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His initial amendment passed on a near-party line vote, with just four Democrats voting for it and 10 Republicans voting against it.

The House legislation still must go through the Senate, which is controlled by Democrats who likely will try to block it or water it down.

The upper chamber has seen its own share of census fights over the years — most recently in 2009 when Sen. David Vitter, Louisiana Republican, tried to attach an amendment that would have had the 2010 census ask a question about citizenship.

Mr. Vitter pointed to studies that found that several states end up getting more representatives in Congress based on counting of their illegal-immigrant population, and he said that was unfair. The Constitution says seats in Congress shall be divided based on total population, not on citizen population or voting-age population.

In addition to defunding the ACS, the House passed another amendment preventing the Census Bureau from imposing fines on those who refuse to fill it out.

But Mr. Groves told Congress earlier this year that this would amount to making the ACS voluntary, which would cut down on the number of people responding and make the survey’s data far less reliable. He also said it would increase costs by $66 million a year.

Mr. Groves said he could find no evidence of anyone actually having been assessed the fine, but said keeping it as an option is a way to compel better compliance.

Still, making the survey voluntary could be one middle ground between the House and Senate when it comes to making a deal later this year.

Some lawmakers in the House have even floated the idea of making businesses pay for the data as a way of recouping costs, while others said the Census Bureau — which has occasionally advocated using statistical sampling for the decennial census — could make use of that technique to correct for any errors in a voluntary census.