Yet one sure lesson of the surprising Congressional result from northern Mississippi is that the use of Mr. Obama as an electoral tactic — Republicans resorted to it heavily in the contest — is at best a double-edged sword. At worst it is a guillotine for Republican candidates in areas with substantial black populations, like the Mississippi district won by Mr. Childers, where 26 percent are African-American. Indeed, Tuesday’s Mississippi vote emerged as a case study in the effects and consequences of focusing on Mr. Obama.

“We realized the Republican machine was on the attack,” said Mr. Buck, the state representative who helped Mr. Childers. “They wanted to say he was tied to Barack Obama. The question we asked was, What’s wrong with that? We wanted to prove to them that there’s nothing wrong in Mississippi with a person being tied to Barack Obama.”