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But House Republicans also have a history of special elections that analysts said they should have won, including races in New York, Pennsylvania and Michigan.

While Mr. Sanford has momentum, history also is on his side. In the 45 special House races held in the past decade, the party of the seat’s former occupant held the seat 76 percent of the time, says the University of Virginia’s Center for Politics.

The South Carolina seat came open after Sen. Jim DeMint resigned and Gov. Nikki Haley tapped Rep. Tim Scott to fill that seat.

Mr. Sanford won a crowded Republican Party primary after a runoff while Mrs. Colbert Busch easily took the Democratic nomination.

Whoever wins Tuesday, though, likely will have the tenure end late next year.

If Mr. Sanford wins, the Republican establishment would be poised to put its full weight behind a formidable opponent in the 2014 GOP primary, political analysts say. Meanwhile, Mrs. Colbert Busch would have a tough time repeating in a full election in a district that hasn’t elected a Democrat since the 1970s.

Larry Sabato, director of the University of Virginia’s Center for Politics, put the odds of a Colbert Busch re-election victory next year essentially at zero “unless the [GOP] renominates Sanford or some kook.”

Analysts say special elections typically aren’t reflective of the national political scene, and instead should be viewed as isolated cases.

As for whether Tuesday’s special election has any deeper meaning on the national political landscape, Mr. Sabato said, it “not a bellwether of anything.”

“This is one of those strange specials,” he said. “It’s all about Mark Sanford. Does anyone believe that any other Republican wouldn’t be winning this in a landslide? It’s a one-party district.”