Dr. Callahan’s sister was growing frantic. “I said, ‘Maureen, just do it,’” Dr. Callahan said. “Sign him out.”

Her sister signed the A.M.A. form that hospitals typically demand before releasing patients against physician recommendations, acknowledging that patients understand and assume the risks, medically and legally.

Mr. Callahan went home and went to sleep, but his physician daughter remains livid.

“He should have been discharged right from the E.R.,” she said. “This was cookbook medicine, done without thinking. It was very adversarial.”

Such events happen more commonly than one might think. Though A.M.A. discharges occur far more frequently in younger patients, a recent study in The Journal of the American Geriatrics Society analyzed a large national sample from 2013 and found that 50,650 hospitalizations of patients over age 65 ended with A.M.A. discharges.

“This is a very conservative number,” said the senior author, Dr. Jashvant Poeran, an epidemiologist at the Icahn School of Medicine at Mount Sinai. Had he included outpatients who left emergency rooms against medical advice and those who simply walked out and never actually signed a form, the total would have been much higher.

And the numbers are rising, Dr. Poeran found: A decade earlier, 45,535 hospitalizations of older patients ended with A.M.A. forms. The proportion has climbed from 0.37 percent of senior hospitalizations to 0.42 percent — an uncommon event, to be sure, but a fraught decision nonetheless.

“It’s always been one of the most difficult ethical dilemmas,” said Dr. Arthur Derse, who directs the Center for Bioethics and Medical Humanities at the Medical College of Wisconsin.