Hoping to set a precedent for other states, Michigan’s labor unions spent months pushing a referendum to amend the state’s Constitution to prohibit the legislature from ever enacting a law that would curb the powers of public employee unions.

But this push to enshrine collective bargaining rights in the Constitution was roundly defeated in Tuesday’s election, 58 to 42 percent — an embarrassing loss for labor in a state known as a cradle of American unionism.

While union leaders have been quick to claim success in other Election Day contests, from the re-election of President Obama to the defeat of a California proposal that would have limited their ability to spend union dues in political campaigns, they have been largely silent on the Michigan loss.

Some political experts say the measure was voted down for the same reason that the four other ballot initiatives to amend Michigan’s Constitution were defeated: voters were wary of tinkering with their state’s Constitution.