To hospitals, billions of dollars depend on how they define a patient's stay. Even though a patient may be in the hospital for a simple surgery, Medicare might classify the procedure as outpatient care and pay a lower fee to the hospital. Or someone may come into the emergency room complaining of chest pain but leave after just a day or two, once the tests have ruled out a heart attack. Does that qualify as an inpatient stay at a higher Medicare payment? With government auditors and contractors cracking down on hospitals for what they contend are overbilling practices, hospitals are paying more and more attention to how they classify care when filing claims to Medicare, the federal health insurance program.