Updated

Democrats are working hard in Arizona in the special election Tuesday to win the open House seat once held by Gabrielle Giffords -- an effort to avoid a second election loss in as many weeks during the final months of the presidential election cycle.

Giffords -- who has resigned to focus on recovering from a gunshot wound to the head last year -- has made several recent campaign appearances for hand-picked candidate Ron Barber, also a former staffer. Barber has a double-digit lead on Republican candidate Jesse Kelly in the 8th Congressional District race, but analysts still consider the contest a tossup.

Democrats were dealt a setback June 5 when they failed to oust Wisconsin GOP Gov. Scott Walker in a recall election.

Democratic leaders tried to replace the first-term governor because they thought he and state Republican lawmakers overstepped their authority when they eliminated labor agreements for most state public employees in an effort to cut a multibillion-dollar budget shortfall.

Some political analysts say the defeat showed voters nationwide still feel passionate about wanting elected officials to keep government living within its means, and that the loss was a setback for labor unions and the Democratic Party that it traditionally supports.

“The (national) importance of this election depends on the outcome,” David Wasserman, who analyzes House races for the non-partisan Cook Political Report, said Tuesday. “If Barber wins, it’s what we expected, Gifford’s aide inherits the seat. If Kelly wins, it sends a message about President Obama’s job performance.”

Wasserman told FoxNews.com the Washington-based firm still considers the race a tossup because special elections are a little tougher to predict than regular elections, in part because voter turnout is less predictable.

Barber's 12-point lead is according to an average of polls by RealClearPolitics.com

Republicans have been trying to make the southern Arizona House race a referendum on Obama and his handling of the economy. Kelly is a former Marine who narrowly lost to Giffords two years ago.

Giffords, 42, resigned from office about six months ago, following the January 2011 incident in which a gunman fatally shot six people and wounded 13 others during a constituent event she was hosting outside a Tucson supermarket.

Gifford has made few public appearances since the shooting, but has returned to Tucson in recent days to support Barber.

She attended a concert Saturday night and listened as her husband, astronaut Mark Kelly, spoke on her behalf in praising Barber, who was wounded in the shooting.

The couple also met Sunday with volunteers to thank them for their work.

"Some of those people used to be her supporters," Kelly said after the meeting. "But now Ron has his own team that's energized to make sure he gets across the finish line on Tuesday, and Gabby is very excited about that."

Jesse Kelly -- no relation Giffords' husband -- continued to make the case in the election's final hours that Barber and Obama are out of touch with people in the district.

Republicans have a 26,000-person edge over Democrats in voter registrations.

"It's time to put a stop to the Barber-Obama team," Kelly's campaign said in its final ad.

Outside groups have spent more than $2 million on the Arizona race.

Barber, 66, had a sizable fundraising lead in late May, but spending from conservative groups helped reduce it.

The district is a rare swing district that is competitive virtually every election.

Giffords defeated Kelly by about 4,000 votes in 2010 when the election focused on immigration and when tea partiers rallied to the tough-talking former Marine.

Now, the economy and jobs are atop voters' concerns.

"Emotions are very high. People are very concerned about the economy and tiring of just limping along," said John Ellinwood, a spokesman for the Kelly campaign.

Kelly, 30, has called for lower taxes and more energy production as a way to improve the economy. He would roll back federal regulations and environmental protections in an effort to boost oil and gas drilling.

Barber also is trying to convince voters that he understands their concerns. He frequently talks about building up the solar industry and the need to cut taxes, but only for the middle class. While Kelly has made it clear he would not support any income tax increases, Barber has said the wealthy need to "pay their fair share."

The Associate Press contributed to this report.