A multilevel model with heterogeneous variances for snap timing in the National Football League
利用贝叶斯多水平模型分析NFL四分卫在开球时机上的同步能力，发现开球时机的高变异性有助于进攻，并给出四分卫排名。
Abstract Player tracking data have provided great opportunities to generate insights into understudied areas of American football, such as pre-snap motion. Using a Bayesian multilevel model with heterogeneous variances, we provide an assessment of National Football League (NFL) quarterbacks and their ability to synchronize the timing of the ball snap with pre-snap movement from their teammates. We focus on passing plays with receivers in motion at the snap and running a route, and define the snap timing as the time between the moment a receiver begins motioning and the ball snap event. We assume a Gamma distribution for the play-level snap timing and model the mean parameter with player and team random effects, along with relevant fixed effects such as the motion type identified via a Gaussian mixture model. Most importantly, we model the shape parameter with quarterback random effects, which enables us to estimate the differences in snap timing variability among NFL quarterbacks. We demonstrate that higher variability in snap timing is beneficial for the passing game, as it relates to facing less havoc created by the opposing defence. We also obtain a quarterback leaderboard based on our snap timing variability measure, and Patrick Mahomes stands out as the top player.