Update: …and, the other shoe drops. PPP (D) reports a tie in Ohio, an eight-point swing from their previous poll. As you can see from the history graph, the result, Obama 291 McCain 247, with a Meta-Margin of 1.58%, is the closest the race has been since mid-June. The next few weeks (VP selections, conventions, McCain’s 72nd birthday) should be interesting to watch, to say the least.

Many individual components of the FiveThirtyEight model seem reasonable. But I also get the impression that Silver has added assumptions one at a time, sometimes at the suggestion of his readers. I do like the idea of one of the early steps, trend adjustment, which helps compensate for the relative infrequency of state polls. I was just discussing how to do this with my colleague Ed Witten. This step is potentially terrific, especially while polls are sparse.

Overall, the procedure appears to have grown piece by piece. Based on my own experience with getting floods of mail in 2004, I’ll guess that his code is somewhat juryrigged, and therefore not in a condition to be used by anyone else. Lack of overall advance design, lack of testability, and lack of transparency. It might be time to simplify and streamline. Every garden needs to be weeded now and then.

* * *

Today let’s deal with one frequently asked question in the last thread - non-independence of state results. A number of you wanted to know how one should deal with the fact that if sentiment in one state moves, others will tend to move in the same direction and by a related amount.

Before embarking on any sort of correction, one should stop and ask whether the correction will increase the accuracy of the ultimate prediction, and by a detectable amount. In the case of taking a snapshot, as I do, second-order corrections should alter the snapshot very little because Election Eve polls do so well (for example see Tanenbaum and me) in predicting Election Day outcomes. To put it another way: outcomes among states are coupled, and this is already contained in the primary polling data.

But for making a distant prediction, how should one proceed?

Go examine Charles Franklin at Pollster.com’s Presidential polling trends in 2000 and 2004. As you can see, the amplitude of swing in sentiment can be large - up to six percentage points of voter sentiment. To guess what will happen on Election Day, one would need to offset all the state margins by a randomly selected 0-6 percentage points. Then calculate the EV outcomes. Silver has a strategy for doing this.

Now, let’s go back to my 2004 analysis. For those of you who were not here then, I put a variable in the MATLAB script named bias that performed the function of shifting voter sentiments in all states at once. Varying bias allowed me to give median EV estimates for no swing, 2% swing to Kerry, 2% to Bush, and so on. I gave the results so you could see what the effect would be for your own favorite swing. A remnant of this calculation is still visible under “Interactive Maps” on the right sidebar (see “+2% for Obama” and “+2% for McCain”).

Does this complication make it necessary to do all those simulations on FiveThirtyEight? No. Let’s consider a simple example. Imagine that you think that national opinion could swing 2% to Obama, 2% to McCain, or not at all. Okay, then do my Meta-Analytic calculation for the three corresponding sets of win probabilities. Then average the three histograms. You’re done.

Now, that’s only three cases. If you want to space it more finely, you can do the Meta-Analysis at 0.1% intervals (2.0%, 1.9%, 1.8%…) and weight the sum according to your swing model. It’s still far easier.

Some of you think that state-state correlations can account for the spikiness of the FiveThirtyEight simulation histogram. That’s probably not true. Basically, the simulations are now drawing from a range of Gaussians instead of just one Gaussian, and the sum of similar smooth functions is still a smooth function resembling the original. Besides, now the simulations are undersampling many distributions, not just one.

But let’s take one more step back - a big step. Under the current win probabilities listed at fivethirtyeight, even the calculation of probability distributions is unnecessary. When most probabilities are between 5% and 95%, the result is a sum of many uncertain outcomes. The swing model then adds variation. Such a situation is in the regime of the Central Limit Theorem and related concepts. In general, when many variable outcomes are summed, the resulting distribution will look approximately Gaussian. One consequence is that you can get an accurate result simply by calculating a sum of state EV weighted by win probability. No simulations, no fancy swing model - just a weighted sum. It could all be done in “closed form,” i.e. one could write a formula for it. It wouldn’t be as intuitive, but it would be about as accurate.

The 95% confidence intervals are Obama [289,359] EV, McCain [179,249] EV.

The Popular Meta-Margin is Obama +2.3%, meaning that if his overall margin dropped by 2.3%, the median would be an exact toss-up. As an independent comparison, some narrowing seems to appear in the averages of national polls as well, as seen on Pollster.com.

Both results represent a narrowing from July 20th. This is surprising since news events featuring Obama (al-Maliki endorsement of his plan, visits to Afghanistan and Iraq) and McCain (several gaffes, including one on the surge) seem to favor Obama. This gets back to the question of which events are the ones that move voter sentiment. It’s not always obvious from our own reactions, especially if we ourselves are strongly decided. To see what seemed to move voters in 2004, click here.