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I understand that in case of separated data, logistic regression via ordinary MLE has an upward bias in the p values, which implies that any penalized MLE designed to reduce this bias will have more power in such cases. Specifically I'm considering Firth's penalization.

Can someone point me to a reference or an argument that compares the power of Firth logistic regression to ordinary MLE in general (that is, without the assumption of data separation)?

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With some more research I found this article where several methods are compared for power amongst other features, and they find that Firth regression has a power comparable to ordinary GLM:

https://www.nature.com/articles/s41598-021-82547-z.pdf

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    $\begingroup$ The too-large-p-value problem applies when unadvisedly using the Wald $\chi^2$ test. If you use the likelihood ratio $\chi^2$ test, complete separation is not a problem, and Firth's approach is not needed. $\endgroup$ Sep 11, 2022 at 18:14

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