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“There were really stark differences,” said Ms. Stainer, 45, a licensed practical nurse.

In the nearly 20 years that Megan Stainer worked in nursing homes in and around Detroit, she could almost always tell which patients near death were receiving care from nonprofit hospice organizations and which from for-profit hospices.

Ms. Stainer, now a private duty nurse and certified death doula in Hamburg, Mich., also found nonprofits more willing to keep patients enrolled and for-profits more prone to “live discharge” — removing patients from hospice ostensibly because they no longer met the criteria for declining health, then re-enrolling them later.

“It seemed like people were being discharged when they still needed their services,” Ms. Stainer said.

In addition, the progression that follows bears a striking resemblance to music by the English composer John Stainer, who used the same dramatic chord and elegant escape in his setting of “O Come, All Ye Faithful” — which concluded his Epiphany anthem “I Desired Wisdom,” published in 1876.