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Updated 29 September, to add Global Strategy poll. Six months ago, I reported that polling results indicated Jon Tester’s campaign was in trouble. Republicans applauded my fairness and perspicuity, while Democrats bemoaned my disloyalty and lack of faith. This report may cause those refrains to be reprised.

Another half year of polling reveals virtually nothing has changed. Tester’s still trailing Denny Rehberg, not closing the gap, and running out of time. At the New York Times’ Five-thirty-eight blog, Nate Silver reckons there’s a 60 percent chance Rehberg will win.

I’ll provide my thoughts on Tester’s predicament after I present a series of graphs displaying how his 2012 campaign differs from his 2006 campaign.

Most of Flathead Memo’s visitors are, I suspect, familiar with the margin of error statistic. But they should also be aware of a second, in some ways more important, statistic: the ballot lead probability; the probability that Smith leads Jones. Kevin Drum, now blogging for Mother Jones, explained the difference four years ago:

The idea of a “statistical tie” is based on the theory that (a) statistical results are credible only if they are at least 95% certain to be accurate, and (b) any lead less than the MOE is less than 95% certain.

There are two problems with this: first, 95% is not some kind of magic cutoff point, and second, the idea that the MOE represents 95% certainty is wrong anyway. A poll’s MOE does represent a 95% confidence interval for each individual’s percentage, but it doesn’t represent a 95% confidence for the difference between the two, and that’s what we’re really interested in.

In fact, what we’re really interested in is the probability that the difference is greater than zero — in other words, that one candidate is genuinely ahead of the other. But this probability isn’t a cutoff, it’s a continuum: the bigger the lead, the more likely that someone is ahead and that the result isn’t just a polling fluke. So instead of lazily reporting any result within the MOE as a “tie,” which is statistically wrong anyway, it would be more informative to just go ahead and tell us how probable it is that a candidate is really ahead. Here’s a table that gives you the answer to within a point or two:

The American Research Group polling firm has online ballot lead and MOE calculators. Praeter Software’s Statistics Pro (now priced at five bucks) calculates confidence levels and MOE.

Most of my graphs display the margin of error as error bars. A second statistic, the probability that Tester is leading, is displayed as a black square. A smoothing curve runs through the black squares; it is not a trend line.

Let’s get started:

During 2006, Tester led Burns by a very small margin, a margin that varied from poll-to-poll, and was usually within the MOE, but the probability that Tester was leading was high.

Six years later, Tester is eating Rehberg’s dust. Again, the differences are often within the margin of error, but overall the probability that Tester is leading is low.

The key to this graph — and this is the key graph — is the concept that a candidate who is trailing in the polls has a negative lead (I know, but please stay with me). For 2006, I subtracted Burns’ percentage from Tester’s percentage to get Tester’s lead. For 2012, I subtracted Rehberg’s percentage from Tester’s percentage. In 2006, Tester’s lead was a positive number. In 2012, it’s usually a negative number. Those numbers are plotted below.

Let’s start with Nate Silver’s analysis, which may understate Tester’s plight:

Republicans have a 60 percent chance of winning in Montana, where the Democratic incumbent Jon Tester is up for re-election. Although the polls are close to being exactly tied there, the model gives the Republican, Representative Denny Rehberg, an edge on the basis of the fundamentals. Mr. Rehberg is a strong candidate, with streaks of moderation in his voting record, which could play well to center-right Montana. And Mr. Tester won his election by an extremely slim margin in 2006 — during a year where the overall partisan climate strongly favored Democrats — which often bodes trouble for re-election.

Silver omits mentioning how weak Republican incumbent Conrad Burns was. Not only had Burns been roughed-up by allegations that he had an unsavory relationship with jailed lobbyist Jack Abramoff, he was north of 70 years old — and his mojo was slipping. He slept at meetings, rambled, and even took a cell phone call while on stage delivering a speech in Lake County. He’d thrown his hat in the ring once too often. Even so, Burns might have won had he not lost conservative votes to Libertarian Stan Jones.

Another Libertarian, Dan Cox, is on the 2012 ballot with Tester and Rehberg, but he lacks the name recognition and political stature of Jones, who was fairly well known for both his politics and his gray-blue skin, which he colored himself by drinking colloidal silver to fend off hostile microbes (under fluorescent lights, his face seemed soaked in woad). But if the race tightens up, Cox could spoil a Rehberg victory.

Thanks to early voting, which starts in two weeks, Tester is almost out of time. He’s ahead in money, not that it much matters. Both candidates have so much cash that they’ll have trouble finding ways to spend it all.