Relation between Canadian and American elections?

You would have to have been frozen in a block of ice not to know that this year there will be a presidential election in the USA. It is very exciting for all involved: the ultimate showdown between right and left, good and evil, corruption and change, and so on.

But, on the other hand, unless you've been unfortunate enough to be frozen in that peculiar block of ice called CANADA, where dragons roam and people live in frozen huts made out of snow (we'll thankfully soon be able to upgrade to mud, thanks to the changing climate), you probably didn't know there's a federal election coming up there too. It's been a real cliffhanger these last couple of sessions of parliament: Will they call an election? Will this be a confidence motion? Will anyone do anything at all? Now there's no turning back: October election. But admittedly not as exciting as the American election.

This year's federal election will be held in the fall. That's a bit unusual, but not very. More unusual, however, is that it falls in the fall in a US presidential election year. I don't know that that's ever happened except in 2000, when Chretien's Liberals were elected in Canada, and the Republican George W. Bush was more or less elected in the US.

Now, these two parties are quite different. The Liberals are a centrist party, a fair bit like the Democrats in the US, and the Republicans are a right-wing party, something like the party that's now called the Conservative Party of Canada. Simplifying greatly, the 2000 elections in the US and Canada were a kind of "mismatch": conservatives in the US, centrists in Canada.

To me, that stands out a bit. After all, I have a vague recollection of moving out of a Republican America and into a Conservative Canada when I moved as a small child. Then, all through the 90's, we had Chretien's Liberals, and the US had the Democrat Clinton as a president. Ignoring the differences between these parties, and the fact that Chretien's Liberals had a majority government the whole time while Clinton had a Republican Congress from 1994 on, and all the various deep differences between Canadian and American politics, we might think there was a link between these two things. And that might be exciting for people trying to forecast the upcoming US presidential election: maybe the Canadian election in October will serve as a kind of pulse-taking for the continent.

I decided to test the theory that there's a relation between these two things. I looked at all the Canadian federal elections that took place within a year of a US presidential election (after all, who really cares about those Congressional elections?), before or after. I checked to see if they "agreed": if the Liberals won in Canada and the Democratic candidate won in the US, or if the Conservatives (whatever name they were under at the time) won in Canada and the Republican candidate won in the US, then they agreed. Of course, even with the vague historical knowledge that I have, I know that the American parties were too different from now for this to mean the same thing before about FDR, so I started at 1932.

Here's the data (sorry about the weird gap here...):

US Election Year	Agreed?	Canadian election was...
1940	Yes	Before
1944	Yes	After
1949	Yes	After
1952	No	After
1956	Yes	After
1968	No	Before
1972	No	Before
1980	No	Before
1984	Yes	Before
1988	Yes	After
1993	Yes	After
1997	Yes	After
2000	No	After
2004	No	Before

The first column gives the year of the US presidential election. The second column gives the result: did the Canadian federal election agree with the US presidential election, or did it come out differently? And then the third column, which I will get to, tells whether the Canadian election was held during the year preceding the US presidential election, or the year following the US presidential election.

So, are they related? If a Canadian election and a US election aren't related, then they should agree with a 50-50 chance. (Okay, we probably know more than that, so we could work in things about who won the last election, the economy, etc etc, but let's keep it simple.)

Probability tells us that if we see a lot of "events" (unpredictable things), like elections agreeing or disagreeing, and we know (or pretend) that each one happens independently of the event previous to it, and we know (or pretend) that we know the probability of each event going one way or the other (like agreeing, rather than disagreeing), then the probability of having so-and-so many of the events go a certain way, should be such-and-such.

In real life, if we see a bunch of events, we can see if the number of events that came out a certain way lines up with the theory that there's no pattern (here, the theory that they're not related), by looking at how often they really came up that way (related). If this looks really unlikely according to the theory that there's no pattern, then we're safe to think there's a pattern. This is called a binomial exact test.

So, what does the binomial test tell us here? Despite first impressions, a mere 57% of the elections agreed. There's a 79% chance of that occurring if there's no relation. So no evidence here.

What if we think the American election is related to a later Canadian election, but not an earlier one (maybe because it influences the results)? Restricting ourselves to the cases where the Canadian election was held during the year following the American election, we find that they agreed 75% of the time, but there's a 29% chance of that happening if there's no relation. That's not good enough proof for a scientist by any stretch of the imagination. And it's not good enough proof for me.

Finally, what about restricting ourselves to the cases where there was a Canadian election before the American election? Here, the elections disagreed 66% of the time, which seems like it might not be a coincidence - but as far as we can tell, it is. There's still a 69% chance of that happening if there's no relation.

So what does all this mean? There's no evidence that watching the Canadian federal election this year will tell us anything at all about the outcome of the US race. Of course, there's all those other factors: Canadians, like everyone else in the world, are watching the US race with bated breath; there hasn't ever been a Canadian federal election right before the US presidential election; etc. But only dreamers will take any of that seriously if they've looked at the history.