A few weeks ago, I offered a model to test the Senate playing field for Democrats. It concluded that, based on the president’s job approval rating alone, Democrats would be heavy underdogs to keep the Senate, with expected losses in the seven-to-nine-seat range.

The model ended with some important caveats. I wrote:

“Now, again, one might decide that, based on candidate quality and other issues, Democrats are poised to systematically end up on the high side of the probability calculus. I actually think this is probably correct. My general view is that this approach gets the odds right in the most commonly discussed races, especially since Republicans don’t seem poised to nominate weak candidates (for now). At the same time, though, weak Republican recruiting in places like Iowa and Minnesota, combined with unusually gifted Democratic incumbents in places like Virginia, probably means that the model overstates Democrats’ chances of losing more than eight seats.”

It’s worth addressing these caveats, as well as a few others that readers raised. There are really two shortcomings of the model that I identified: The first is the possibility that Democratic incumbents might fare better than the model suggests simply by virtue of being incumbents. The second is the possibility that the Republicans might nominate an unusually weak challenger in some of these seats, a la Christine O’Donnell.

Readers raised a few other points. Some questioned my usage of the president’s job approval rating dating back to his first inauguration for my simulation, since that “honeymoon” period is unlikely to repeat itself (it could, but that seems unlikely). Others questioned whether Democrats would need 50 percent of the vote to win a seat, since there are usually third party candidates that set the “line of victory” lower.

Finally, a few noted that Joe Manchin’s 2010 bid was not included in my calculations, and that he would seem to be an exception to the rule that no Democratic Senate candidate in a competitive race has run more than 10 points ahead of, or behind, the president’s job approval in his or her state.

So I’ve updated the model to respond to those concerns. I ran a regression analysis with the Democrats’ vote share in the 31 competitive races in 2010 and 2012 as my dependent variable. I once again estimated the president’s job approval rating in these states as an independent variable.

I added an independent variable for the presence of Democratic and Republican incumbents. I also used a dummy variable for cases where an outsider candidate or unusually weak candidate (the two aren’t necessarily coterminous) was nominated in an otherwise competitive race. The list of such candidates is as follows: Rand Paul in Kentucky, Sharron Angle in Nevada, Linda McMahon in Connecticut, Ken Buck in Colorado, Alexi Giannoulias in Illinois, Richard Mourdock in Indiana, Todd Akin in Missouri, and Shelley Berkley in Nevada.

Finally, I ran the model with and without Joe Manchin. To be honest, I’m not sure why Manchin wasn’t included in the original data set; I either took the list of competitive races from a Cook Report before Robert Byrd passed away or just overlooked him.

Regardless, we end up with a pretty nice model if we don’t use Manchin; our r-square is .71. The only races it really struggles to deal with are Heidi Heitkamp’s (it thinks she should have lost badly) and Kelly Ayotte’s (it thinks she should have won by about 10 points instead of 20). Overall it gets the arrow right in every case except Heitkamp’s (and would still miss her race even if we characterized Rick Berg as a “problem” candidate).

If we do include Manchin, however, our r-square drops to .45 and our standard error increases by about 33 percent. Dropping an outlying observation is one of the most difficult choices for an analyst to make. But there are two reasonably compelling reasons to do so here. The first is simply what we observed previously: We have a case of a single observation badly distorting the model. The actual result is about 3.4 standard deviations from the predicted, meaning that we have a 1-in-2,000 event occurring in a relatively small data set.

Second, and more importantly, we have good reason to believe that there will be no Joe Manchins in this election. Manchin rode into the election with an incredibly high job approval: 79 percent. It speaks to our current level of polarization that a substantial portion of those who approved of Manchin nevertheless voted against him.

Among the competitive races, the closest counterpart that we have this cycle is Virginia’s Mark Warner, who sports a personal approval rating around 60 percent (a case could be made that Mitch McConnell represents the inverse of this phenomenon).

The preferred outcome would be to build a model that incorporates candidate job approval, but unfortunately, we don’t have job approval questions for all of the Senate races in the exit polls. For now, it’s probably just a good idea to remember that Warner perhaps has a better chance of victory than the model predicts. It’s also good to remember that the model has him as the likely winner except at relatively low job approvals for Obama.

Before modeling out 2014, it’s probably a good idea to check ourselves using a so-called “out of sample set.” Basically, we want to see how our model would perform using real-world data before using hypothetical data.

I chose to use the 2006 elections, which featured 16 competitive races (according to Cook). If we set George Bush’s job approval rating at the 39 percent he recorded in the exit polls, and then run the simulation (using Conrad Burns and George Allen as damaged candidates) we end up with Republicans losing between five and eight seats 95 percent of the time, with six seats as the most probable outcome.

Now we can proceed with the simulation. Because we don’t yet know for certain where Tea Party candidates emerged victorious, I assigned a probability of an upset for each race. These are somewhat arbitrary, but I think are roughly correct, if not a bit overly generous to the Tea Party candidates: 40 percent in Alaska; 10 percent in Colorado; 50 percent in Georgia; 30 percent in Kentucky; 30 percent in Louisiana; and 50 percent in North Carolina. (Tea Party upsets are possible in states like Kansas, Mississippi, and South Carolina, but those races aren't listed as competitive and are Republican/polarized enough that they seem unlikely to flip in the event of an upset.)

I use 48 percent as the amount needed for victory, since third parties have averaged around 4 percent of the vote in Senate races over the past two cycles (excluding races like Maine’s in 2012). In some cases the threshold for victory will fall below this, but in others it will exceed this threshold. The errors should cancel each other out over the course of 10,000 simulations.

Applying the model to 2014, here are the probabilities for Democrats winning each individual seat at various job approval ratings.

For a summation of the results, see the following table:

Given the “bonus” that Democrats receive for potential Tea Party upsets and incumbent advantages, its unsurprising that they fare a bit better in each “bracket” than they did in the earlier iteration of this model. Again, even this may be a touch too generous to Republicans, given the strength of incumbents like Warner and (possibly) Jeanne Shaheen.

But because we’ve also downgraded Obama’s chances of scoring an unusually high job approval by using only data from his second term, the overall probabilities using randomized job approval scores look a lot like they did before: Republicans win the Senate about 80 percent of the time, they gain seven-to-nine seats about 45 percent of the time, more than that 25 percent of the time, and less than that 30 percent of the time:

Why is this different from outcomes predicted by other modelers, such as Alan Abramowitz and John Sides/Eric McGhee? Part of it is that the predicted outcomes really aren’t that different. Abramowitz’s most likely outcome is a GOP pickup of six, while this model’s most likely outcome is a pickup of eight. This has great substantive importance, but in statistical terms, the findings are well within the confidence intervals of the various models.

These other models also take a much broader swath, putting results from back to the 1950s into their data set. One of the assumptions behind this model is that something has substantially changed in the past few cycles as we’ve become increasingly polarized. Red states don’t vote for blue senators except in exceptional circumstances, and vice versa. There’s some support for this in the Sides/McGhee models; if they base their predictions off of findings from 1980 to the present, instead of from 1952 to the present, they find that Tom Cotton’s chances of winning in Arkansas skyrocket.

At the end of the day, it’s important to remember that these models are largely heuristic devices, especially this far out. There’s also still a lot we don’t know: While the model predicts 2006 quite well when you know who the “problem” senators will be and that Bush’s job approval will be at 39 percent on Election Day, it suggests that Republicans are likely to lose only three Senate seats when you randomize his job approval and don’t yet know that Burns or Allen will be badly damaged incumbents on Election Day.

But here’s the important thing: All the modelers seem to agree that the Democrats’ Senate majority is in real trouble, and that they may even be underdogs in their quest to keep it. The current polling certainly bears out this view, with Democrats behind in seven seats, below 45 percent in three seats, and below 50 percent in another three. Of course, there is still a lot of football to be played: The president’s job approval could improve, or the GOP could implode in the primaries. But for right now, it doesn’t look like that will be enough.