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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 $\pi_{ij}$?

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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.

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  • $\begingroup$ Thanks, it was helpful. $\endgroup$
    – Dihan
    Mar 1, 2022 at 17:11
  • $\begingroup$ $OR= \frac{\pi_{00} \pi_{ij}}{\pi_{i0}\pi_{0j}}$, 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? $\endgroup$
    – Dihan
    Mar 1, 2022 at 17:33
  • $\begingroup$ @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. $\endgroup$
    – EdM
    Mar 1, 2022 at 17:54
  • $\begingroup$ 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. $\endgroup$
    – Dihan
    Mar 1, 2022 at 18:11

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