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The odds for logistic regressions can be computed as:

$$e^{x_i^{T} w}$$

If we thus only vary one regressor by one unit (e.g. a dummy variable) while holding constant the other variables, the odds ratio from comparing two observations yields:

$$e^{w_i}$$

But is there also a possibility to calculate the respective odds ratio from the contingency table in the multivariate case?

I know that in the simple case where we only regress on the dummy variable, we can relatively straightforward calculate the odds from the contingency table and compare it to $e^{w_i}$, which must yield the same outcome.

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    $\begingroup$ Well, technically, the answer must be "yes" since the computer does things we can do. But I'm guessing that there is no practical way to do it. $\endgroup$
    – Peter Flom
    Mar 31 at 10:11
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    $\begingroup$ Suggested title: something like "Can one calculate the odds ratio from the contingency table in the multivariate case?" $\endgroup$
    – rolando2
    Mar 31 at 18:28

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You can only do this with a saturated model, i.e., one with only categorical predictors that all fully interact with each other so that there are as many coefficients as there are cells of the design matrix. Otherwise, the logistic regression makes smoothing assumptions (i.e., that certain odds ratios are equal) that are not reflected in the contingency table itself.

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