Annie Krause moved into a nursing home in Detroit in 2015, when she was 98 years old. She had grown frail. Arthritis, recurrent infections and hypertension had made it difficult for her to manage on her own.

When the facility’s doctor examined her, he found a mass in Ms. Krause’s breast and recommended a biopsy — standard procedure to determine what sort of tumor this was and, if it proved malignant, what treatment to pursue. Once diagnosed, breast cancer almost always leads to surgery, even in older women.

“If she were a passive person, she would have had a lumpectomy,” said Ms. Krause’s granddaughter, Dr. Mara Schonberg, an internist at Beth Israel Deaconess Medical Center in Boston. “But my grandmother was very strong-willed. She said no, no, no, she didn’t want any procedure.”

That didn’t stop the doctor from recommending a biopsy, however.

Having spent years studying how best to inform older women about breast cancer, Dr. Schonberg said that patients’ decisions — about screenings and treatments — have proved stubbornly resistant to change.