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I am working on a project where I have to study the link between hospital competition and mortality (but also the duration of hospital stays). More precisely I want to determine whether the more a hospital faces competition the better it performs, and therefore the lower the mortality rate. There is an index (Herfindahl-Hirschman Index or HHI) that can be used to calculate the degree of competition a hospital faces. It can be calculated for each hospital. I am looking for an appropriate model to study this link. My first thought was a multi-level logistic regression model. So I want to proceed according to the following equation (That's just illustrative, it's not a mathematically correct equation:

Mortality= Intercept + a*Patient characteristics + b*HHI+ c*Hospital's other characteristics+ residuals.

Mortality (coded by yes or no) and patient characteristics (age, sex, diagnosis, severity of disease, etc.) are lower level variables. The Herfindahl-Hirschman Index (the hospital-related predictor) and other hospital characteristics (hospital status, overall volume of cases treated by the hospital) are higher level variables. In a multilevel model, predictors are most often lower-level variables (individual level), but in my case, the predictor is a higher-level variable. Does this change anything in the multilevel model? Or is another type of model more appropriate for my analysis? Finally, if my outcome variable is a continuous variable (such as length of stay, in number of days), do I need to run a multilevel linear regression?

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In a multilevel model, predictors are most often lower-level variables (individual level), but in my case, the predictor is a higher-level variable. Does this change anything in the multilevel model?

No, this is precisely one of the situations where a multilevel model is advantageous. The multilevel model splits the total variance in the outcome into that which is within clusters (hospitals) and that which is between clusters (hospitals). This means that hospital level predictors act only on the between-hospital variance in the outcome. Predictors at the lower level can explain both within- and between-hospital variance if those predictors' average value differs across hospitals.

Finally, if my outcome variable is a continuous variable (such as length of stay, in number of days), do I need to run a multilevel linear regression?

Yes, you can run these as linear models. With an outcome such as length of stay, you may be able to get away with a linear model, but you should probably also consider a model more appropriate for count data such as a generalized linear model, either straight Poisson or Negative Binomial flavor. Looking at the residuals and predictions from a linear model should help you see whether you can get away with treating the outcome as continuous.

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