Vulnerable House Democrats, touting their independence in their re-election efforts, might be breathing more easily this week after Republicans failed in their bid to make Speaker Nancy Pelosi (D-Calif.) a political albatross.

In the first test-case for the strategy, the special election Louisiana’s 6th district, Democrat Don Cazayoux pulled out a win, making him the first Democrat to hold the seat in 32 years.

The victory came after the National Republican Congressional Committee and a conservative group ran a pair of ads linking Cazayoux to national Democratic leaders, including Pelosi, and what they called their radical liberal agenda. Democrats immediately seized on the results as a signal that GOP efforts to nationalize Congressional races as referendums on Pelosi’s leadership are doomed.

Yet, some vulnerable Democrats interviewed last week hemmed about a possible Pelosi visit to their district. Most said they will run on their own records — rather than that of party leaders — and any attempt to wrap the Speaker around their necks will fail.

Rep. Jason Altmire (D-Pa.) pointed to his voting record as one of the least supportive of Democrats as a defense against potential GOP attacks linking him to Pelosi. “I think I’m sixth from the bottom,” he said. “It’s going to be hard for [the NRCC] to tie me in with her.”

Asked if he would welcome a Pelosi visit to his northeastern Wisconsin district — where he faces a rematch of his closely fought 2006 race — freshman Rep. Steve Kagen ducked. “I’m focused like a laser on my district right now. I have no idea what other people are doing.” Does that mean he’d welcome a Pelosi visit? “It means just what I said.”

Likewise, Rep. Zack Space (D-Ohio) said he “has not asked for any surrogates to come to my district, and I don’t intend to.” But Space described the NRCC strategy as “grasping at straws.”

“The people in my district are much more concerned with what I’m doing. I will be judged by my own performance, not somebody else’s,” he said.

Rep. Bill Foster (D-Ill.), who won an upset victory in March to take the seat of former Speaker Dennis Hastert (R), said mailers in his race linking him with Pelosi made no impact. “We shrugged it off,” he said. “People know who she is, but I don’t think it’s a big issue.”

Republicans believe otherwise. NRCC Chairman Tom Cole (Okla.) last week told reporters that the party is reviving its failed 2006 strategy that a vote for a Democratic candidate was a vote for Pelosi as Speaker. The difference this year, Cole said, is that Pelosi is now nationally known and seen as a polarizing liberal.

The divergence of views about Pelosi’s impact on local races means that even in a year when presidential contenders at the top of ballots will set the tone of the campaign, she will figure prominently.

On the ground, she has proved herself a potent fundraising force. In the air, she will star as a bogeyman in Republican ads.

An NRCC ad in the Cazayoux race said the election was really about Pelosi and Sen. Barack Obama (D-Ill.). In it, a narrator said, “The Obama-Pelosi team needs Don Cazayoux to win this special election. See, Cazayoux will be a vote for their agenda, so they fund his campaign. Is Obama right for Louisiana? Is Pelosi? You decide.”

In another, paid for by the conservative Club for Growth, a narrator argued the consequences of a Cazayoux vote. “A vote for Cazayoux is a vote to increase Hillary Clinton and Nancy Pelosi’s liberal majority in Congress.”

For his part, Cazayoux put some distance between himself and Pelosi by suggesting last week he would consider supporting someone else for Speaker.

Democratic strategists said when their candidates run in the South and West — home to many of the most vulnerable House Democrats — they must define themselves as Independents who won’t march in lock step with party leadership, a fact that leaders themselves recognize.

“Running in the South, you have to reach a threshold comfort level with voters that you’re not a national Democrat,” said John Anzalone, a pollster with Anzalone Liszt Research, which advised Cazayoux’s campaign. “Nancy Pelosi understands that and so does anyone in her position.”

Added Dave Beattie, with Hamilton Campaigns: “Throughout the South and West, it’s not about being a federal candidate. It’s about how do they localize who they are. You’re not going to see a lot of going out on the stump by the leaders of either party. But [candidates] will be happy to use them for fundraising.”

Pelosi raised $62 million for the Democratic Congressional Campaign Committee last cycle, and so far this cycle has raised $27 million, said Jennifer Crider, a former Pelosi aide now serving as communications director of the DCCC.

From Pelosi’s political action committee, PAC to the Future, she has doled out $465,000 directly to candidates, according to CQ MoneyLine. Instead of clipping her wings, Pelosi appears to be keeping a campaigning schedule roughly on par with that of other House leaders. She has visited 24 states to date, compared with 24 states visited by Minority Leader John Boehner (R-Ohio), and 25 states visited by Majority Leader Steny Hoyer (D-Md.), according to tallies compiled by their offices.

Nevertheless, Ed Goeas, a Republican strategist and president of The Tarrance Group, said Pelosi will provide GOP candidates “value-added” in their races. He pointed to results from the Harris Poll in February that showed Pelosi’s negative ratings have jumped 12 points, to 57 percent, while her positives have dropped 13.

He said the survey showed more than 80 percent of Americans can identify the Speaker. “And if they don’t know now, it sounds to me like Tom Cole is going to make sure they do by the election,” he said.

In Louisiana, Republicans insist the strategy helped their candidate, Woody Jenkins, narrow the margin with Cazayoux and said they will continue to deploy it. In a memo sent out later Saturday night, the NRCC said “a Democrat was clearly favored to easily win this election before Republicans invoked the names of Barack Obama and Nancy Pelosi. This should come as a warning shot to Democrats.”

Victorious Democrats were dubious, saying the strategy could not keep a GOP-leaning district from falling into the Democratic column.

“The Republican argument is simply not supported by the facts,” Crider said.