PITTSBURGH — Gov. John Kasich often speaks about mental health in his campaign for president. He has defended his decision to expand Medicaid in Ohio by highlighting its benefits for mentally ill residents.

He is probably the only Republican candidate this year to ask a crowd, “Do you know what it’s like for somebody to live with depression?”

The question, posed at a rally in upstate New York recently, threw a hush over a room of 1,000 people. Mr. Kasich went on:

“There are people here who know exactly what I’m talking about.”

Mr. Kasich is one who knows.

His only brother, Richard, 59, has struggled with depression disorders since college. He was occasionally hospitalized and today receives disability benefits for mental illness.

The Kasich brothers have taken vastly different paths from their hometown, McKees Rocks, Pa., an industrial suburb of Pittsburgh.