On the way to winning back the majority in the House on Tuesday, Republican challenger after Republican challenger overcame significant fund-raising gaps to beat their opponents, bolstered by a deluge of spending by independent groups.

They are arguably the most clear-cut examples of the impact that these Republican-oriented outside groups had on the midterm elections, spending record sums on dozens of races in the wake of a landmark Supreme Court ruling in January that eased restrictions on corporate political spending.

Most of these groups, which are able to take in contributions of unlimited size from individuals and corporations, do not have to disclose their donors. While it is hard to sort out the exact difference they made, their success rate, particularly in races in which Republican challengers would have otherwise been badly outgunned, raises the prospect that a relatively small number of deep-pocketed donors exerted an outsize influence on Tuesday’s results.

Democratic incumbents raised more than twice as much as their Republican opponents over all in three dozen House races The New York Times classified as competitive, yet Republicans won about half of those contests on Tuesday, according to a Times analysis. In all but a handful of those races in which fund-raising was lopsided, Republican outside groups significantly outspent Democratic ones.