Do NHL referees call "make up" penalties? Part III

The last two posts talked about how NHL referees are more likely to "even-up" their calls, issuing the next penalty to the opposing team 60% of the time.

This post, I'll show a regression I ran to quantify the effect a bit better. If you're not interested, just skip the technical parts (smaller font). If you're *really* not interested, you can probably just skip this entire post, since the results are pretty much the same as shown in the previous posts.

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Technical notes 1:

Even though we're interested in whether the referee called the next penalty to the "other" team, I set up the regression to predict whether the referee called the next penalty to the *home* team. That just makes everything easier to interpret, but, as I'll describe, it still lets us estimate the "even-up" effect.

In the study, I ignored all misconduct penalties, all first penalties of the game, and all penalties where the other team had a player called at the same time. (I treated those penalties as if they didn't exist, so they didn't interrupt "consecutiveness" of the two surrounding penalties.)

Non-dummy variables I used: Time gone in game. Time since last penalty.

Dummy variables I used: PP goal on last penalty. SH goal on last penalty. Home team lead, from -3 to +3, except 0 (that is, six dummy variables), where anything more or less than 3 goals was coded as 3. All eight of the previous variables interacted with "whether the last penalty was to the home team." And, of course, the dummy for "whether the last penalty was to the home team" itself.

The regression shows the home team percentage diminishes during the game, by about 1 percentage point per period. In all the numbers in this post, I just used the beginning of the game. If you want the middle of the game, subtract about 1.5 percent from each "home team" percentage (or add 1.5 percent to each "visiting team" percentage) if you want to adjust to the middle of the second period.

Also, the regression says you have to subtract about 1 percentage point for every 20 minutes since the last penalty. I didn't bother for this post. If you assume penalties are usually around 5 minutes apart, feel free to subtract 0.25 percentage points from each of the "home team" percentages.

Those two time adjustments won't affect the "even-up" numbers, just the raw percentages of home team penalties.

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OK, first, let me show you the percentage of penalties taken by the home team, by game score. Clearly, teams are more likely to take penalties when they're ahead in the game.

After the visiting team took the last penalty, the home team took:

46.3% when down by 3+
49.2% when down by 2
53.1% when down by 1
61.2% when tied
64.3% when up by 1
67.9% when up by 2
66.8% when up by 3+

And after the home team took the last penalty, the home team took:

34.5% when down by 3+
32.8% when down by 2
32.3% when down by 1
35.0% when tied
42.6% when up by 1
47.6% when up by 2
52.9% when up by 3+

Obviously, you can do this for visiting teams just by subtracting all the percentages from 100.

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Now, we can calculate the "even-up" effect as the difference between the lines of the two tables. When the score was tied, the home team took:

61.2 percent after visiting team penalty
35.0 percent after home team penalty
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26.2 percent difference

You can convert to visiting team just by subtracting the first two numbers from 100%. The difference has to come out the same. I'll do that anyway. When the score was tied, the visiting team took:

38.8 percent after visiting team penalty
65.0 percent after home team penalty
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26.2 percent difference

It turns out the "even-up" difference is highest for tie games. Here's the full breakdown:

11.8 percent difference down by 3+
16.4 percent difference down by 2
20.7 percent difference down by 1
26.2 percent difference tied
21.7 percent difference up by 1
20.3 percent difference up by 2
13.9 percent difference up by 3+

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Tango suggested there might be an extra "compassion" effect when the team scores a PP or SH goal. He seems to have been right. The effect is small, relative to the overall effect, but still enough to affect the games:

-- If the home team scored a PPG on the previous penalty, add 1.2 percentage points to the above differences.

-- if the visiting team scored a PPG on the previous penalty, subtract 3.2 percentage points to the above differences.

-- if the home team scored a SHG on the previous penalty, add 2.8 percentage points from the above differences.

-- if the visiting team scored a SHG on the previous penalty, subtract 0.7 percentage points from the above differences.

The PPG numbers are statistically significant. The SHG numbers aren't, but they go in the right direction and are about the right magnitude, so I think it's reasonable to consider them as decent estimates.

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So, as I promised in the second paragraph: the results of the regression seem to match what we found in the previous posts.

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Technical notes 2:

For full disclosure, here are the coefficients for all the variables in the regression. I'll present them in "here's how to calculate the percentage of home-team penalties" format. (If you prefer a table, the full computer output is here (pdf). You'll be able to tell what the variables represent by matching the coefficients to what's below.)

Start with 0.6119 (constant).

Add -0.2619 if the home team took the last penalty.

Add -8.15E-06 for each second that's passed in the game. (About -.01 per period.)
Add -7.39E-06 for each second that's passed since the last penalty (not significant, p=.104, but magnitude is reasonable and has the right sign).

Add -0.1483 if the home team is down by 3 or more goals.
Add +0.1436 if the home team is down by 3+ and also took the last penalty.

Add -0.1196 if the home team is down by exactly 2 goals.
Add +0.0983 if the home team is down 2 and also took the last penalty.

Add -0.0807 if the home team is down by 1 goal.
Add +0.0545 if the home team is down 1 and also took the last penalty.

Add +0.0309 if the home team is up by 1 goal.
Add +0.0450 if the home team is up 1 and also took the last penalty.

Add +0.0668 if the home team is up by 2 goals.
Add +0.0591 if the home team is up 2 and also took the last penalty.

Add +0.0559 if the home team is up by 3 or more goals.
Add +0.1228 if the home team is up 3+ and also took the last penalty.

Add +0.0115 if a PP goal was scored on the last penalty
Add -0.0438 if a PP goal was scored on the last penalty and that penalty was to the home team.

Add -0.0072 if a SH goal was scored on the last penalty
Add +0.0350 if a SH goal was scored on the last penalty and that penalty was to the home team.

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