Yellow fever: an investigation into referee consistency in the ‘Big 5’ leagues of European football using a bivariate mean-parameterized Conway–Maxwell–Poisson copula model
研究使用双变量均值参数化Conway-Maxwell-Poisson Copula模型分析2018-2022年欧洲五大足球联赛黄牌数据，评估裁判和球队的异质性，发现裁判和球队异质性相似，意甲裁判变异性最小。
Abstract Scrutiny of referees in football is ever-increasing and likely at an all-time high. Is this fair? How consistent are modern day referees? In this work, the number of yellow cards given to the home and away teams over four seasons of data from 2018 to 2022 in each of the ‘Big 5’ European leagues in men’s football is modelled. This allows us to make an assessment of the heterogeneity amongst both referees, primarily, and teams, secondarily. The underdispersed nature of the data from the small counts, and likely dependence of the cards issued within a game, leads to a bivariate mean-parameterized Conway–Maxwell–Poisson copula model to analyse the data. We also model home advantage, and examine whether this was diluted during COVID-19, and league effects, allowing for an assessment of how consistent referees are across leagues and which league has the most heterogeneous ‘men in the middle’. We find that teams and referees are, perhaps surprisingly, similarly heterogeneous and that referees in Serie A appear to be the least variable. The underlying model has wider use in other fields when the researcher is analysing small multivariate counts with possible bidispersion.