As Agresti says (page 54, Categorical Data Analysis, 2nd edition)

For $$I \times J$$ tables, it is rarely possible to summarize association by a single number without some loss of information.

For rows $$a$$ and $$b$$ and columns $$c$$ and $$d$$, you can write the odds ratio as a generalization of the $$2 \times 2$$ formula:

$$\frac{\pi_{ac} \pi_{bd}}{\pi_{bc} \pi_{ad}}.$$

You can describe all such odds ratios in terms of a set of $$(I-1)(J-1)$$ odds ratios. For me, the simplest is to set cell $$IJ$$ as a reference ($$b=I,d=J$$ in the above) for the $$(I-1)(J-1)$$ combinations of rows and columns not involving either row $$I$$ or column $$J$$.

There are summary single measures of association of nominal variables in $$I \times J$$ tables, explained for example in Sal Mangiafico's R Handbook . Cramér's V is frequently used; its Wikipedia page has links to other such measures.

• Thanks, it was helpful. Mar 1, 2022 at 17:11
• $OR= \frac{\pi_{00} \pi_{ij}}{\pi_{i0}\pi_{0j}}$, where $\pi_{00}$ is the reference cell, and $i=1,2, \ldots, I$, and $j=1,2, \ldots, J$. Does this make any sense? Can I write it like this? Mar 1, 2022 at 17:33
• @Dihan close, but you have to be careful with your indexing. With your setting $\pi_{00}$ for the reference cell and with $i$ and $j$ each running from 1 to $I$ and $J$ respectively, you might seem to have an $(I+1) \times (J +1)$ contingency table. Indexing of contingency table rows and columns typically starts with 1, not 0.
– EdM
Mar 1, 2022 at 17:54
• Yes, I understand. I am working on a paper, in that paper, the author starts indexing from 0 instead of 1, which makes me think like this. Though I do not like to start with 0. Thanks for your feedback. Mar 1, 2022 at 18:11