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I'm conducting a population study analyzing a rare outcome (prevalence 2-3%) with a rare exposure (~225 exposures, 800 000+ non-exposed). This gives me 5-8 cases among the exposed. Data on 7-8 covariates are available among both exposed and controls.

My plan is to perform logistic regression, adjusted for covariates without interaction terms, to get an OR (with a confidence interval) for the outcome with respect to the exposure.

Does the low number of cases among the exposed pose a problem for the logistic regression, apart from me possibly getting very wide confidence intervals? I'm thinking problems perhaps in the form of non-convergence of estimators, creating significant bias, or something else. If so, is it still sensible to perform the regression but interpret it with care or should it not be performed at all in this setting?

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The most likely problem, beyond "getting very wide confidence intervals," will be perfect separation, with some combination of predictors exactly distinguishing the individuals with the rare outcome. That's not a reason to avoid doing the analysis, it just means that you will have to use one of the penalization approaches noted in that page and elsewhere (e.g., Firth regularization or ridge regression) to get reliable coefficient estimates.

You seem to be as skeptical as I am that you will find any significant association between your rare outcome and your extraordinarily rare exposure, but you don't know until you try. If you do find a "significant" result you might want to see if you can repeat it on bootstrapped samples of your data.

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  • $\begingroup$ 1) Is perfect seperation always related to overfitting? I have 20 000 individuals with the rare outcome (but without exposure), and only 7-8 covariates (2-3 binary), and thought overfitting wouldn't be a problem. 2) Does the rare exposure add to the problem of perfect seperation or is it only dependent on how rare the outcome is? $\endgroup$ – Peter Z Aug 31 at 10:08
  • $\begingroup$ @PeterZ perfect separation can be related to overfitting but isn't always. If you happened to find a perfect way to distinguish class memberships, the fitting process wouldn't converge. I can't say how much the rare exposure issue will contribute. This has to do with how combinations of predictor values are related to class membership, so general statements are hard to make. Fear of perfect separation shouldn't prevent you from modeling, just use one of the approaches suggested on the page that is linked in the answer if you find it. $\endgroup$ – EdM Aug 31 at 12:27

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