Four by-elections are taking place today, in three different provinces. The Liberals will most likely win 3 of them, but we should remember that only one riding will likely change colour today: Brandon-souris.

I covered these by-elections and polls before, and interested readers can find the post here.

The latest Forum poll (who has been polling these four ridings for weeks now, something that I think is quite unusual for by-elections) shows a slightly different story from the first three polls. We could say that we finally saw some changes!

Let's take a look riding by riding and let's compare these polls to the projections we should expect based on the latest national/provincial polling numbers put into the model/simulator.

**1. Bourassa.**

The former riding of the new Montreal Mayor Denis Coderre. The LPC has won there easily over the last decade and more. And since the last election, Justin Trudeau has boosted his party support in Quebec to new highs (in recent years), even actually taking first place in most polls. We shouldn't therefore expect any suspense in this riding.

The Forum polls have all shown the new Liberal candidate largely ahead. Dubourg is almost polled as high as what we would expect if Denis Coderre was still running in this riding. Or at least he was a month ago. With recent polls showing the LPC as high as 35-38% in la Belle Province (see the Crop poll from last week for instance), it seems Dubourg is doing slightly worse than what we would expect from Coderre. I estimated the Coderre effect to be around 13-points here and this is pretty much what we're observing here.

The big talk talk over the last day of course was the drop of 7 points from Dubourg and the gain of 10 points by the NDP. If we use the last three polls, we actually see the Liberals down 13 points (from 56% to 43%). So, should the Grits be worried? Maybe, but let's look at two key elements. First of all, the sample size of these polls is really small. In Bourassa, Forum had 238 observations. In the other polls, it varied between 280 to 400. A party at 43% in a poll with 238 observations has margins of error equal to 6.3%. It means the 95% confidence interval is 36.7%-49.3%. Quite an imprecise estimate. For the NDP, at 31%, its margins of error are 5.9%, giving us confidence interval of 23%-36.9%.

So it seems that it'd indeed be possible for the NDP to be ahead of the Liberals. Except that this isn't how it works. That would be true if there was only two parties. In this case, the "twice the margins of error" would be techniclaly correct. But it's not the case. In this specific example, the margin of error for the difference between the Liberals and the NDP is 10.8 points. It means that the actual lead of 12 points is outside and is thus significant. In other words: statistically speaking, at a 95% confidence level, the last Forum poll in the riding of Bourassa shows that the Liberal candidate will win. Of course, it does leave 5% chances that this poll is completely wrong. Remember that the "twice the margin of error" rule is wrong for Canadian politics where we have multiple parties.

The other element here is time. While the Liberals clearly experienced a negative trend during the last month (the difference between the projected support for Dubourg in this poll is significantly different from the poll published on November 5th), there might be a race if the campaign was lasting a couple more weeks. But given that the election is today, it seems unlikely the trend for the Liberals (as well as the new upward trend observed for the NDP) will be enough to create a huge surprise. Again, keep in mind that with sample sizes so small, variations are expected and can even be important. After all, the 7-points drop during the last two polls is not even significant!

All in all, anything but a Liberal win would be a huge surprise. However, Dubourg must win with at least a 10 points lead. Anything else and it would/should be seen as a failure. After all, even during the disaster that was the 2011 election for the Liberals, they won this riding with more than 8-points. The loss of Coderre should be more than compensated by the new popularity of the LPC in this province.

Conclusion: polls say Liberals (relatively easily despite a drop in the end), model says Liberals easily. So anything but a Liberal win would constitute a big surprise. The real story will be to see how easily (or not) does Dubourg win.

**2. Toronto Center.**

Forum even published two polls in this riding during the last week. Similarly to what we observed in Bourassa, the Liberals' lead has decreased continuously. Indeed, we moved from a 15 points lead in October to a small lead of 8 points yesterday. The sample size is greater than in Bourassa (544 observations). This 8-points lead is actually still significant, but barely. The minimum lead to be significant is 7.8%.

However, remember that margins of error work both ways. So while it is possible for the NDP and LPC to be actually tied in this riding, it's just as likely for the Liberals to have a huge lead. Don't confuse what is possible and what is likely. If you look at the numbers from Forum, you see the Grits very stable. The NDP's rise is done at the expense of the Conservatives. Is it a local effect or an indication of a broader effect in Ontario? Hard to tell. Recent polls haven't really shown a NDP rise in this province or nationally (except for one poll by Ipsos-Reid). The latest numbers from Forum put the Liberals first in Ontario with 41%. head of the Conservatives with 31% and the NDP at 23%. Put into the simulator, you would expect the Liberal candidate to be around 55%, largely ahead of the NDP.

So are the Liberals actually lower than that in Ontario (and the NDP higher), or is it simply a local effect? Again, hard to tell. But as in Bourassa, the LPC won this riding in 2011 by 11 points. They must hold this riding by at least this margin (and actually, we should really expect them to win bigger than this). The simple fact that we could have a race in this riding is puzzling.

Conclusion: polls say Liberals (but getting closer), model says Liberals. We should therefore expect a win by the Grits, but again the margin will be what we should be looking at.

**3. Brandon-Souris.**

This riding should be very easy to project. The Liberals received 5.4% of the votes in 2011! Even with Justin Trudeau as the new (and popular) leader, it should be a no brainer. Yet, the Forum polls have all showns a likely Liberal victory! Clear local effects are in play here (in particular, some controversies regarding the CPC nomination). And as opposes to the first two ridings, the Liberals lead hasn't decreased in the last polls (it actually increased and the change is significant). So expect a Conservative upset there. This is definitely a riding that Harper should never has lost.

Conclusion: polls say Liberals, model says Conservatives. But the model is most likely useless in this case as the strong local effects make the provincial numbers and trends meaningless. The Liberals should gain a seat. Only possible problem: the turnout. But if Conservative voters are upset at their party, it might not be that hard to get them out to vote against it.

**4. Provencher**

The Forum polls have consistently shown a healthy Conservative lead. Despite a slight drop in the last poll, it is very difficult to imagine (and statistiscally very unlikely) to see an upset here. Given that the CPC won the riding with more than 70% of the votes in 2011, we shouldn't expect anything less. The numbers are still meaningful though. Provincial polls (that aggregate Manitoba and Saskatchewan though) usually show the Liberals up quite a bit compared to 2011. Still, polls have the CPC ahead and the projected results in Provencher show that the Tory candidate should have a bigger lead than what Forum shows. And the NDP shouldn't be that low.

This riding could actually be the most interesting of the four. Not in term of suspense, but as far as giving us information regarding where each party stands in the Prairies. A strong Liberal showing could indicate that this party is back as the main opposition in this region, relagating the NDP in third place. Ironically, while most of the coverage will likely be about the three Liberals wins and the gain in Brandon-Souris, the most interesting number could well be the margin of defeat in Provencher.

Conclusion: polls say Conservatives, model says Conservatives. A Liberal win here would be as much of as surprise as a Liberal loss in Toronto-Center or Bourassa.