KOGELO, Kenya — Waving her walking cane and smiling broadly, the stepgrandmother of Barack Obama celebrated Wednesday as this tiny Western Kenyan village danced and rejoiced after the US president won four more years in the White House.

Kogelo, a dirt-road town where children play soccer in bare feet, was the home of Obama’s father, and claims several relatives of the president among its population. The family matriarch is Sarah Obama, who was married to the president’s late grandfather.