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Does it make sense to estimate a confidence interval for the odds ratio / logistic regression model when the sample size is nearly equal to population size? For example, if the sample size is 50 and the population size is 60.

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In general, there is no problem with forming a confidence intervals when your sample is near the size of the (finite) population. You typically just need to use the finite population correction:
\begin{align} {\rm finite}\ SE &= SE \times \sqrt{\frac{N-n}{N-1}} \\ \\ &= SE \times\sqrt{\frac{60-50}{60-1}} \\ \\ &= SE\times.41 \end{align} In your specific case (i.e., logistic regression), it is worth noting that this would be a Wald confidence interval, and Wald CI's rely on large sample theory. (There is some relevant discussion here: Why do my p-values differ between logistic regression output, chi-squared test, and the confidence interval for the OR?) Your sample of 50 should probably not be considered large, so the CI will probably not really have the coverage it promises.

Nonetheless, I think a confidence interval would add some legitimate information to help interpret your results, so I might still do it (with the appropriate caveats explicitly noted for readers).

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