An Indiana Department of Education analysis of the state’s school voucher program released this week estimates it cost the state $16 million last year, but advocates say the calculation is flawed.

An expansion of the voucher program in 2013 led to a flood of new students enrolling in the program last fall. A total of 19,809 students used vouchers in 2013-14, more than double the prior year’s total of 9,324. The first-year number was 3,919. Indiana’s voucher program is the fastest growing in U.S. history and the nation’s second largest.

Proponents say vouchers offer opportunity to poor students at low-performing schools to attend usually higher-scoring private schools, and help families with children whose parents feel they might fit better in private schools to afford it. Voucher opponents argue they drain away tax dollars intended for public schools while benefiting just a small subset of children.

Until 2013, the vast majority of students using vouchers had to first attend public school for two semesters under state law. But that rule was loosened last year, allowing siblings of students using vouchers and students assigned to F-rated schools to avoid attending public school first. In all, about half the students who joined the program — more than 5,000 — would not have been eligible without the changes.

The key question is how many of those children would have gone to private schools anyway and how many were persuaded to choose private school instead of public school.

Vouchers redirect to private schools a dollar amount less than the full per pupil state aid for each student that uses them. So a small amount is returned to state coffers and shared among all public schools after all the program’s expenses are paid. In 2012-13, $4.9 million that was saved was redistributed statewide. That’s up from about $4.2 million the prior year.

Voucher opponents argue that the new rules allow many children, such the siblings of those already in the program, to use public dollars for private school even if they intended to enroll in private school all along. In that case, they actually cost the state money by using state aid they otherwise would have foregone without the voucher program. The state only saves money when a student who would have gone to public school instead goes to private school because of the opportunity to use a voucher.

In calculating the cost to the state for 2013-14, state officials had to estimate how many of the program’s new participants would have gone to private school anyway. Their estimate is too high, voucher advocates say.

That was the message from the Institute for Quality Education, formerly School Choice Indiana.

“If the 19,809 voucher students were instead being served in their traditional public school, it would have cost taxpayers at least $130.7 million (as compared to the nearly $81 million distributed to voucher families,” Tosha Salyers, communications director, said in a statement. ”That is a minimum savings of $49.7 million for the Hoosier taxpayer in the 2013-2014 school year alone.”

But Greg Porter, D-Indianapolis, said the education department’s study proves opponents right.

“What we are seeing is the continued fallout of an unfettered expansion of a program without any measure of accountability to determine its ultimate effectiveness and no concern about the long-term impact upon a public school system that does not have the ability to pick and choose who it can educate,” he said. “Within a very short time, the voucher program has become a zero sum game for our public schools. Every student they had that they lost to a voucher costs them. If there is a student living in their district that has gone into the voucher program, the school district loses funding as well, even though that student never went to the school.”

Vouchers offer more tuition aid for poorer families. Eligibility depends on family size and annual income. The income guidelines allow a family of four with annual income less than $43,500 to receive up to 90 percent of the state aid for a child’s public school education toward tuition. Families of four making more than that amount but less than $65,250, can receive 50 percent of the state aid amount.

Per-student state aid varies by district. In Indianapolis Public Schools, for example, the figure is about $8,000 per student. A maximum of $4,700 can be spent on private school tuition for elementary schools. There is no such cap for high schools.

The creation of vouchers in Indiana brought intense opposition from teachers’ unions. The program, approved by the legislature in 2011, survived an Indiana State Teachers Association lawsuit challenging its constitutionality. The Indiana Supreme Court turned the suit away last year and declared vouchers legal. Before she ran for state superintendent, Glenda Ritz had been a plaintiff in the lawsuit.