When a hospital (or a surgeon) has a high death rate, the first response is "but we treat sicker patients". At which point risk adjustment is called for.
Risk adjustment is basically a multiple regression analysis, with 'place of treatment' as the predictor of outcome, and everything else as a confounder. The 'everything else' in a risk adjustment is a set of indicators of severity (risk of death). That's ok if your measure of risk is something very objective such as blood pressure but it's not ok if you pick a measure such as '%age of patients admitted as an emergency'.
At first sight, it seems obvious that hospital A, where most patients are admitted as emergencies, must be treating sicker patients than hospital B, where there are few emergency admissions. But there is a problem: policies on emergency admission may differ. So the data item 'emergency admission' does not mean the same in both hospitals (whereas blood pressure 100/60 does). You can't use emergency admission in risk adjustments.
Here is an interesting worked example of all this using real data. The statistical warning sign is that a predictor (emergency admission) signifies a higher risk of death in hospital B than hospital A. The predictor carries a different meaning or value in the different places, so technically this is an interaction between the two predictors.
Moral: Don't measure something till you understand it thoroughly. (Kelvin famously thought the opposite!)