Most studies which model athlete behavior try to control for score differential as a mechanism for accounting for player aggressiveness.
In many cases, including the raw score differential makes sense; in soccer, for example, a team trailing by two goals is twice as likely to need another score as a team trailing by one goal only. For many sports, like soccer, the raw score differential makes sense as a control.
A recent paper, published in a Economic Inquiry, followed suit. In this study, Carl Kitchens had an interesting hypothesis – that in the NFL, the position change of its official unknowingly caused an increase in the percentage of offensive holding penalties. Kitchens’ results are statistically significant, and tie into economic theories of crime and spatial distribution that frankly, I know little about.
What I’m more familiar with are the assumptions in modeling binary outcomes, in this case, the likelihood of a holding penalty on each play. Kitchens’ model includes the following variables:
A model of holding outcomes as a function of score differential and score differential^2 makes an assumption on the relationship between score and holding likelihood which can’t possibly be true. Specifically, in football, an offensive team’s aggressiveness is hardly any different with a 4 point deficit or a 6 point deficit. In both situations that team needs one touchdown to take the lead. However, the difference between a 7 point deficit and a 9 point deficit is much larger, as its the difference between a one and a two possession game. Adjusting for raw score differential, however, treats all point changes equally; the difference in the log odds of holding from a 4 point deficit to a 6 point deficit equals a 7 point deficit to a 9 point deficit, which, sadly, also equals the change from a 1 point lead to a 1 point deficit. In my opinion, an ideal model of football behavior would adjust for score differential using categorical variables (i.e., down 21+ points, down 9-20 points, down 1-8 points, etc). Would this changes Kitchens’ results? I’m not sure, but I’d certainly want to know the answer.