Different Discharge Dispositions in a Competing Risk Model I am building a data set on which to estimate, via regression, the impact of covariates on the cause-specific and subdistribution hazards for time to discharge from a hospital.  The main competing risks are death and discharge.  However, besides death, there are varying types of discharge that also preclude one another.  A patient may be discharged home, home with home health, or to a nursing facility.  Each of these different discharge dispositions is likely to be associated with a different mean time to discharge.  Is the correct method for treating the data to run the estimations with four competing risks?  Namely: death, discharge home, discharge home w/ home health, discharge to nursing facility.  As discharge is an outcome, I don't see a way of also including it as a covariate.

If you need to distinguish among the different types of (non-death) discharge, then what you describe seems to be a competing-risks situation with 4 mutually exclusive outcomes (plus censoring for those not yet dead or discharged).
The multi-state vignette of the R survival package illustrates how to approach this type of modeling. It also points out some potential limitations in Fine-Gray subdistribution hazards, as you envision. You have many choices to make, for example whether covariates have shared or separate associations with the 4 potential outcomes, so work carefully and try to avoid the temptation of using the observed outcomes when making those choices.
