WASHINGTON — The Republican defeat in a special Congressional contest in Mississippi sent waves of apprehension across an already troubled party Wednesday, with some senior Republicans urging Congressional candidates to distance themselves from President Bush to head off what could be heavy losses in the fall.

The victory by Travis Childers, a conservative Democrat elected in a once-steadfast Republican district on Tuesday, was the third defeat of a Republican in a special Congressional race this year. In addition to foreshadowing more losses for the party in November, the outcome appeared to call into question the belief that Senator Barack Obama of Illinois could be a heavy liability for his party’s down-ticket candidates in conservative regions.