Hi Martin:
First of all, thank you for your interest in my problem.
I will answer and asks stuff in the same order:
I am aware of the issues with the log link. There are 2 reasons to use it. First, My outcome is common for some of the strata, which means that if I want to present my results with prevalence ratios (cross sectional study) I need to convert to logit exp(a + bx)/(1+exp(a + bx)). Becasue the disease is common, it is quite described this method will overestimate the prevalences (or risk if this would be a cohort or experimental study). The most common option to skip this problem described in Epi is the log link specially with categorical covariates only as in my case. Leaving some references: https://www.frontiersin.org/articles/10.3389/fvets.2017.00193/full https://www.ncbi.nlm.nih.gov/pmc/articles/PMC521200/. The only reason to prioritize prevalence ratios instead of OR is the intuitive interpretation of the former. The discussion about the problems with the log link have been discussed here as well: Log binomial random intercept model
I checked the wikipedia link. Great resource! thank you. I have 2 concerns about moving to another functions: the typical interpretation of coefficients and secondly, the behavior of the estimated p considering again, the common outcome. In the first case, I do not want to use something that is unfamiliar and nobody understands. The second one, I have no idea what would be the behavior of the posterior pred distribution given a common outcome using these alternatives.
Yes! I definetely take the advice on the priors.
the cauchy comes from here http://www.stat.columbia.edu/~gelman/research/published/taumain.pdf
Although I can definetely takes suggestions on this.
Thanks again