As the voting for the Republican presidential nomination gets underway, super PACs have already emerged as a big factor, spending more than $13 million so far in Iowa and other early-voting states. Yet a growing number of the independent expenditure groups active on behalf of presidential candidates has made moves to ensure that information on their donors will likely remain hidden until votes are cast in three more states.

Four of the biggest spenders—the pro-Romney Restore our Future PAC, the pro-Santorum Red, White & Blue Fund, the pro-Huntsman Our Destiny PAC, and the pro-Gingrich Winning Our Future, recently changed their ...