Running head: ASSESSMENT OF REFEREE BIAS IN ASSOCIATION FOOTBALL 1 Bayesian networks for unbiased assessment of referee bias in Association Football

Objectives: To assess referee bias with respect to fouls and penalty kicks awarded by taking explanatory factors into consideration. Design: We present a novel Bayesian network model for assessing referee bias with respect to fouls and penalty kicks awarded. The model is applied to the 2011-12 English Premier League season. Method: Unlike previous studies, the model takes into consideration explanatory factors which, if ignored, can lead to biased assessments of referee bias. For example, a team may be awarded more penalties simply because it attacks more, not because referees are biased in its favour. Hence, we incorporate causal factors such as possession, time spent in the opposition penalty box, etc. prior to estimating the degree of penalty kicks bias. Results: We found fairly strong referee bias, based on penalty kicks awarded, in favour of certain teams when playing at home. Specifically, the two teams (Manchester City and Manchester United) who finished first and second appear to have benefited from bias that cannot be fully justified by the explanatory factors. Conversely Arsenal, a team of similar popularity and wealth and who finished third, benefited least of all 20 teams from referee bias at home with respect to penalty kicks awarded. Conclusions: Among our conclusions are that, in contrast to many previous studies, being the home team does not in itself result in positive referee bias. More importantly, the model is able to explain significant discrepancies of penalty kicks bias into non-significant after accounting for the explanatory factors.