# Am I doing hierarchical bayesian regression?

I'm doing a Bayesian logistic regression to predict the probability of my dependent variable Y with two predictors, one continuous (X) and the other categorical (C). I deal with C by building 3 models of Y ~ X where 3 is the number of levels of C (levels=a,b,c).

Now, I also have a second categorical variable C2 which is a sub-category of C. C2 has sub-categories a1,a2 (parent=level a of C), b1,b2,b3 (parent=level b of C) and c1,c2 (parent=level c of C). To deal with C2, I'm doing 7 logistic regressions where I use as my priors the ones I derived from the 3 models above.

For example, if model 1 (which models Y ~ X for level a of C) shows gaussian traces with mean m and standard deviation s, I use m and s for the normal priors of the two sub-models (which model Y ~ X for sublevels a1 and a2 respectively).

Questions: does it make sense to proceed like that? if yes, is it what is called hierarchical bayesian regression?

## 1 Answer

No, what you are describing is not a hierarchical model. Hierarchical model is a single model (all at once) that describes such hierarchy, in this case this is a random slopes model, where the slopes vary among groups. Moreover, the procedure is incorrect, because you are using the same data multiple times to calculate same things (first to estimate higher-level parameters, then use them as a "prior" and use same data combined with this prior for estimating new parameters etc.), this will lead to your model being overconfident, because it would see the same information multiple times. If you want to learn about hierarchical regression model (with emphasis on Bayesian approach), check the Data Analysis Using Regression and Multilevel/Hierarchical Models book by Andrew Gelman and Jennifer Hill.

• Thanks @Tim, very appreciated. What is not clear for me is the following: suppose I conducted an analysis last year but only on the highest order grouping (groups A,B,C). So three models with 2 parameters each ($\beta_{0}$ and $\beta_{1}$) have been built. Now suppose I want to extend these models to include the subgroups. Can't I use the information gained from the first analysis? I thought that bayesian learning was exactly that: incorporate new insights to refine models, isn't that true? – Patrick Apr 17 at 16:08
• @Patrick first, you would be using same data twice, so the model would be overconfident. It's like you found some information on encyklopedia Britannica and cross-referenced it with Wikipedia that quoted as a source Britannica... Second, you would gain nothing by this as you are using the same data, so why not simply train a different model? What would be the benefit of using such "priors"? – Tim Apr 17 at 18:06
• Understood @Tim. And if it were new data, it would be ok, right? And to answer your questions, I thought that it would be a good idea, after having estimated parameters for each group A,B,C to reinject this information into the submodels so if submodel (A,a1) for example does not have a lot of data, at least it is "guided" by top level parameters of model A. – Patrick Apr 17 at 18:26
• @Patrick if it was separate data, then it would be typical case of Bayesian updating. I get the argument with small subgroup, it makes sense. But standard Bayesian way would be to use hierarchical model that estimates both, higher and lower level effects, that can deal with such cases. Why not use it? – Tim Apr 17 at 19:40
• simple answer: because I don't know how to do hierarchical modelling! I'm just starting to read about it (and I'm not sure I really understand the concept of hyperpriors). Stay tuned for future questions! :) Thanks again Tim. – Patrick Apr 17 at 19:59