Researchers also examined what happened when one of EmCare’s top competitors — TeamHealth — took over a handful of mostly nonprofit emergency departments. There, they found a smaller increase in out-of-network billing and virtually no change in hospital admissions, testing or coding.

Analysts point out that hospitals, despite any patient complaints, can benefit financially from the increased testing and admissions EmCare has delivered. In the study, surprise bills were more common at for-profit hospitals than at their nonprofit competitors.

“They’d have to have their heads in the sand to be totally unaware” about the out-of-network billing, said Leemore Dafny, a professor at Harvard Business School, who reviewed the research.

EmCare’s emergency room management has come under scrutiny before. The company was named in a 2011 whistle-blower lawsuit against Health Management Associates, a for-profit hospital chain. The suit alleges that both EmCare and the hospitals pressured E.R. doctors to increase admissions and tests, even when the physicians believed they were not medically necessary. The company “repeatedly terminated physicians and E.R. medical directors” who pushed back, the suit says. The case, which was brought by a hospital chief executive and a former EmCare executive, is still pending. Envision said it does not comment on pending litigation.

Hospital emergency departments, which must take all comers regardless of their health insurance, were once viewed as financial drains. Then hospital leaders started to see the E.R. as the front door, critical to attracting paying patients. In the early 1990s, emergency departments accounted for a third of admissions to hospitals; today, they account for half, according to a RAND study.

As in so many other parts of the modern economy, turning operations over to large outside contractors has been a big part of the transition. Nearly a quarter of all emergency room doctors now work for a national staffing firm, according to a 2013 Deutsche Bank report.