Too often the statistical results in papers are presented as if the authors have uncovered the Holy Grail of whatever they are investigating. A detailed reading can then reveal something like their findings only hold for games under floodlights, in the rain, in the month of November, in the North-East of England, and when the team is wearing the away kit. To add insult to injury, the reader has to plough through a range of models, estimated with a variety of techniques, to find that the variable of interest explains something like 0.31% of the variation.
Vladimir Hlasny and Sascha Kolaric cannot be accused of overselling their results in their paper titled 'Catch Me If You Can: Referee-Team Relationships and Disciplinary Cautions in Football'. Hlasny & Kolaric do follow the rules for the statistical estimation. They inform the reader that these "regression models explain a very small portion, 0.2-3.3%, of overall variation in the dependent variable" where the dependent variable is yellow cards. Specifically they test four hypotheses: (i) the importance the difference in the number of previous times the referee has officiated at a game involving the teams; (ii) the distance between a referee's hometown and the participating teams; (iii) the likely impact of the refereeing decisions on the result; and (iv) the importance of the match for the home team.
It is clear that the authors do not oversell their result.
The authors are more interested in convincing the reader that the interaction between the referee and the teams are an important consideration. They succeed - possibly because they don't try to suggest it is the only, or most important, influence on a referee.
The paper has a neat literature review section and this is enhanced by the way the results are discussed. For example, the authors raise the possibility of media attention playing a role. But there is a slightly strange twist in the paper. Having struggled to find statistical significance of any size, the authors separate their data by the tier of football. Differences appear between the upper and lower tiers. This draws the authors into a discussion about the way referees are appointed and how it depends on age/experience. It seems that when it comes to home bias the relationship between referees is as important as the relationship between referees and the teams.