Yesterday, Patrick Roy pulled the goalie with 11 minutes left in the game between the Avalanche and the Predators. The Avs were down by 3 and on the power play, so there was probably good justification for it, even though it didn’t work out. But when is pulling your goalie justified statistically? With the plethora of new stats available on NHL.com, and my physicist’s ability to make plausible assumptions that actually don’t correspond to reality at all, I decided to investigate.

**Comparing Goals For and Shots Against**

When your goalie’s pulled, every shot against is a goal, so to compare the likelihood of scoring versus getting scored against with an empty net, you can start with goals for and shots against statistics. Let’s take the Canucks as an example. According to NHL.com’s stats on 5-on-4 power play time (285 min) and goals (32), the Canucks have a PPGF rate of 6.74 goals/60 minutes. It’s harder to get a PPSA rate, but looking at Lack and Miller’s stats, they have together faced 36 shorthanded shots, while only giving up one shorthanded goal. The Canucks in total have given up two, from which I conclude that the second goal was an empty netter, and also (obviously) the only empty net shorthanded shot on goal the Canucks have given up. Thus, a total of 37 shots against in those same 285 minutes (assuming no shots against in 5-on-3 seems reasonable), for a rate of 7.79 shots against/60 minutes. These numbers are surprisingly close – pulling the goalie as a general strategy should clearly not be as tempting as it seems. They are, however, highly skewed in favour of going for an empty net. The difference in the real world is that (a) the man advantage you get is 6-on-5, which is considerably less of an advantage than 5-on-4, and (b) the opposing team is probably more likely to try to shoot at your net if it is empty.

We can also just count the number of goals for and goals against with the net pulled. It’s harder to get statistics on this, but from reading this kind of analysis, it seems that the ratio of empty net goals to goals with the goalie pulled for an extra attacker are anywhere from 2 : 1 to 3 : 1.

**The Likelihood of Scoring at Even Strength**

No matter the exact ratio, we know it will not be favourable, and so ideally, every team would like to score without pulling the goalie. But how likely are they to do that? To figure that out, we calculate the even strength goal scoring rate. For the Canucks, again using NHL.com stats and excluding power play and shorthanded time, I get around 2.41 goals/60 minutes (this is a slight overestimate because I didn’t factor in overtime).

So how likely are the Canucks to score in a given amount of time? To figure that out, we use the Poisson distribution

where k is the number of goals for in a certain period of time, and λ the expected number of goals for. We have assumed that goals in hockey are approximately Poisson distributed because they are random, rare and (sort of) memoryless (this paper by Alan Ryder verifies that assumption). Thus, for example, the likelihood of the Canucks scoring exactly one goal in a period played entirely at even strength would be the above expression with k = 1 and λ = (2.41/3) = 0.80 (the amount of even strength goals the Canucks score in an average period) – about 36 %.

**Comparing Likelihoods**

In order to decide whether to pull the goalie, a coach must decide what is more likely – that the team can score at even strength, or that they’ll have the first goal in an empty net situation. To figure out the latter, we simply treat “goals in an empty net situation” as balls in a hat, and ask which is more likely to be picked out. So, if the ratio of empty netters to extra attacker goals is 3 : 1, then that (25 %) is also the probability that the team with the extra attacker scores first. (Is this reasonable? It sounds too simple, but I can’t see a problem with it at first glance. Someone with some knowledge of stats would be useful here). To figure out the former, we simply use the Poisson distribution (well, actually the Poisson distribution’s cumulative distribution function because in actuality we’d be fine with the Nucks scoring 2, or, say, 7 (if last year’s Islanders could do it in one period…), goals in whatever period of time we pick). Figure 1 shows the likelihood of scoring n goals in the remaining time for the Canucks at even strength. Whenever the Canucks are down by that amount of goals and that likelihood dips below the 25% mark we have set for likelihood of scoring first with an empty net, our analysis says that they should consider pulling the goalie. That means that if they are down by 1 goal, according to Fig. 1, they should consider pulling goalie with 7 minutes left. If down by 2, they should do so late in the 2nd, and if down by 3, late in the 1st. If they’re down by 4 or more, they should pull the goalie immediately regardless of the situation.

**Problematic Assumptions**

Do I actually think that this calculation is correct? No. There are some very obvious problems. The most clearly problematic assumption is that in calculating the Canucks ESGF/60 we neglected to consider the other team. After all, they’ll be trying to score, too. So even if the Canucks are down by a goal, don’t pull the goalie, and proceed to score, this doesn’t guarantee that they will have tied the game. And what’s more, the quality of the other team clearly matters. A subtler problem is that the probability of extra attacker goals is inflated by the fact that if you only have the net empty for a short time, you can keep your best offensive forwards on the ice for that time. You may have a 25% chance of scoring first if you have the Sedins on all the time against the other team’s penalty killers. But when your four forwards are Higgins-Vey-Dorsett-McMillan or whatever, that probability reduces to something more like 2.5%. So of course this is not what a coach should actually do. But I wanted to point out that there is probably a way to figure out when to pull the goalie that’s an alternative to “let’s look at when we pulled the goalie before at this time in the game and whether we scored then,” and to point in the direction of what such a way could be.

Where did you get 2 to 1 or 3 to 1 for empty net to non empty net? That seems way off. Intuitively it’s more like 7 to 1, which I’m guessing would bring your suggested goalie pulling time to 1 minute being one down, 3 or so minutes being two down, and whatever being 3 down.

Like I said, it’s hard to find competent stats on this, but for example in 2010 the ratio of empty net goals against to goals for overall in the NHL regular season was 204 : 86 (http://www.theglobeandmail.com/sports/hockey/rethinking-when-to-pull-the-goalie/article4318101/?page=all). The Poisson statistics paper I listed also gives a ratio of roughly 2 : 1 for empty netters vs. reduced lead goals in 2004 (they need to investigate this because the time of a goal stops being random during the last 3 minutes of a game and Poissonian stats is no longer a good assumption overall). Another article (http://www.uscho.com/2015/01/22/extra-attackers-add-wackiness-at-games-end/) gives ratios between 2 : 1 and 4 : 1 over several seasons in college hockey, but the lower ratios seem to include delayed penalty goals, so it’s not a very convincing stat. 3 : 1 seems a safe ratio. It surprised me, too.

As for why teams that are good at two point conversions don’t default to two point conversions, there isn’t a statistics reason. I’m assuming it has to do with some kind of loss aversion (http://en.wikipedia.org/wiki/Loss_aversion). Probably coaches think of the one point as already earned and think of going for the extra point as possibly losing them a point, so it’s a psychological issue. Also in the wake of the Wilson-to-Lockette playcall there were a lot of articles written about how it offers you a game theoretic advantage in football to do unpredictably stupid things sometimes, so maybe it’s that…

How did you make that graph?

If you mean what software I used, it was Igor Pro (which I don’t recommend for this purpose. Using pyplot would have been a lot easier, in retrospect).

If you mean what is the function of each curve, then it’s the Poisson function with k given by the number of goals they need, and λ given by 2.41*(3600-t)/3600, where t is the number of seconds elapsed in the game, 3600 is the total number of seconds in the game, and 2.41 being the average ES Goals/60 for the Canucks in February 2015, I guess.