How to fit the correct multilevel (logistic) regression? I want to study the link between hospital competition and mortality. The competition a hospital faces is measured by the Herfindalh-Hirschman index (HHI). So I want to know if a hospital's HHI is lower (that is, if it faces more competitors) if its mortality decreases. I have three types of variables. Patient-level variables (age, gender, diagnosis...), hospital-level variables such as the HHI (The Predictor), the public or private status of the hospital, etc., and variables related to the patient's area of residence (Social Disadvantage Index of the patient's municipality of residence). I therefore decided to do a 2-level logistic regression. The outcome (mortality=yes/no) as well as all other variables at the patient level are level 1 variables. The variables at the hospital level are level 2 variables. I now ask myself the question: what level should the variables at the area-level be? Should they also be level 2 variables like the hospital-level variables, knowing that the hospitals where patients are treated and the municipalities where patients live are not the same kind of clusters and that a given patient belongs to both kinds of clusters at the same time. Or should there be a hierarchy between hospitals and the patient's area of residence? Finally, I don't know if I should do a fixed-effects model or a random-effects model (with only a random intercept or with a random intercept and slope). Note that I have hundreds of municipalities (areas) and hundreds of hospitals. Hundreds or even thousands of patients in each municipality and each hospital.
 A: There's no reason why you have to have a strict hierarchy. In principle you could consider area clusters for patients that are separate from the area clusters for hospitals. You would lose the hierarchical structure but might gain some precision. If you feel compelled to have a single area cluster, you will need to make that choice based on your understanding of the subject matter. Or instead of area clusters for patients you might consider something like the distance from the patient's home to the hospital, which might be associated with outcome.
You will quickly lose power with a large number of fixed effects, and random-effects nicely pool information among groups, weighting by the numbers in each group. Nevertheless, some variables might best be incorporated as fixed effects. With your primary interest in HHI, that should probably be a fixed effect. For others, think carefully about whether a particular nesting structure makes sense. For example, do you expect the relationship between outcome and gender to vary among hospitals, among areas, etc? For random effects, the choice of intercept-only or intercept plus slope models (and any associated choices of intercept-slope correlation structure) is a question for the subject matter.
With such a large study it would be wise to enlist some help from a local statistician who has experience in such modeling.
