The ebb and flow of official calls in water polo

1Introduction

Fans, players, and coaches alike are well acquainted with the important role played by officiating in sports. A call made (or missed) at a critical moment can drastically alter the evolution of a game and as a consequence, investigating causes, instances, and impacts of foul calling bias is an important topic in sports analytics. The most extensively researched factor influencing referee decisions is home team bias where the literature suggests that not only does home team bias exist, but that its effect can be amplified by external factors such as crowd noise or whether a game is nationally televised; see Neville and Holder (1999) for a survey of results on this topic. In contrast, other studies have focused on how various game-state factors, such as previous calling or scoring patterns, can influence referee decisions. For example, studies of foul calling patterns in NCAA basketball suggest that referees tend to favor the losing team, but also exhibit “sequential bias” in their calls, meaning referees tend to downplay consecutive calls against the same team (Anderson and Pierce (2009); Noecker and Roback (2012)). Similar sequential biases have been observed in soccer penalty kick calls (Plessner and Betsch (2001)). A final example occurs in baseball where it has been shown that strike count can significantly impact an umpire’s “strike zone”, with a smaller strike zone for 0-2 counts than 3-0 counts (Moskowitz and Wertheim (2011); Green and Daniels (2014)).

This study extends previous research on referee bias by examining foul calling patterns in the sport of water polo. There are two kinds of major fouls in water polo: exclusions, in which a defensive player is temporarily excluded from play resulting in a 20 second 6 on 5 power play for the offense, and penalty shots, in which a severe goal-preventing infraction against the defense results in a penalty shot. These two types of fouls play an undisputedly important role in the sport. For example, Fig. 1 below shows the distribution of goals scored from 68 elite men’s water polo contests spanning 2012–2014. Notice that power plays and penalty shots together account for about 56% of all goals scored in these games.

Considering the importance of exclusions and penalty shots in generating goals, one would expect that the number of foul opportunities received by each team is a good classifier of the outcome of water polo games with winning teams receiving more opportunities than losing teams. Recent research, however, has shown that there is no significant difference between the number of exclusion opportunities obtained by winning and losing teams in elite men’s water polo with slightly more than 50% of all games ending with more opportunities for the losing team (Graham and Mayberry (2014)). This counterintuitive result may suggest that water polo referees follow the principle, stated in Askins (1978), of being “fair” (giving equal opportunities to both teams) as opposed to being “objective” (calling fouls based on severity of infractions alone). This paper aims to investigate game-state factors which affect foul calling rates in water polo using game data from three recent international contests: the 2012 Olympics, the 2013 World Championships, and the 2014 European Championships. In addition to the two major types of defensive fouls, we also include offensive foul rates in our analysis. While there have been a number of previous studies which investigate the impact of major fouls on the outcome of water polo contests (Enomote et al. (2003); Hughes et al. (2006); Escalante et al. (2011); Escalante et al. (2012); Lupo et al. (2012); Graham and Mayberry (2014)), this is the first study which takes a dynamic look at foul calling patterns in the sport and seeks to identify situations in which a team’s chances of drawing a foul may significantly differ from their baseline rates. We conclude by exploring whether statistically significant foul calling biases can be explained by differences in playing styles or whether referee bias is a more likely explanation for these effects.

2Results

3Results and discussion

Tables 5 and 6 below show the fitted parameters from our models for defensive and offensive foul rates, respectively. In both scenarios, the level 1 models were significant improvements over the null models (DFR: χ62=102.249,p<0.001; OFR: χ62=51.645,p<0.001) showing there are significant dependencies between foul calling rates and game-state variables. For the defensive model, all game-state variables except foul differential were significantly associated with foul calling rates. In particular, it was predicted that the odds of drawing a defensive foul decrease by 100* (1 - exp(-0.388)) ≈32 % when the offensive team is winning/tied as opposed to when they were losing; see Fig. 5. The odds of drawing a defensive foul were negatively correlation with each of the three variables scoring momentum, foul momentum, and lead, but positively correlated with game-time suggesting that the odds of drawing a defensive foul increase over the course of a typical game.

State, foul, and scoring momentum had similar associations with offensive calling rates: an offensive foul was significantly more likely to be called if the offensive team was winnign/tied or had momentum in their favor. In contrast to defensive fouls, however, offensive rates were not significantly correlated with either game-time or lead, but were negatively correlated with foul differential.

Both level 2 models were significantly better predictors of foul calling rates than their level 1 counterparts (DFR: χ112=542.761,p<0.001; OFR: χ102=135.773,p<0.001). In the defensive model, all five significant game-state variables from level 1 remained significant after accounting for the new play selection and location variables. Comparing coefficients in Table 5, we can see that the roles played by state, lead, foul, and scoring momentum were impacted very little by the inclusion of additional terms whereas game-time took on a more significant role after we accounted for these additional variables: the level 2 model predicts that the odds of receiving a defensive foul more than double over the course of a game. To demonstrate the persistence of the losing team effect across play selection variables, Fig. 7 compares the losing and winning/tied DFRs across different offensive and defensive tactics. Note that foul calling rates are biased in favor of the losing team across all categories. To confirm that any differences in foul calling rates were not significant, we tested for interactions between game state and play selections, finding no significant benefit from the inclusion of such terms in our model (Interaction with First Defense: χ32=2.642,p=0.450; Interaction with Initial Attack: χ62=4.699,p=0.583). Furthermore, while losing teams did run significantly more plays per possession on average than offensive teams (two-sample t-test: x¯L=1.759,x¯W/T=1.673,t4618=2.770,p<0.001), their per play defensive foul calling rates were still significantly higher than winning teams (2-proportion z-test: p_L = 0.199, p_W/T = 0.167, z = 3.595, p <0.001). Similarly, in the offensive model, the coefficients of state, scoring momentum, foul momentum, and foul differential were not greatly altered by the inclusion of play selection and location variables in level 2.

Game-time, however, had an interesting interaction with state in modeling defensive foul calling rates. While Fig. 8 below shows that losing team bias was present across the spectrum of a game, there was a moderately significant interaction between these two variables (estimated coefficient of interaction = -0.458, z = -1.786, p = 0.074). In particular, the interaction model predicts that at the beginning of a game, the odds of drawing a defensive foul are only about 12% lower for winning/tied teams as opposed to losing teams, but near the end of the game, the odds of drawing a foul are 44% lower for winning/tied teams. We also tested for an interaction between the state and size of the lead to confirm that losing team bias was equally present in close and unbalanced games and found no significant interaction between these two predictors (χ12=0.254,p=0.6143).

Although the impact of game-state variables is the primary aim of this study, the play selection variables in our model exhibited some noteworthy correlations with defensive and offensive foul rates as well. First, defensive selections were not significantly associated with DFR, but were significantly associated with OFR. Split and zone defenses yielded smaller OFR than press or transition defenses. The offensive team’s choice of attack was significantly associated with both types of foul calls, but in different ways. DFR was higher for what would be considered more aggressive offensive tactics (center, counterattack, double post, and movement) than for more conservative, outside shooting tactics (perimeter and direct shots). OFR was also higher for center, movement, and double post tactics, but in contrast to DFR, was higher for direct shots than for counterattacks or perimeter shots. Thus counterattacks appear to be the best case scenario for an offensive team, yielding high defensive, but low offensive foul calling rates. This observation is not too surprising since counterattacks usually result from “fast break” scenarios in which offensive players greatly outnumber the defensive players on the offensive side of the pool. The fact that center and movement based tactics yield high OFR and DFR is also not surprising since both typically involve even situations with a great deal of player interaction. Somewhat more surprising is the fact that direct shots, which have been suggested as an undervalued tactic in Graham and Mayberry (2014) yield low defensive foul calling rates, but high offensive foul calling rates. Finally, note that the number of plays run in a possession was positively correlated with a team’s chances of drawing a defensive, but not an offensive foul. This may be related to the idea that if an event has a positive probability p of occurring in a single experiment, then the probability of obtaining at least one occurrence in n independent experiments is 1 - (1 - p) ⁿ ≥ p. Of course, the outcomes of successive plays will not be completely independent, but the dependencies may be weak enough to ensure that the probability of obtaining at least one successive is a strictly increasing function of the number of plays run.

To address location, our models suggest that offensive foul calling rates were roughly consistent across the three events in our database, but that defensive rates were significantly lower at the WC than the other two events. Figure 6, however, demonstrates that the losing team bias was consistent across the three tournaments. A formal test confirmed that there was no significant interaction between state and event in our model (Defense: χ22=0.546,p=0.761; Offense: χ22=4.728,p=0.094).

4Conclusions

Our analysis shows that foul calls in water polo are highly dependent upon the state of a game. Losing team bias is particularly prevalent in the sport with offensive teams who are winning/tied being about 31% less likely to get a defensive foul called in their favor and about 32% more likely to get an offensive foul called against them than losing teams. Offensive teams are negatively affected by both scoring and foul calling momentum as well, with the odds of drawing a defensive (offensive) foul decreasing (increasing) by around 10% for each consecutive point scored or foul called against them. Defensive foul calling rates (but not offensive) tend to be higher in close games, decreasing as the size of the lead increases. It also appears that defensive foul rates tend to increase over the course of the game while offensive foul rates remain constant. Table 7 below further elucidates these results by comparing the odds of getting called for a foul in several opposing game scenarios as predicted by our level 2 models. For example, when an offensive team has had two consecutive goals scored against them, the odds of drawing a defensive foul are 100 [1 - exp {4 (-0.110)}]≈36 % greater than when they have scored the previous two goals if we hold all other variablesconstant.

There are two potential explanations for why foul calling rates depend on the game-state variables mentioned above:

In other words, the explanation could either lie with the players (1) or the referees (2). Although we cannot completely rule out (1), the inclusion of player choice variables into our level 2 models provides some evidence in favor of (2) as an explanation: even after we account for differences in offensive and defensive choices, scoring momentum, foul momentum, and state remain significant predictors of foul calling rates on both sides of the pool. Furthermore, there were no significant interactions between state and player choice variables in modeling foul rates suggesting that losing team bias is not an artifact of tactical choices made by teams. We therefore argue that referees are the main source of game-state biases in foul calling rates. In particular, referees consciously or subconsciously tend to favor teams who are losing and attempt to minimize sequential calls against the same team. Since there is no significant interaction between the sign and size of the lead, it would appear that this losing team bias is not just due to a “sympathy” effect which comes into play once a team is down by a certain amount. There was also evidence of an interaction between game-time and game state with losing team bias increasing in magnitude over the progression of a game.

Foul calling biases may help explain why Graham and Mayberry (2014) found that exclusion conversion rate (ECR), defined as the fraction of exclusion opportunities converted to goals, is one of the best classifiers of game outcomes in international men’s water polo while exclusion opportunities is no better than flipping a coin. In fact, the team with a higher exclusion conversion rate wins almost 90% of all contests while the team with more exclusion opportunities loses slightly more than 50% of all contests. The results of this present paper in regard to both losing team effects and sequential biasing further illuminate the importance of this fact. A losing team is more likely to get an exclusion opportunity, but consecutive opportunities are discouraged and hence those team’s which can convert exclusions are at a significant advantage over their opponents.

Although the ratio of observations to variables in our data set is too low to yield a meaningful analysis of all higher order interaction terms, this would be an interesting project for future research. One particularly interesting question is to see how interactions between pairs of game-state variables persist across events. As an example, Fig. 9 suggests that there is a significant interaction between scoring momentum and state at the Olympics and World Championships with scoring momentum being positively correlated with offensive foul calling rates for winning/tied teams, but not losing teams. There appears to be no such interaction at the European Championships, however, and a test for a three-way interaction confirms the significance of these differences (χ72=19.061,p=0.008). After another round of each tournament, it would be interesting to see if these patterns persist.

Another limitation of our analysis is the lack of time specific information related to possessions in our database. For example, we did not have a record of how long each possession lasted and hence, were forced to measure game-time as a fraction of the total number of game possessions which had elapsed by the end of the current possession. As more possession specific water polo data becomes openly available, the impact of possession length and game-time should be reexamined. In addition, to further rule out team play as an explanation for losing team bias, it would be helpful to have a method for quantifying the aggressiveness of a possession. Our player choice variables provide a potential surrogate for the absence of such measurements (press defense could be considered as a more aggressive choice than zone; counterattacks, center plays, and movement based tactics could be considered more aggressive than direct shots and perimeter plays), but cannot account for differences in aggression in the execution of said tactics.

Finally, we would like to mention that the inclusion of random effects for teams and games did not impact our results and that the variance related to these effects was neglible. This suggests that the reported biases are widespread and not restricted to any particular team or game. Consistent with the findings of previously mentioned studies in basketball, soccer, and baseball, game-state based referee biases appear to be a uniform phenomenon in elite men’s water polo, favoring “fair” over “objective” patterns of foul calling behavior. Thus our study extends previous research on officiating call patterns to a new sport and contributes to the growing body of analytic research in water polo.