NHL has a refereeing problem and their own data shows it

Yes, the game's next call usually goes to the team that's been called less

What the data bears out is that a team that has 1 fewer penalty called against them than their opponent has the next penalty called on them 60% of the time. If the gap between them and their opponent is 2 penalties, that number jumps to 69%. If the gap is 3 or more, it jumps to 76%.

These are model-adjusted for score, game time, strength on the ice and season, and averaged over home and away. The raw shares are almost identical (61% at a gap of 1, n = 5,021), so the adjustment isn't driving the result. The pattern holds when all strength-changing penalties are included (13,863 calls).

The trend is even more pronounced in rates

Let's take a look at this using rates, which controls for how long various teams spend in each state. The way to read the below plot is that a team with one fewer penalty than its opponent is called 2.85 times per 60 minutes, versus 2.20 times per 60 when the penalty distribution is even. This is a perfectly clean trend from a penalty differential of -3 up to +3. That's a clean fingerprint of referee influence.

Refs swallow the whistle at the end of close games

So how does this bear out over the course of a game? Well, simply put, the likelihood of a penalty being called decreases as the game goes on. Do you believe the likelihood of a player actually infracting upon the rules decreases as the game goes on? I don't, and so I'd hypothesize that this is again referee game management influence.

The last two minutes are too sparse to read. In close games most of that time is played with an empty net, which is excluded because it's a different game state. Hover over the intervals there to see how wide they are.

Make-up calling also fades as the game goes on.

The penalty-differential slope is steepest in the 1st period (-0.63 log-odds per penalty) and flattest in the final two minutes (-0.15). It isn't stronger in close games: the interaction is -0.04 (p = 0.21).

Leading teams get called more

Calls go against the leading team 54% of the time with a one-goal lead and 56% of the time with a two goal lead. I represent it as a rate in the plot below, which interestingly shows pretty minor effects in penalty call rate for a team that is down in a game to tied, versus a pronounced effect for teams with the lead, and as that lead gets larger. Now it's fair to note that trailing teams push play, hold the puck more, and ostensibly as a result of that draw more real penalties.

That part is behaviour on the ice, not necessarily the referee, so the penalty-count effect in Finding 1 is better evidence of what referees do. Home teams took 47.9% of all calls.

All referees do this

There are 33 referees who have worked at least 100 games in the NHL over the last two years. All 33 of them show a statistically significant behavior towards using penalties to balance the game. Francis Charron has the highest estimate, but that estimate's interval overlaps every other referee's, so the data can't single out any one referee.

Their estimated slopes run from -0.31 to -0.51 around a league value of -0.42: roughly 58% to 63% of calls going against the team with fewer penalties at a 1-penalty gap. Each referee's estimate is pulled toward that league value, which is part of why every one clears zero. The spread between referees (SD 0.10) is about a quarter of the average effect.

Crews aren't fixed: the 2,612 games used 732 different referee pairs, and only 1 pair worked 15+ games together. So crew-level estimates would mostly reflect noise. Individual referees are the meaningful unit.