# Bayesian Logistic Regression and model comparison

I asked how to conduct a Bayesian logistic regression in another post. I have been advised to use the brms R package.

I have conducted this analysis, but being a beginner in statistics and coding, I need to know if my analysis is right.

I have a data set with one binary outcome (“Object”). Participants had to choose between two objects. I want to check if this choice was not influenced by a continuous predictor (“PHQ”).

First, I have fitted a bayesian logistic regression model with Non-Informative Priors following tutorial of Fang & Van de Shoot (https://www.rensvandeschoot.com/tutorials/generalised-linear-models-with-brms/).

Could you help me to interpret these outputs?

• I see that the credibility interval contains the 0. Does could be in favour of an absence of effect of the predictor on the outcome?
• How to interpret estimate in a report?

Second I have repeat the procedure to fit e null model without predictor.

I have used the LOO function to compare the two models.

How can I interpret that? How is it informative for my hypothesis that PHQ (predictor) does not predict the binary outcome (Object)?

Thanks.

Only looking at the results of `M1`, I’d say that the evidence you have doesn’t really show a relationship between `Objet` and `PHQ`. Having said that, your credible intervals are pretty wide, and I wouldn’t be surprised to have only positive credible intervals if you include a 19th observation.

It doesn’t seem particularly informative. If you make a density plot of the posterior samples for the coefficient for `PHQ`, you will see that there is more probability for positive values, but the estimate for the parameter is compatible with negative, zero, and positive values. Since you are interested in a particular point hypothesis (that the coefficient is exactly zero), then you would hope to see the posterior distribution for the coefficient for `PHQ` to have statistics like the mean near to zero and a small standard error. How close to zero and how small a standard error is a question of how precise you need to be for your specific task. It seems that you only have 18 observations, so as @tomas.rossetti says, your posterior distribution for the coefficient for `PHQ` may very well change quite a bit with more observations. Since getting exactly zero won’t ever happen in the real world with real data given enough observations, then given an ever increasing amount of observations, your estimate for the coefficient will converge on some non-zero value with some decreasing SE, but will perhaps be for all practical purposes considered a non-existent effect. Say for example that you collect 999,982 more observations and re-run the model with more data and find now that the estimate is 0.002 (0.001-0.003). Maybe that’s small enough to be of no consequence, but it is measured more precisely. Trying to determine that something has no effect would seem difficult to me, because how far from exactly zero is good enough for ‘no effect’?