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I performed a logistic regression using Stata's bayes: wrapper and obtain the following histogram from 10,000 posterior distribution samples of the log(odds) of my exposure of interest. I also obtained the standard, frequentist distribution of the log(odds) ratio (solid red line).

As a frequentist, I could report that the p-value for the estimate as >0.05 and the 95% confidence intervals include an estimate of 0 (OR 1.0) or no effect. This is essentially a non-conclusion. I am tempted to use the Bayes estimates to make further statements about the association between exposure and outcome, but it's unclear to me what I would say that is fundamentally different. An odds ratio of 0.95 or less is an interesting effect size and the probability of this is 0.538. This seems like an effect worth knowing more about when stated in the Bayes framework.

My question: With the Bayes distribution in hand, what can I say about the effect size I couldn't say with the standard frequentist approach? What can I conclude from Bayes that is distinct from standard logistic regression?

Histogram of log(odds) for exposure.

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    $\begingroup$ Usually we use "exposure" to denote the "X" variable and "outcome" or "event" to denote the "Y" variable. All the same, the log-odds as you say, is the intercept term and summarizes no association: the significance test is whether the prevalence in default group is 0.5 or otherwise: useless. If you summarize the posterior of the odds ratio (or better yet), you can credibly summarize what the effect might be because probability is now your belief (provided you have the right prior). When you can do this, significance testing is pointless. $\endgroup$ – AdamO Mar 15 at 19:21
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The interpretation doesn't differ that much. The difference is that instead of point estimates, you end up with posterior distribution, so now, unlike in frequentist case, interpretations like "given the assumptions we made and the observed data, with 95% probability, the result lies between X and Y" etc.

As a frequentist, I could report that the p-value for the estimate as >0.05 and the 95% confidence intervals include an estimate of 0 (OR 1.0) or no effect.

With Bayesian estimate you would know what is the probability that the result is actually greater then zero.

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