Like lots of others, I’m anxiously watching forecasts of the US election outcome. But it’s hard to figure out what’s going on, with Biden way ahead in the polls, behind in the betting markets and rated a 70 per cent chance by the model at 538.com. Inspired by this post from Andrew Gelman, who is working on the Economist model (Biden currently a bit over 80 per cent), and an informative tweet from Nate Silver, I’ve managed to improve my own understanding a bit. At least I think so.
Silver’s tweet confirms that the Electoral College system gives Trump a significant advantage relative to an election by popular vote. He syas
Chance of a Biden Electoral college win if he wins the popular vote by X points:
0-1 points: just 6%!
1-2 points: 22%
2-3 points: 46%
3-4 points: 74%
4-5 points: 89%
5-6 points: 98%
6-7 points: 99%
With that information, it’s easy enough to fit a normal distribution to the margin, and get an estimate probability of winning. By fiddling with the numbers, it’s easy to replicate the 538 probability estimate and also to get a probability distribution looking fairly similar to those displayed on te site. My best estimate is N(5,4), that is, the mean value for the margin is 5 points and the standard deviation is 4. The mean value is consistent with the description of the state level estimates on the 538 site, which (very roughly speaking) take the existing polls (which currently have Biden ahead by 7.4 nationally) and then give Trump 1 point for an incumbency advantage (reducing the margin by 2 points).
Looking at the Economist model (which doesn’t necessarily agree with 538 on the exact distribution of the Electoral College advantage) it fits pretty well with N(6,3)
The standard deviation is a big deal here. N(5,4) implies a 95 cent range of, roughly, -3 to 13. I can’t say I find this plausible, at least assuming the election proceeds without armed intervention. Short of personally inventing a vaccine and hand-delivering it to the entire US population, I can’t imagine anything that would give Trump a 2.5 per cent chance of winning the popular vote. And it’s equally hard to see what would push him much lower than he is now.
If you would like a more optimistic story, you can get one by focusing exclusively on the polls where Biden’s lead has been consistently between 7 and 9 points, consistent with a distribution like N(8, 0.5), which puts Biden at 99 per cent.
I should alert readers that I don’t always get this kind of calculation correct, so feel free to check it out and correct it if necessary.