Ceclilia Noecker & Paul Roback penned an interesting piece on the Journal of Quantitative Analysis in Sports, which I summarize below. The article is linked here, although it may be require academic access.
The authors gather all fouls from selected NCAA basketball conferences in the 2004-05 and 2009-10 basketball seasons. For each, they expand the logistic model provided from a previous study by using a generalized mixed effects model, with random effects for each team and each game. The primary model is shown below. Other variables included below include the score and foul differentials at the time of the infraction, the team with the lead, and conference indicators.
Random effects are important with this data to account for the correlation between the fouls within a game, as certain teams (and certain games) are more prone to more or less fouls. The result of the inclusion of random effect terms is an inflated estimate of the effect of previous foul differential. The author’s estimate “each extra foul on the visiting team relative to the home team increases the odds that the next foul will be on the home team by 19.9%.”
What sells me on the viability of make-up calls in NCAA basketball (which most fans have claimed for years) is the dose-response shown in Figure 4: the higher the foul differential, the more likely the chance of the team with fewer fouls being called for the next foul (especially shooting and offensive fouls).
Random notes and conclusions:
(1) One downside of this study is that there might be an issue with the ordering of possessions which accounts for part of the association found. Let’s say the two teams are Team A and Team B. If Team A is called for a foul, its more likely that Team B is called for the next foul simply because the Team B is going to playing defense on the possession after. This ordering of possessions isn’t perfect – offensive fouls, and non-shooting fouls before the bonus complicate things by increasing the chances that the same team is called for two consecutive fouls – but I suspect that after most fouls, teams switch possessions. In turn, this possession switch alone increases the odds of future fouls being called on the opposite team. One way to account for this would be to look at play-by-play data. This might require a multinomial logistic model with three outcomes for each play (offensive foul, defensive foul, no foul), and would certainly require the random effects provided in this model.
(2) I’d love to see evidence of how foul differential – and the inevitable evening-up of calls – effects game outcomes. There are different ways to do this, and its certainly an area for future work.
(3) Do fans care? Officials are possibly prone to bias (and make-up calls) to increase the perception of unbiased behavior in the minds of the fans. It might be the case that fans are happier when officials issue make-up calls, as opposed calling each play independent of the plays prior, as refusing to even up the number of infractions could yield large disparities in the number of fouls on each team.