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.
Pete Philipson (Tue,) studied this question.