Link functions and interpreting credible intervals

Question

I am pretty new to statistics, and was trying to interpret credible intervals from a bayesian analysis I had preformed. Some of my models are glms, and so have a link function. I know that to correctly interpret my parameter estimates on the scale of my question I need to transform them appropriately.

Let's say though as part of your criteria for interpreting whether or not a parameter influences an outcome you have a credible interval threshold (say does not include 0 at .95CI). Do you base this judgement on transformed CIs, or untransformed CIs if you've included a link function in your model?

Answer 1

GLMs try to answer a nonlinear question by transforming it into a linear problem. If within the linear model the credible interval does not include $0$ than you can assume, that this paramater has a non-zero influence in the linear part of the model. Whatever has an influence on the linear part has influence on the non-linear part as well. To just detect, whether there is an influence, you only need to look at the linear part. If you want to measure the importance in terms of the dependent variable, then you will have to take the link function into account.

Answer 2

It actually doesn't matter, but what will change is "not including 0". For example, a CI not including 0 in an untransformed negative binomial or poisson model, actually doesn't include 0 - meaning the effect is either positive or negative (whatever your values are). Now if you exponentiate (log-link) these values (estimates and CI), then you are in multiplicative terms, which means the "0" is now relative to "1". if your estimate is 1, that means that the response Y changes by a factor of 1 every unit increase in predictor X. Anything * 1 = Anything right, so 1 is now 0. Does this make sense? So if your exponentiated CI = 0.15 - 2.15 this is overlapping 1, meaning you don't know if the effect is positive (>1) or negative (<1), and in this case the untransformed CI will be overlapping 0.